Episode 71 - Emily Howard

Evelyn Lamb: Hello and welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Evelyn Lamb, a freelance math and science writer based in Salt Lake City but currently podcasting from my parents’ house in Dallas, which is actually not any warmer than Salt Lake City right now, unfortunately. This is your other host.

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. I'm in my faculty office. I'm usually — I'm the chair of the department. But I'm hiding out in the faculty office today. Actually, I was looking for better wireless. And it seems to be working a little better in here. But it's so weird because I have nothing in this office, like nothing. It's very strange. So anyway, how are things going for you?

EL: Not too bad. Yeah, seeing my family, which is nice, and very excited about today's episode. So let's get right to it! We're happy today — both of us are music lovers, and we're very happy to introduce our guest, Emily Howard, a composer. Emily, do you want to tell us a little bit about yourself?

Emily Howard: Yeah. So I'm based in the UK, in Manchester. I'm originally from Liverpool. And I'm a composer. I love writing for large ensembles, large acoustic ensembles, such as the orchestra. I also write vocal music, choral music, and also chamber music. So a lots of different areas. And I suppose probably the reason that you've got me on here is that I've got a real interest in mathematics. And actually, I have a degree in mathematics and computer science, my undergraduate is in mathematics and computer science. And I suppose that, you know, definitely it's one of the key influences on my work.

EL: Yeah, I was listening to — it might have been the BBC Proms a few years ago — I was listening and saw this piece that I think was called Torus. And I thought, “You don't accidentally name something Torus.” So I decided to try to find out more about this person who had named her composition Torus. And so yeah, I found out that you had a math background, and thought it would be just really fun to talk to you on the podcast. So yeah, can you talk a little bit about the — I know, you've done some collaborations with mathematicians, you know, written pieces, like kind of in conversation with mathematicians in the composition process, and I would love to hear about that.

EH: Yeah, so I mean, I suppose actually, in 2015, I had I think it was a Leverhulme fellowship at the University of Liverpool, working within the mathematics department. I had been invited by Lasse Rempe-Gillen. He's a professor in dynamical systems. And I think he'd been in touch because he had himself played the violin in an amateur orchestra in Liverpool, and they had performed a piece of mine called Mesmerism. Actually, it was after Ada Lovelace. Ada Lovelace used to dabble in all types of things, including mesmerism, sort of a form of early hypnosis. And the piece, I mean, that piece was a piece for solo piano and orchestra, and he was playing in it. And I think he thought it will be great to invite me as someone with a mathematics background back — after 15 years away in the music world— actually back into the maths department. And, I'm so glad that this happened, because going back in and speaking to lots of different mathematicians in a different way, rather than, I don't know, having to take exams and study, I was more being an observer, having amazing conversations about people's research. Regularly I'd speak to someone in one area of mathematics and another, and I suppose I felt — what I realized was that mathematicians often don't understand what each other are speaking about. So I didn't feel so bad about it, that I could sort of dip in and I suppose it helped me to take a more global approach, or to take in more general ideas, because I think that's one thing, perhaps, you lose if you don't practice mathematics regularly, you know, you lose this very detailed approach, and that's kind of annoying. It’s really annoying in many ways, but I also find it annoying when you can’t completely understand something because you'd have to spend quite a few years really thinking about it in depth. But then something is also gained from sitting back and taking a look at everything and absorbing it in a very different way.

So anyway, we worked together on a set of chamber pieces. You can find them on my website. One’s called Leviathan, another is called, well, these are all pieces, exploring ideas from lattice research in dynamical systems. And I suppose they're all based on thinking about perturbations, and thinking about — I’ve got some etudes as well, they’re called Etudes in Dynamical Systems — just trying out very particular things with datasets, but also conceptual ideas about reaching stasis and loops from something. I also spoke a lot with colleagues in mathematical biology and also in topology. And I remember, I think this is where, kind of, talking about Torus, it was kind of born, the idea, because this period of time gave me more confidence to actually do something with mathematics. Just to go back back a bit, after my undergraduate in mathematics, I'd spent a couple of years doing a Masters at the Royal Northern College of Music in composition, and then more years at Manchester University doing a PhD in composition as well. And I suppose, when I was doing this, I really wanted to use these ideas or concepts from mathematics, but I didn't really have a musical technique to do that. I suppose that grew. And at this point, it sort of all merged, and I was able to perhaps achieve something that I wanted to achieve with these mathematical ideas through music. And so yeah, just to go on, you mentioned Torus. And I'm immensely proud of Torus. It's a big orchestral work, it's like 23 minutes long, based on the mathematical shape, a torus, a doughnut.

And this is when I had actually just met a mathematician, Marcus du Sautoy, who I’ve worked with quite quite a lot. He's based at Oxford, and we met because of this thing. And when we spoke first, actually, he was writing about, no, he’d written a play about a torus. And I was sort of saying, “Well, actually my piece is about a torus,” so we really bonded on this. And one of the things that interested me most was, you know, the, the BBC Proms are held at the Royal Albert Hall in London. And the first thing he said was, “Well, that is a torus.” And I just thought, How funny. It really is, because you can run round and round it, you know, it's just just really nice to, you know, hear this.

KK: So I'm especially fascinated here, because my son is a composer. Well, he just finished his undergraduate degree, and now he's in graduate school for composition. And so I'm always — first, I'm encouraged that composers can actually make a living, right, because whenever I would tell a mathematician, one of my friends, “Oh, my son's studying composition,” they say, “Well, how’s he going to live?” And I say, “Well, I don't know,” but clearly it’s possible to figure it out. So I'm super encouraged. Yeah, so Evelyn, you had a real question, though.

EL: Oh, well, I was just saying this is an existence proof then for your son.

KK: Yeah, that's right. That's right.

EL: Yeah, well, I was — you know, music is a very non representational art form, unlike a canvas where you could — I mean, it's still hard to represent mathematical ideas on a painting. But can you say anything about how you do use the form of music to represent mathematical ideas?

EH: I suppose — it's really interesting that you say that, because I completely agree with you actually, that it is very difficult to do that. And I wonder if, even if I, especially in abstract music, if we take aside — you know, if it has text in it, that's a very different thing — but in an abstract form, actually, I wonder if I ever tried to absolutely represent something from mathematics within music, whether in fact, anyone would know. And I suspect, actually, there are cases when I've worked very closely with people and they really know what I'm doing. I think they can certainly tell, but actually, I don't think that that's necessarily happens. And also, it's definitely not what I'm trying to do. Absolutely not. So I suppose my aim is not to represent mathematics. My aim is to — I mean, I love hearing about mathematics, and I'm completely inspired by processes and systems and patterns. And I suppose what I'm doing is taking them, and that’s a catalyst for my creative process. And so I do think something comes through, but I think it's more that I couldn't make what I'm making without doing this. And it's more that something new is occurring that came from thinking about these things. But it's definitely not a representational thing. I mean, that's definitely the case for maybe the more large-scale pieces we could talk about, like Torus.

But I think also there have been a couple of pieces when I have tried my very best to represent ideas, and one of them would be this set of dynamical system etudes. That was one of them. And then another one would be the music of Proof, which is, I wrote a string quartet because Marcus and I were discussing proof and different ways you can prove things: proof by contradiction, by induction. He wanted to, to put those to me, and then I would create responses to different kinds of proofs. And so I suppose that’s as representational as I've been, and actually, it was really useful, because in doing that, then I gained a whole set of like, I thought about — I've never tried to write music before as though I'm solving a mathematical proof. But actually, in doing that, that led me to new places. So that's kind of what I'm trying to do, find new ways to do things and make new sounds.

EL: Yeah, maybe respond more than represent you?

EH: Definitely.

KK: I mean, certainly, some music is sort of, you can tell, it's not deliberately mathematical, I don't think. I think of like, like Steve Reich's minimalism percussion pieces, right?

EH: Clapping Music.

KK: Yeah, they’re so cyclical. And you know, you can kind of, if you think about it as a mathematician, you can kind of imagine, well, if I saw this on the page, it would almost look like — our listeners can't see my hands doing this — but you can imagine sort of, you know, intersecting sine waves and things like that. So I can see how you could do that.

EH: And to take that further, I wonder if all music in some way can be — well, I say reduced, and I don't like to reduce — but you could certainly represent ideas in very complex music, I think mathematically probably. Not that anyone necessarily has. But perhaps the secret to lots of things is in really complex music represented. I mean, I don't know.

KK: Well, there’s a whole journal of mathematics and music. I mean, you could you could certainly, I sat on a PhD committee for composer on campus here, who really was trying to do these very strange time signatures that were sort of approximations of pi and things like that.

EH: Wow.

KK: And I put the question to him. I said, “Okay, I mean, you can make a machine do these. But, you know, can a human do this?” And he said, “Well, no, not really.” But it was interesting.

EL: Can humans do math anyway?

KK: This is not a philosophy podcast. I don't know.

EL: Yeah. Thankfully,.

EH: Just to say on that on that subject. I mean, I've got certain things — there are certain things you do put in notation that are absolutely beautiful mathematically, you know, but in fact, yes, they're not really possible. But they they do make a performer think in a certain way, and it will give you certain results. So I mean, I think there can be, I think they're beautiful, and they can be there. But you just can't expect perfection, perhaps.

EL: Yeah. All right. So we always like to ask our guests, what is your favorite theorem?

EH: I mean, we've kind of been speaking about it. That's, that's the main problem with this. I mean, actually, I would say, rather than a theorem, it's definitely these shapes. So I mean, let's take the torus, but I've got this fascination, I'd say, with thinking about mathematical shapes and thinking about, you know, them being far away, and also thinking about being on them. And I suppose, I mean, so you've got a torus. I mean, so, if you think about the difference between, say, flat geometry and a sphere with the spherical geometry, and then, I mean, there's a pseudosphere, and then I would call, like a negatively curved geometry, I've got a piece actually called Antisphere because I've worked out that an antisphere, there's a word antisphere, which is the same as the pseudosphere, and the word antisphere is a lovely word, and I don't like the word pseudosphere. So I called the piece Antisphere. And I suppose I've just got this — because, you know, my music is usually notated. So I find it very interesting as a process to be in my mind wandering around these shapes. And I suppose, I also think it’s very useful for music because they don't necessarily need to be 3d, I mean music, potentially, you could perhaps hear really high-dimensional mathematics. Because, you know, you could be traveling around and you're not necessarily visualizing it.

So my answer to your question is that I like these mathematical shapes. And I've been thinking about them and really studying them and using them to influence these large-scale orchestral work. So that's Torus, Sphere, and Antisphere. And going forward, I'm now looking at, well, I'm trying to look at Thurston’s eight geometries and pushing myself in this direction with the aid of a number of very kind mathematical friends, because they're very difficult. And yeah, so that's kind of the direction where I’m going.

And I find, you know, I suppose that's what we were saying earlier about the representation thing. So there are a number of stages. So I'm thinking about this tours, I'm thinking about traveling around it. Shall I give an example of the compositional process for writing Torus?

KK: Sure, please.

EL: Yeah.

EH: Okay so Torus is a piece, yes, it was for the BBC Proms. It was performed first in 2016, by the Royal Liverpool Philharmonic Orchestra with Vasily Petrenko. Now I’d say, I wrote it over at least a year or so. And I'd also been thinking a lot about this just previously, thinking about traveling on this torus. And what I was thinking about when I wrote it was the idea that you're traveling round and round one direction of the torus, you know, and I took these very consonant chords, see, I've got major sixths. If you hear it, they’re a very constant chord. And I start with this major sixth, and I travel up one and down one and up another — they're going up in semitones, and down, and they reach a point and they come back again. So that the harmony is very much also shaped like a torus all the way through. And it's skewed, when you hear it, it's more complex than just listening to it timed because I've got a real combined fascination with exponential functions and the idea of really big things becoming absolutely minuscule, but also kind of being related. So, you know, this kind of journey around the torus, the first loop in this orchestral work is about five minutes long. And if you do listen, you'll hear it in the strings. And to my mind, I've got this sort of ever-expanding enclosing torus idea in the strings, but, but maybe on the surface of it, as you go round, you hear different things each time you go round. And that's perhaps in the wind and in the percussion, and in the brass. So it’s almost like, you're on a landscape and it's changing.

Now, the piece works by this — suppose the radius of the doughnut-shaped Torus, sort of shrinking. So you go around it, and it gets quicker and quicker. And you'll hear — I think it's around about maybe 17 minutes in, there’s a viola solo. Because it's really big, and it sort of shrinks down, I think there are seven, sort of around this this way of the torus, and you get this viola solo that encapsulates these major sixth idea because you can just play all of that on one solo viola. And then there's this sort of huge shift in thinking in the piece. And, again, that almost came from thinking about dynamical systems and just completely changing the goalposts, and then you're traveling very fast. And in my mind, we've flipped to that we’re now thinking about going on the other direction of a torus forever and ever. Do you know what I mean?

EL: Okay.

EH: You’ll hear that. It’s a huge moment. And that music seems to completely change. And I don't think anyone listening to it will hear a torus shape, because you can’t, because you're hearing my ideas about journeying around it. Do you see what I mean?

EL: Right.

EH: But actually, it really helped me to think about this thing and you know, it is absolutely about this sort of journeying around on it.

EL: Yeah, I don't know, I'm almost imagining like you could, maybe — I wish I had like a, some kind of inner tube or something. But like, maybe the first loop around is on the top where it's further apart, and then you're kind of almost falling into the middle where the You know, the two sides of the torus, or the hole of the tours is kind of small. And then you you kind of flip the other way and start. The next time I listen to it, I'll have to imagine that kind of journey.

EH: The thing is, I’m sure now I've told you that, you probably will hear it, you know? And I think as well, I mean, it's not, it’s never so obvious, because I'm also doing a couple of other things as well. So each journey around this torus — so I said the first one's about five minutes — you’ll also hear these major sixths alternate in the string, so it’s almost like there's one side and there's another side. So you hear this sort of, they appear to travel like this, and then on the other way, they're sort of all, everything is together, so it's rhythmic unison. And it and there's a sort of written-out rall, so it gets slower and slower. And then you'll hear, it comes around again. So I mean, there’s that and there, and then sometimes as well, one side of the torus. That is going on in the background in my mind, and that's definitely happening. But in fact, there's something else on the surface, like there are these very loud things going on sometimes. And they obliterate this thing, but it's always there. And a bit like, say, a painter might have a layer of something to start the piece, this is my layer, this sort of journey. And then it builds up.

KK: And you totally, you had Evelyn at viola solo.

EL: Yeah, yeah, I do play viola.

EH: Great. You’ll hear that.

EL: Yeah, you're pushing all our buttons. Kevin’s son’s composer thing, my viola thing. You’re just excellent. Yeah. What I was wondering is, do you know, why were you drawn to the torus specifically? Do you know?

EH: It’s a really good question. No! I’m not sure I do.

EL: Well, you sort of mentioned the Sphere and Antisphere and Torus are kind of this set. And those can be the three different two-dimensional geometries. But I'm wondering whether you kind of already liked the torus and just were excited that you could use it here, or if you kind of sought it out because it is this model of flat geometry.

EH: That's very interesting, because I think I was trying — when I was writing Torus, I was already thinking about writing Sphere as well. And, you know, in many ways, there's something too perfect about a sphere.

EL: Yeah. It constrains you. It’s very limiting.

EH: It's a problem. And the other thing is, I was having to think of a title. And actually, the title Torus is just the most beautiful word. And there, you know, I think it's the fact that the sphere was too perfect. And there was a way to sort of have two very different things on this torus. But I think it's also true to say that I hadn't thought until later on to do the set. It emerged, you know. So I was sort of playing around and suddenly, I thought, when I'd written Torus, I thought, “Well, actually, that's quite a really wonderful way to think about things.” And then then came Sphere. And that's a shorter piece. It’s a five-minute piece for chamber orchestra. I suppose, I was thinking more about spherical geometry. As I said, I found that one more difficult because it had to come straight after I was writing Torus. However, a couple of years later, that's when I decided to write this Antisphere. And that's for the London Symphony Orchestra with Simon Rattle. And it was just performed a few months before lockdown. So I was very lucky to have that happen. I'm really grateful. And I'd say that, I mean, it's interesting, because somehow, I feel that my link to the maths in it is stronger with Antisphere. And I think that's just because I've gained experience and maybe confidence. I suppose with Antisphere, I was thinking about harmony. I like using quarter tones. They’re present in a lot of my music over the last, say, 10 years. We’re used to maybe the 12 semitones of the scale, and the quarter tones are just in the middle. And I suppose I was thinking about, you know, the classic angles in a triangle, which add up to 180. And then on the sphere it’s more, and then on the negative curves, it’s less. And actually in Antisphere, I use that so that you get these chords that we might all recognize, like a major chord, but actually it's been shrunken in some way.

EL: Oh!

EH: And so it's weird. And it sounds weird. And also, there's a section in there, it’s a very fast section of a kind of circle of fifths idea. But actually, I think it's a circle of 4 3/4. And they're great, because they last longer. Obviously, you know, these players are amazing, because they can do this in an orchestra ensemble.

EL: I was going to say how did you get — I think I would struggle to play quarter tones on purpose. I’m sure I’ve played some by accident.

EH: I mean, I was really blessed with this string section, but it is an amazing experience. And I think you do feel because you do feel this — I mean, the association of a major chord, I mean, I don’t think I’m using major chords, but whatever I’m using — you can feel it, but then you can certainly feel that something's happened to it. And I really liked that. And another thing I did was the rhythmic thing. We've mentioned Steve Reich, and I like the thought that you've got a regular pulse in our world. But if you're somewhere else, or looking in on a different kind of — well, the thought is that you're the pulse might, if it was like 1-1-1. But in fact, it could go 1-2-4-8. So actually, therefore our perception of this piece of music is not one of a regular pulse. In fact, I have created it as though it is, but it isn't. So we hear a short section and a longer one. And then and by the time you get to here, it's so long, you perceive it as something completely different. And I love that as a compositional process. So I did that. In Antisphere, I've actually taken a chord sequence that I used in Mesmerism, this this early Piano Concerto I wrote in 2011. And that’s, quite regularly, you know, there are a few chords, at the opening, you hear solo piano playing these chords, and they're quite regularly spaced. But in Antisphere, what I've done is I've taken them as the basis, these piano chords, as the basis of these sort of chords that start off like a bit and actually end up as three minute sections. And loads of weird things going on in between you like this with the percussion, there's loads of weird metals and sort of resonant sounds. And so, yeah, as I said, it gives a completely different sort of perception of what's going on.

EL: Yeah, I don't think I have listened to Antisphere yet. But I am now going to seek it out because I really want to hear this — especially if I've got in my mind this idea of, like, hyperbolic triangles with their, you know, curved in — or they look to me like they're curved in. If I actually believed enough, they would look like straight lines to me, because I would really embody the hyperbolic metric.

EH: But and this so this is happening in pitch, but it's also happening in sort of rhythmically and timbre in lots of ways as well. I did write an article, Orchestral Geometries, which I will post, you know, we could put there. And actually, I've been lucky enough to have wonderful recordings of the three pieces, and also the scores as well alongside so yeah.

EL: Oh, wonderful. So yeah, another thing we do in this podcast, which we kind of already done, is we ask our guests to pair their theorem, or mathematical object, with something. And so yes, I kind of assume that you would be pairing it with your compositions based on these, on the torus and these other shapes. But yeah, do you want to add anything else about that, or any other pairings? You know, if it's just a nice cup of tea or something?

EH: Or a doughnut? I was going to pair them with that. And actually, I was going to pair them perhaps with, you know, we could we could play a little clip from one of them. So this clip is a little bit from the orchestral work Torus, and I think, it’s sort of about two thirds through where everything changes. So before then you've been rotating round, the kind of doughnut shape of the sphere, no sorry, the doughnut shape of this torus. And then you go into this viola solo and then you hear the huge perturbation, and that's when you change as though you're rotating. At least when I was writing, I was thinking about rotating around the other part of the torus.

[music clip]

KK: So I really, now I'm sort of curious to know how you’re — I mean, I know you haven't thought about it yet. But pursuing the sort of 3D geometries, that's going to be weird. Have you seen these, people have tried to visualize these things using VR. Have you seen any of these explorers?

EH: Yes, yeah, I have. And actually, if you have any more, I would love to have them because it's very helpful to see them. We've been looking online, and there are quite a few. You know, there are some amazing websites and people, actually programs for doing it. And we were kind of exploring it. But I am interested. And it's a new pursuit, you know, so I'm really interested to, well, think about it more. I think it'll take some time as well.

I'm currently writing a piece. And I was thinking that it will be very nice, now having written these orchestral geometries, to embody sort of a process of moving between the different geometries a bit. So that you could, I mean, no one need necessarily know this, but musically, there might be a difference between this spherical and the — but they could, so I'm really interested in that. So I'll do that. But I also I'm interested in this, is it H3 [hyperbolic 3-space]?

EL: Yeah.

EH: So yeah, I'm interested in that. And it's kind of frustrating, because I'd really like to understand why. And it's quite difficult, I think. I'm not sure. I think it's probably well beyond me. But, you know, the thought of this dodecahedral space and moving around and the twist, I'm really interested in the fact that it would you know, that the spherical that these twists, change what dimension, it’s really interesting.

EL: Yeah, yeah. And I'm wondering how you can use like, the specific — what am I trying to say, like, the opportunities that an orchestra gives you to sort of, you know, can you, like, pass off ideas from one section to another, can that give you a twist? Or, you know, something like that, how you could, I guess, use the tools that you have at hand to kind of explore it in a different way than you might if you were trying to draw it on a piece of paper?

EH: Yeah, I think, I mean, it’s probably better doing that than drawing on a piece of paper actually. Like, I feel there must be something about the way I think that I like — these thoughts, and these kinds of systems and shapes immediately present musical ideas to me. So, absolutely, you know, what I would be interested in is a process that you could audibly hear becomes something different when it's spherically curved, and becomes something different when it's, you know, hyperbolic, and making that more and more extreme. So the math underlies that, and perhaps, maybe it's a process of me imagining going through this, these these dodecahedron and things. But in fact, that will be a layer, and then after that, perhaps then, I do something more musical, as in this has given me this data, but I accentuate it in ways, and perhaps also go against it and things. Because I do think as well with art, usually, you're asking a question rather than answering anything, which is a real difference, I think. Because mathematicians, you're so interested in truth, or mathematical truth, and you’re really bound by it; it’s really important. But I always feel that I'm more interested in what your process is in these wonders, and then I'm just using that to sort of leap somewhere unknown.

KK: Well, so are we. We're leaping into the unknown all the time, we just, then want we want the answer once we get there, right?

EL: I do think there's a similarity in that, you know, the way mathematicians approach things a lot is, you kind of set up these axioms, and this is the rule system I'm going to be working within. And I think that forms of art do that as well, you know, say, this is the aesthetic system that I'm working in, or this is, and maybe they're not as rule-bound as mathematics is, because when you say like, these are the axioms, or these are, you know, whatever, I'm doing, then you're just really stuck with them. And with art, a lot of the times it's about breaking the rules. But I do think, you know, you kind of set up these, sometimes composers can set up compositional rules where, like, Okay, well, I'm writing a fugue, which means that I have, you know, I have this kind of structure. And the allowed things to do are like transposing it or flipping it backwards, or things like this, and say, like, well, I'm working within this form, in this way. So, I mean, I did a lot of music in college, and I was kind of torn between going into math and music. And I think the way that I thought about them kind of tickled the same part of my brain, that’s why I was interested in both things, and ended up, you know, in math instead of music professionally, but I do think we've made these aesthetic or form kind of rules in music or art. And now we're going to work within them, just the way mathematicians do with axioms.

EH: Yeah, I mean, I completely agree with you on that. And I also would say as well that I agree, and actually working closely with professional mathematicians has really kind of opened my eyes to how much they are going, because I don't think you learn that, you know, these are the answers, actually it is a lot of guesswork. And it's really, yeah, leaping, as you say.

EL: Yeah, well, this, this has been a lot of fun to talk with you about this. I really hope our listeners will go find your pieces. And we'll definitely link to your website and the article you talked about and everything. Is there anything else that you want to share, you know, things you want to suggest that they look into or read or any concerts coming up that people could actually attend or livestream?

EH: Yes. So actually, next week, the BBC Philharmonic is playing sphere in Manchester. And that should be on Radio 3, as well. I’m based at the Royal Northern College of Music in Manchester. I’m a professor of composition, and I run, I direct, the PRiSM lab. PRiSM is a research center for Practice and Research in Science and Music. And I'm lucky enough to work there also with colleagues. Marcus du Sautoy joins us and David De Roure, as well. And I suppose we're interested in mathematics meets music meets science meets AI, and there are lots of different types of composers. We have a blog, and that's where this orchestral geometries blog will be. And lots of very exciting, very different things going on there. So I just wanted to mention that and maybe I could also give you a link for that as well.

KK: Sure.

EL: Yeah, that would be great.

KK: We will include it. All right. This has been fantastic. Thanks for joining us, Emily. It's really been great.

EH: Thank you so much for having me. It's been a pleasure.

[outro]

On this episode of the podcast, we were delighted to talk to composer Emily Howard, who uses her mathematics background in her compositions, about her favorite mathematical object, the torus, and the orchestral work it inspired. Below are some links you may enjoy after you listen to (or read) the episode.
Emily Howard's website
Her page about the composition Torus, including a recording by the BBC Radio Orchestra
Her article Orchestra Geometries
The November 11, 2021 BBC Radio concert featuring Howard's composition Sphere
PRiSM, the Royal Northern College of Music Centre for Practice and Research in Science and Musicthat Howard directs
A website visualizing the eight Thurston geometries for 3-dimensional space
An article by Evelyn about the pseudosphere (or antisphere)
Our episode with Emily Riehl, who is relevant to this episode because she is both an Emily and a violist

Episode 70 - Joel David Hamkins

Evelyn Lamb: Hello, and welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Evelyn Lamb, one of your hosts. I'm a freelance math and science writer in Salt Lake City, Utah, and this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. I like your background, Evelyn. Our listeners can't can't see it. But it looks like you had a nice camping trip somewhere in Utah.

EL: Yes, that's actually a couple years ago — we’re going to describe this in great detail for all the listeners who can't see it, no — this was a camping trip in Dinosaur National Monument on the Utah-Colorado border, which is a really cool place to visit.

KK: Excellent. It’s beautiful over there. Yeah. So how's things?

EL: Oh, not too bad. A bit smoky here, we're getting a lot of wildfire smoke from the west coast. And it, you know, makes some of those outdoor activities that are so fun, a little less fun. So I hope it clears out.

KK: Right. Yeah, well, my big adventure lately was getting my son settled into his new apartment in Vancouver. And it was my first time on an airplane in a year and a half. That was weird. And then of course, Vancouver, such a lovely city, though. We had a good time. So he's all set up and starting grad school and his nice new adventure.

EL: Yeah. And he's kind of, almost as far as he can be in a populated place in North America from you.

KK: That’s correct. That's, like, almost 3000 miles. That's far.

EL: Yeah.

KK: It’s great. Anyway, enough about that.

EL: Yeah. Well, today, we're very happy to have Joel David Hamkins join us. Joel, would you like to introduce yourself and say a little bit about yourself?

Joel David Hamkins: Yeah, sure. I am a mathematician and philosopher at the University of Oxford. Actually it’s something of an identity crisis that I have, whether I'm a mathematician or a philosopher, because my original training was in mathematics. My PhD was in mathematics. And for many, many years, I counted myself as a mathematician. But somehow, over the years, my work became increasingly concerned with philosophical issues, and I managed somehow to turn myself into a philosopher. And so here in Oxford, my main appointment is in the philosophy faculty, although I'm also affiliated with the mathematics department. And so I don't really know what I am, whether a mathematician or a philosopher. I do work in mathematical logic and philosophical logic, really all parts of logic, and especially connected with the mathematics and philosophy of the infinite.

KK: Well, math is just kind of, it's just applied philosophy anyway, right?

EL: This might be a dangerous question. But what are — like, do you feel a big cultural difference between being in a math department and a philosophy department?

JDH: Oh, there's huge cultural differences between math and philosophy, I mean, on many different issues, and they come up again and again, when I'm teaching especially, or interacting with colleagues and so on. I mean, for example, there's a completely different attitude about reading original works. In philosophy, this is very important to read the original authors, but in mathematics, we tend to read the newer accounts, even of old theorems, and maybe for good reason, because oftentimes, those newer accounts, I think, become improved with greater understanding or more connections with other work and so on. Okay, but one can certainly understand the value of reading the original authors. And there's many other issues like that, cultural differences between math and philosophy.

EL: Yeah, well, I, when I was at the University of Utah, I taught math history a couple times. And I was trying to use some original sources there. And it is very difficult to read original sources in math. I don't know if part of that is just because we aren't used to it. But part of it, I do feel like the language, and the way we talk about things changes a lot really quickly. It makes it so that even reading papers from the 1920s or something, you sometimes feel like, “What are they talking about?” And then you find out Oh, they're just talking about degree four polynomials, but they're using terms that you just don't use anymore?

JDH: Yeah, I think that's absolutely right. And oftentimes, though, it's really interesting when there's an old work that uses what we consider to be modern notation. Like if you look at Cantor's original writings on the ordinals, say, it's completely contemporary. His notation, he writes things that contemporary set theorists would be able to understand easily, and even even to the point of using the same Greek letters, alpha and beta to represent ordinals, and so on, which is what we still do today. And so it's quite remarkable when the original authors’ notation survives. That's amazing, I think.

EL: Yeah. So we invited you here to share your favorite theorem. What have you decided to share with us today?

JDH: Well, I found it really difficult to decide what my favorite theorem is. But I want to tell you about one of my most favorite theorems, which is the fundamental theorem of finite games. So this is the theorem, it was proved by Zermelo in 1913. And it's the theorem that asserts that in any two player finite game of perfect information, one of the players has a winning strategy, or else, both players have drawing strategies if it's a game that allows for draws. So that's the theorem.

EL: So what is the definition of a game? This is, like, the most mathematical thing to set, you know, like, yeah, okay, games, we've all played them starting from when we were, you know, two years old or something. But now we need to sit down and define it.

JDH: Yeah. Well, that's one of the things that really excites me about this theorem, because it forces you to grapple with exactly that question. What is a game? What is a finite game? What does it mean? And it's not an easy question to answer. I mean, the theorem itself, I think, is something that you might think is obvious. For example, if you think about the game of chess, well, maybe you would think it's obvious that, look, either one of the players has a winning strategy, or they both have drawing strategies. But when it comes to actually proving that fact, then maybe it's not so obvious, even for such a game as chess. And then you're forced to grapple with exactly the question that Evelyn just asked, you know, what is a finite game? What is a strategy? What is a winning strategy? What does it mean to have a winning strategy, and so on?

So there's a wonderful paradox that surrounds the issue of finite games called the hypergame paradox. So if you have a naive account of what a finite game is, maybe you think a finite game is a game so that all plays of the game finish in finitely many moves or something. Okay, so that seems like a kind of reasonable definition of a finite game. And then there's this game called hypergame. And the way that you play hypergame is, say, if you and I are going to play hyper game, then the first player, maybe you go first, you choose a finite game. And then we play that game. And that's how you play hypergame. So the first player gets to pick which finite game you're going to play. And then you play that game. And if we said every finite game is a game, so that all plays end in finitely many moves, then it seems like this hypergame would be a finite game, because you picked a finite game, and then we play it, and then the game would have to end in finitely many moves. So it seems like hyper game itself is a finite game. But then, the paradox is that if hypergame is a finite game, then you could pick hypergame as your first move.

KK: Right.

JDH: Okay, but then we play hypergame. But we just said when playing hypergame, it's allowed to play hypergame as the first move. So then I would pick hypergame as my first move in that game. And then the next move would be to start playing hypergame. And then you could say, hypergame again. And then I could say hypergame, and so on, and we could all just say hypergame, all day long, forever. But that would be an infinite play. And so what's going on? Because it seems contradictory. We proved first, that every play of hyper game ends in finitely many moves, but then we exhibited a play that didn't end. So that's a kind of paradox that results from being naive about the answer to Evelyn's question, what is a finite game? If you're not clear on what a finite game is, then you're going to end up in this kind of Russell paradox situation.

KK: Exactly. Yeah. That's what I was thinking of, the sets that don't contain themselves. Right.

JDH: Right, exactly. It's actually a bit closer to the what's called the Burali-Forti paradox, which is the paradox of the class of all ordinals is well ordered, but it's bigger than any given ordinal. And so it's something like that. So if you think a lot about what a finite game is, then you're going to be led to the concept of a game tree. And a game tree is the sort of tree of all possible positions that you might get to. So there's the initial position at the start of the game, and the first player has some options that they can move to, and those are the sort of the child nodes of that root node. And then those nodes lead to further nodes for the choices of the second player, and so on. And so you get this game tree. And the thing about a finite game is that — well, one reasonable definition of finite game is that the whole game tree should be finite. So that in finite infinitely many moves, you're going to end up at a leaf node of the tree, and every leaf node should be labeled as a win for one of the players or the other, or as a draw, or whatever the outcomes are. And so if you have the finite tree conception of what it means to be a finite game, then hypergame is not a finite game, because at the first move, there are infinitely many different games you could choose, and so the game tree of hyper game won't be a finite tree, it will be an infinite tree. It will be infinitely branching at its first node.

And so if you have this game tree conception, then a finite game can mean a game with a finite game tree. And then we can understand the fundamental theorem of finite games. So my favorite theorem is the assertion that in any finite game like that, with a finite game tree, then one of the players has a winning strategy, or both players have drawing strategies. And what does it mean to be a strategy? I mean, what is a strategy? If you think about chess strategies or strategies as they're talked about conventionally, then oftentimes, people just mean I kind of heuristic, you know, the strategy of control the center or something, but in mathematics, we want a more precise notion of strategy. So controlling the center isn't really a strategy; that’s just a heuristic. It doesn't tell you actually what to do. So a strategy is a function on the game tree that tells you exactly which moves to make whenever it's your turn. And then a play of the game is basically a branch through the game tree. And it conforms with this strategy if whenever it was your turn at a node in the game tree, then it did, what the strategy was telling you to do at that node. So a strategy is winning if all the plays that conform with that strategy end up in a win for that player.

KK: Okay, so is this the point of view Zermelo took when he proved this?

JDH: So yes, Zermelo didn't have, he didn't quite have it all together. And this is maybe related to the fact that we don't actually read Zermelo’s original paper now when we want to prove the fundamental theorem of finite games because we have a much richer understanding, I think, of this theorem now. I mean, for example, in my in my proof-writing book, I gave three different proofs of this theorem. And Zermelo didn't have the concept of a game tree, or even a finite game in terms of game trees, like I just described. Rather, he was focused specifically on the game of chess, and he was thinking about positions in chess as pictures of the board with the pieces and where they are. But nowadays, we don't really think of positions like that, and there's a kind of problem with thinking about positions like that. Because if you think about a position in chess, like a photograph of the board, then you don't even know whose turn it is, really, because the same position can arise, and it could be different players’ turns, and you don't know, for example, whether the king has moved yet or not, but that's a very important thing, because if the king has moved already, then castling is no longer an option for that player.

KK: Right.

JDH: Or, or you need to know also what the previous move was, in order to apply the en passant rule correctly, and so on. So if you just have the board, you don't know whether en passant is allowed or not. I mean, en passant is one of these finicky rules with the pawn captures where you can take it if the previous player had moved two steps, and your pawn is in a certain situation, then you can capture anyway. So I mean, it's a technical thing, it doesn't matter too much. But the point is that just knowing the photograph of the board doesn't tell you doesn't tell you whose turn it is, and it doesn't tell you all the information that you need to know in order to know what the valid moves are. So we think of now a position in a game is a node in the game tree, and that has all the all of the information that you need.

So Zermelo’s proof was concerned with games that had the property that there were only finitely many possible configurations, like chess. There are only finitely many situations to be in, in chess. And he argued on the basis of that, but really, it amounts to arguing with the finiteness of the game tree in the end.

KK: Yeah. So is this at all related to Nash's equilibrium theorem? So I was scrolling Twitter the other day, as one does when one has nothing else to do, and I saw a tweet that had, it was a screenshot. And it was the entire paper that Nash published in the Proceedings of the National Academy proving his equilibrium theorem. I mean, it's a remarkable thing that it's just a few paragraphs. Is there any connection here?

JDH: Right, so actually, this kind of nomenclature, there are three different subjects connected with game theory. There’s game theory, which, Nash equilibrium and so on is usually considered part of game theory. And then there's another subject standing next to it, which is often called combinatorial game theory, or it's sometimes also called the the theory of games. And this is the the study of actual games like Nim, or chess, or Go, and so on, or the the Conway game values are part of this subject. And then there's a third subject, which I call the the logic of games, which is a study of things like the fundamental theorem of finite games, and the sort of the logical properties. So to my way of thinking, the Nash equilibrium is not directly connected with the theory of games, but rather is a core concept of game theory, which is studying things like the stability of probabilistic strategies, and so on, whereas combinatorial game theory isn't usually about those combinatorial strategies, but about sort of logically perfect strategies, and optimal play, and so on. And that isn't so much connected to my way of thinking with the Nash equilibrium.

KK: So fine, I hand you a finite game. It has a winning strategy. Can you ever hope to find it?

JDH: Oh, I see. Is it computable? Well, yeah, this is the big issue. Even in chess, for example, I mean, chess has a finite game tree. And so in principle, we can, in the sense of computability theory, there is a computable strategy you can prove, because it's a finite game. So the strategy is a finite function, and every finite function like that is definitely computable. And we can even say more about how to find the strategy. I mean, one of the proofs is this backpropagation proof, you look at the game tree, and you propagate. You know that the leaf nodes, the terminal nodes are labeled as a win, and you can propagate that information up the tree. But it turns out that the game tree of chess is so enormous…

KK: Right.

JDH: That it wouldn't fit in the universe, even if you used every single atom to represent a node of the tree. And so in that sense, you can never write a computer program that would compute the perfect chess play. It's just too big, the strategy. If you're really talking about the strategy on the whole game tree, then the game tree is just too enormous to fit in the universe. And so it's not a practical matter. But theoretically, like in terms of Turing computability or something, then of course it's computable. There are some other issues. For example, I've done a lot of work with infinitary games. I mean, this connects my interest with infinity. And it turns out that there are some, some positions, say, in infinite chess, which I've studied, we identified computable positions in infinite chess, for which White has a winning strategy, but but there's no computable winning strategy. So if the players play computably, in other words, according to a computable procedure, then it will be a draw. So it's very interesting, this interaction between optimal play, and computable optimal plays not always, they don't always align.

EL: So I've got to ask you to back up a little bit. What is infinite chess?

KK: Yeah.

JDH: Oh, I see infinite chess. So imagine a chessboard without any border. It just goes forever in all four directions.

EL: So you start with the same, you know, don't start with infinite number — or, yeah, I’ll let you keep going.

JDH: So of course, infinite chess. I mean, we can imagine playing infinite chess, you know, at a café or something, but it's not a game that that you sit down in a café to play. It's a game that mathematicians think, “What would it be like to play if this if the board looked like this, or if it looked like that?” And so there's no standard starting position. You present as a starting position, and you say, “Well, in this position, it has a very interesting property.” So Richard Stanley asked the question, for example, on mathoverflow. And that sparked my interest in this question. Well, one of the things that we proved was that you can have positions in infinite chess that White has a winning strategy, so it's going to win in finitely many moves, it’s going to make checkmate in finitely many moves. But it's not made in n for any n, for any finite n. In other words, Black can make it take as long as he wants, but it's hopeless. In other words, Black can say, “Oh, this time, I know you're going to win this time, but it's going to take you 100 moves.” Or he could say, “This time, it's going to take you a million moves.” For any number, he can delay it that long, but still White is going to win infinitely many moves following the winning strategy. So these are called games with game value omega, and, and then we produce positions with higher game, higher ordinal game values, omega squared, omega cube, and so on, the current record is omega to the fourth. So these transfinite game values come in. And so it's really quite fascinating how that happens.

KK: Well, knowing that you like logic and philosophy so much, now I see why you like these games, right? This is kind of a logic puzzle, in some sense. Yeah. Yeah.

EL: Well, and what made you choose this theorem as your favorite theorem?

JDH: Well, I mean, there's a lot of things about it to like. First of all, what I mentioned already, it forces you to get clear on the definitions, which I find to be interesting. And also it has many different proofs. I mean, as I mentioned, there are already three different proofs, three different elementary proofs. But some of those proofs lead immediately to to stronger theorems. For example, there's a slightly more relaxed notion of finite game, where you have a game tree, so that all plays are finite, but the tree itself doesn't have to be finite. So this would be what's called a well-founded tree. And then these would be called the clopen games, because in the in the product topology, the winning condition, amounts to a clopen set in that case. And then the Gale-Stewart theorem proved in the ‘50s, is that infinitely long games whose winning condition is an open set, those are also determined: one of the players has a winning strategy, open determinacy. And then Tony Martin generalized that to Borel determinacy. So Borel games also have this property. So infinite games whose winning condition is a Borel set in the product space are determined, one of the players has a winning strategy. And then if you ask, Well, maybe all games are determined, in the sense that one of the players has a winning strategy, you whether or not the winning condition is Borel or not. And this is called the axiom of determinacy. And it's refutable from the axiom of choice for games on omega, say, you can refute it. But it turns out that if you drop the axiom of choice, then the consistency strength of that axiom has enormous strength. It has large cardinal strength. In large cardinal set theory, the strength of the axiom of determinacy is infinitely many wooden cardinals, if you've ever heard of these large cardinals. And for example, under AD, the axiom of determinacy, it follows that every set of reals is Lebesgue measurable, and every set of reals has the property of Baire. And so there's all these amazing regularity set theoretic consequences from that axiom, which is just about playing these games.

And so what I view as the whole topic, the fundamental theorem of finite games just leads you on this walkway to these extremely deep ideas that come along much later. And I just find that whole thing so fascinating. That's why I like it so much.

EL: So I guess I want to pull you out of these extremely deep things into something much more shallow, which is, are there games like, games that kids play, or that like, people play, that are infinite games by their nature?

JDH: Oh, I see. I mean, I've studied quite a number of different infinite games. For example, well, I have a master's student now he just wrote his dissertation, his master's dissertation on infinite checkers, but we're in the UK, so we call it infinite draughts. But I guess ordinary checkers is usually just on an eight by eight board and so that doesn't count. So, I've done some work on Infinite Connect Four, I mean, infinitary versions of Connect Four and infinite Sudoku and infinite Go. And there are infinite analogs of many of these games that are quite interesting.

EL: I guess what I'm wondering is, so, in my mind, maybe there's a game where you can, like, go back and forth forever. And there's no rule in the game that says you can't just like walk towards the opponent and then walk backwards. And then the game tree might not be finite, or maybe I don't quite understand how the game tree would work.

JDH: No, no, you're absolutely right about that. Yeah, in that situation, the game tree would be infinite. I mean, it's related — for example, in chess, there's this threefold repetition rule. If you repeat the situation three times, then it's a draw. But the actual rule isn't that it's automatically a draw, but that either player is allowed to call it a draw. Right? And that's a difference. Because if you don't insist, if both players choose not to call the draw, then actually the game tree of chess would be infinite then because you could just keep moving back and fourth forever. And there's also another rule, which is not so well known, unless you're playing a lot of chess tournament play, which is the 50 move rule. And this is the rule in chess tournaments that they use, where if there's 50 moves without a pawn movement or a capture, then it's a draw. And the reason for that is — I mean, of course, I view both of those rules as kind of practical rules just to have an end of the game so that it doesn't just go on forever in the way that you described. But when we were deciding on the rules for infinite chess, we just got rid of those rules, and we thought, look, if you if you want to play forever, that's a draw. So any infinite play is a draw is the the real rule, to my way of thinking. And and the reason why we have the threefold repetition rule, and the 50 move rule is just those are proxies for the real rule, which is that infinite play is a draw. But it doesn't quite answer your question. I'm sorry, I don't know any children's games that are just naturally infinite already. Name the biggest number.

EL: Yeah, well I’ve also been sitting here wondering, like, can you make a game — like other children's games that aren't maybe on a board? Tic tac toe, of course, is the first thing I think, but then what about Duck Duck Goose [for the Minnesotans: Duck Duck Grey Duck] or Simon Says, or these other children's games that aren't really board games? You know, can you even work those into the framework of these finite games, or infinite games, or not? I don’t know.

JDH: There is this game of Nim where you play with stacks of coins and you remove coins from one stack or another. It's a beautiful game with a really nice resolution to it in terms of the strategy, but that game has infinitary versions where the stacks are allowed to have infinite ordinal heights. And basically, the classic proof of the Nim strategy works just as well for ordinals. And so if you if you think about the sort of the balancing strategy, where you look at the things base two, and so on, well, ordinals have a base two representation in ordinal arithmetic also, and you can still carry out the balancing strategy even when the stacks have infinite height.

EL: It never occurred to me to think about infinite ordinal height Nim.

KK: Well, you never have that many toothpicks, right?

EL: Yeah. Yeah. limited by my environment, I guess. I must say that I've spent a lot of my mathematical career sort of avoiding things with with too much infinity in them, because they're very intimidating to me. So maybe we have kind of different mathematical outlooks.

KK: Right. Yeah. So the other thing we like to do on this podcast is invite our guests to pair their theorem with something. So what pairs well, with the fundamental theorem of finite games?

JDH: Well, there's only one possible answer to this, and I worry that maybe I'm cheating by saying, of course, I have to pair it with the game of chess.

KK: Sure.

JDH: Because Zermelo’s theorem was really focused on chess, and he proved that, look, in chess, either White or Black has a winning strategy, or else both of them have drawn strategies. And I never played a lot of chess when I was a child, but when I had kids, they got involved in the scholastic chess scene in New York, which is quite hyperactive, and fascinating. And so my kids were playing a lot of chess, and I went to hundreds of chess tournaments and so on, and so I started playing chess and I learned a huge amount of chess from, they had such great coaches at their schools and so on. But actually, I'm a pretty mediocre chess player even after having played now for so many years. And one of my coauthors on the infinite chess papers that I wrote, is quite talented chess player. He's a national master, Cory Evans. He was a philosophy graduate student when I met him at the City University of New York, which is where my appointment was at the time. And so I got to meet a lot of a lot of really talented chess players, and it was really great working with him on that infinite chess stuff, because I realized that that actual chess knowledge is really focused on the 8 by 8 board, and that once you go to these much bigger boards, the the chess grandmasters even become a little bit at sea. And so I would know what I'm trying to do mathematically to create these positions with high game values, and I would show them this crazy position with, you know, 20 bishops and hundreds of rooks and so on. And I would talk a little about, and he would say, hang on, this pawn is hanging here, it's totally unprotected. And it would completely ruin my position. So their chess ability, their chess reading ability was such that they could look at these crazy infinite positions and point out flaws with the position. And that was really something that was important for our collaboration. These chess positions are so finicky, these huge, infinite ones. And so many details are running on whether the things are protected properly, and whether — because oftentimes, you have to argue that the play has to proceed according to this main line. And if you want to prove the theorem, you have to really prove that. And if there's some little upset that means that the flow of play isn't exactly like what you thought, then the whole argument is basically falling apart. And so it really was depending on on all of that. So I really had a great time interacting with a lot of these talented chess players. It was really fantastic.

KK: I’m a lousy chess player.

EL: A lot of interest among chess players in the mathematical study of the game of chess? Even leaving aside the infinite versions, but, you know, the finite version. I assume there are theorems being proved about regular chess. Do players care about them much?

JDH: Well some of them definitely do. And actually, there's a huge overlap, of course, between chess players and mathematician

EL: Oh, yeah, that's true.

JDH: And so maybe maybe a lot of the interest is coming from that overlap. But, for example, there was a problem that I had asked, I think I asked it on Mathoverflow. Take chess pieces. On an empty board, take a full set of chess pieces and just throw them at the board. So you you get some position. What's the chance that it's a legal position? So in other words, a random assignment of pieces. And you can you can make some calculations and prove some interesting things about the likelihood that it's a legal position. In other words, a legal position, meaning one that could in principle arise in a game, in a legal game, right. And

EL: Do you happen to recall any ballpark, you know, is this, like a 1% chance?

JDH: It’s way less than 1%. It’s exceedingly unlikely.

EL: Okay.

JDH: If you allow, if you insist on all the pieces, because then there haven't been any captures. So the pawns have to be sort of perfect. There has to be one column and opposing. And already just because of that, that already makes it extremely unlikely to happen if you have all the pieces. And then some other people answered on Mathoverflow, I think giving better bounds when you don't have all the pieces and so on. But it wasn't quite open. But I think the general conclusion was that it's extremely unlikely that you get a legal position.

KK: Well, that makes sense. Given the complexity of the moves, it would be pretty remarkable if a random placement would would actually work.

JDH: There are some amazing — there’s a book by Raymond Smullyan, about the “chess detective,” and he has these many instances. It’s sort of like he gives you a chess position, and and you have to deduce, what was the previous move? Because these positions are often extremely strange. Like you think, “How could that possibly arise?” So there's sort of logic. I mean, he's a logician. And so there are sort of chess logic puzzles to figure out what the previous move was. And there's often a story associated with the game that, you know, so it must have been Black, who was the murderer because… It’s really some fascinating work that way. I really like that.

KK: Well, this has been informative. I've certainly learned a lot today.

EL: Yeah.

KK: So we like to give our guests a chance to advertise. Where can we find you online? And if there's anything you want to promote, we're happy to let you do it.

JDH: Oh, I see. Well, you can find me online, I have a blog, jdh.hamkins.org. And also, I'm on Twitter, and also on Mathoverflow. And I just published a number of books. So one of them I mentioned already, it's called Proof and the Art of Mathematics. And this is a book for aspiring mathematicians to learn how to write proofs, Proof and the Art of Mathematics with MIT Press. And I have another book, a philosophy book called Lectures on the Philosophy of Mathematics, also with MIT Press. And that is a book that I use for my lectures on the philosophy of mathematics here in Oxford. And it's, I would say, a kind of, grounded in mathematics perspective on issues in the philosophy of mathematics.

EL: Yeah, and if I can praise you a little bit, I will say something that I have enjoyed ever since I've been following you is that you — some of some of the things you write are about, like very technical, you know, deep mathematical things. But you've also had some really neat, like, puzzles that you've shared with children and stuff like that. I remember I was working on a Math Circle project one time about paper folding and cutting and you had a fun, I think it was like you show someone a configuration of holes in a piece of paper and say, can you fold the paper so that you just have to punch one hole in this folded paper to get the holes looking like this? Or something like that. And so it kind of spans a big range of mathematical sophistication, and what level you want to jump into something. So I think that's something fun and other people who might be looking for activities like that might enjoy it.

JDH: Thank you so much. I'm so glad to hear you mention that project. Those projects are all available on my blog if you click on the math for kids link, which is one of the buttons on my blog. And they all arose because I was going into my daughter's school every year, or a couple times a year, with these different projects, including that one and a number of other ones. So have about a dozen or more math for kids projects on my blog.

KK: Very cool.

EL: Well, thanks for joining us. I enjoyed talking about chess, a game that I have probably played, you know, 10 times in my life.

JDH: Well, it's a pleasure to be here. Thank you so much for having me.

KK: Yeah. Thanks.

[outro]

In this episode of the podcast, we were happy to talk with Joel David Hamkins, a mathematician and philosopher (or is that philosopher and mathematician?) at the University of Oxford, about the fundamental theorem of finite games. Here are some links you might enjoy perusing after you listen to the episode.

His website, Twitter, and Mathoverflow pages
On his website, check out Math for Kids for some fun activities for all ages
His books Proof and the Art of Mathematics and Lectures on the Philosophy of Mathematics
The Wikipedia page about the fundamental theorem of finite games
The PBS Infinite Series episode on infinite chess
The Mathoverflow question and answers about legal chess board positions

Episode 69 - Ranthony Edmonds

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, where there is a family of quail that live outside my window. And they don't know that I'm here so I can watch them scurrying around in the bushes. There are at least five young ones right now.

KK: Cool.

EL: They are so cute. Oh, it's just like, sometimes — they're not here right now, which is good. Because otherwise, I would just be like, staring out my window. Looking at these cute little quail.

KK: Oh, see, so in Florida, we're in actually the boring birds season because you know, it's just, this is the locals. So I see my cardinals and the titmice and all of that, but it's still fun. I still feed them. I'm out there every day. They’re eating me out of house at home. It's true. It's a good thing. All right, well, so today we are very pleased to welcome Ranthony Edmonds. Why don't you introduce yourself, please?

Ranthony Edmonds: Hi. Yes. So I'm Ranthony Edmonds. I'm a postdoctoral researcher at The Ohio State University.

KK: The Ohio State.

RE: The “the” is very important in Columbus. We take this very seriously. And I've actually become one of those people who corrects and makes sure that they add the “the” in conferences and notes and things like this. It’s very obnoxious. Yeah. So I'm a postdoctoral researcher at The Ohio State University. I study commutative ring theory, classically, specifically factorization theory. And I'm in the midst of this sort of interesting transition, where I am looking into applications of algebraic topology. So I spent the last year learning a bit about topological data analysis, and I’m specifically interested in applying that to redistricting. So I, you know, I my interests are kind of broad, but specifically, to kind of give you some keywords, commutative ring theory, topological data analysis, redistricting. And, of course, you know, my general mission is to increase access to mathematics for Black Americans and members of other traditionally underrepresented groups in the mathematical sciences. And I'm trying to do that through a combination of inclusive pedagogy. academic research, and community engaged scholarship.

EL: Nice, and I was perusing your website before this to familiarize myself a little bit, and I saw that you're working on a kind of a history-related project about Black mathematicians at The Ohio State University historically?

RE: Yes. I think a lot of people had a lot of different reactions to what happened last summer with with George Floyd and the protests that swept over the country. And one thing that I sort of questioned is this idea of, well, if we're going to try to improve access to mathematics for Black Americans, for other traditionally underrepresented groups, how do we really begin to do impactful work if we're not really aware of what's happened historically? And I've always been interested in history. I think a lot of this comes from a previous project I'm still doing with the Hidden Figures story and sort of using that to center discussions about diversity and equity in the discipline. But yeah, I just love math history. I have a team that's very interdisciplinary, and we're looking at the history of the math department, specifically at Ohio State this summer, and then on into the fall. So there are really two things that we want to do. One is, you know, a lot of the narratives of these the pioneers who graduate from the department with PhDs, with master's degrees, they're kind of just hidden. There’s not a lot of recognition about the work that they've done. But we have discovered that there are seven Black PhDs who have graduated with a doctorate degree in math from Ohio State. And we’ve got two former university presidents among that midst. We've got lawyers, authors, you know, program officers in the NSF, just people who have gone on to do really prolific things, and yet are still somewhat kind of unacknowledged by the university themselves, and then just in the wider math community, and I think that there are a lot of hidden stories out there. And I think when I reflect on the Hidden Figures story, this is what made that so impactful is because people didn't know. So I think that there's a lot of work out there that's being done by wonderful people that people just don't know about. And so what we're trying to do is to highlight and amplify those stories, one. And then two, examine and contextualize their experiences at the university. So what was happening when they were students here? What influenced their trajectories after graduation, where they went to work, if they went to industry or academia? I think if we think about trying to get more people in graduate school or get more people at the Faculty level, well, we should start by thinking about how we're serving our undergrads who are in that actual population and how we've done that historically.

So we're doing a lot of things. I'm working with some people in strategic communication, some people in our Office of Diversity and Inclusion here at Ohio State. Also we have a connection with the National math Alliance. So they are an organization that I'm very intimately familiar with from my time in graduate school at the University of Iowa, but they are really focused on trying to increase the number of minorities entering PhD programs in the mathematical sciences. And this is pretty broad, right? It's not just math. It’s statistics, it’s economics, it’s something that requires quantitative training as an undergraduate. So we're working with them. And we're also working with a local museum, the Ohio History Connection, and there is a specific branch which is, they call themselves Afro-Am, but the official title is the National Afro American Museum and Cultural Center. And they're located in Wilberforce, Ohio. We're working with them with some of our archival research, and also our community programming. So we've learned a lot of really interesting things. We’ve sort of broken up the history of the department starting at 1963, when the first Black male PhD, his name was William McWorter, graduated from Ohio State up to the present and just identifying individuals who earned degrees during that time period and interviewing them, as well as sort of contextualizing what are the big things? How did selective admissions affect Black student enrollment in general and specifically in the math department? How did you know the protest of the ‘60s and ‘70s impact the campus environment? We have students who are wonderful who are helping us look at these different questions. And then there's actually like a lot of cool math. For instance, the the first Black male PhD from Ohio State, his name was William McWorter. And he was part of this camp along with like Axler and others that was like “death to determinants.” We don't need them, why are we teaching them to students? He felt like it was a very crippling tool pedagogically, in that students just used them for computations and had no idea what they were. And which makes sense, because I think I've experienced that on the student end of things.

EL: I’d say guilty as charged there.

RE: So he wrote a couple of papers for that were published in Math Magazine about determinant-free methods in linear algebra. And specifically he competed came up with an algorithm for computing the characteristic polynomial of a matrix and computing eigenvalues and eigenvectors of a matrix without using the determinant. And I say, “a matrix,” there are obviously conditions imposed upon it, but it was really cool. So I'm working with a student this summer, and we're reading through this paper, and then we want to create a lesson plan related to that algorithm. Because it mainly focuses on dependency relationships, like do you understand the difference between, like, given a list of vectors, can you determine if they're linearly independent or dependent? And then it requires doing that via elementary row operations. So it's just sort of hitting some of the high points from introductory linear algebra, without getting into the weeds of what the determinant really is. So we're working to create a lesson plan from that, and then hopefully, that'll be incorporated back into the honors track here. It was when he taught here. And disseminating that. Our main goal is learn it and then disseminate it, you know, so other people can can learn from what we're figuring out. So there's this history component, and then there's a lot of math that we're uncovering from the history that's just really interesting in its own right, that we hope to, over time turn into lesson plans that other people can use for their classrooms.

EL: That’s really cool. Like, so I've done a little bit of dabbling in math history and stuff. And it's always really interesting to me how much the language has changed. And you'll see an abstract for a paper written it decades ago and realize, like, we just talk about things differently now, and it's kind of hard to dig down and figure out, Okay, how would I think about what they're doing here? You know, they have these names for different curves that aren't names I use anymore, and like, how do I translate it? It's like almost a translation project. Even going back just to the ‘60s, maybe.

RE: Yeah.

EL: So that must be really interesting. And I think that's a great project for students to do. So yeah, that sounds so cool.

KK: It is. You're very busy. And you know, your list of mathematical interests is super interesting to me too. I mean, I'm not a ring theorist, but the whole TDA and redistricting.

RE: Yeah. We’ll have to talk a bit after the podcast. But yeah, it is really interesting. And I think, you know, it's part of this whole approach of just trying to humanize mathematics. We're studying it, and we're getting into the nitty details, but we're also thinking about how people came to be mathematicians, and how this has actually been affected historically, especially for Black Americans, by policies. You know, a lot of the PhDs that we're studying about were supported by NSF fellowships, and this is a direct response to the space race that was happening. And they saw the influx of federal funding. And so it's all really interesting. I feel like I'm learning a lot about — even though it's focused on Ohio State — I feel like I'm learning a lot about the math community. And one thing that is really cool about us is sort of how we do our lineage. Right? And so the math genealogy websites are really cool, because you can sort of track back, very easily, Oh, this person who would who would they have been working with? You know, I think in another discipline, if you are trying to figure some information out about the person that you want to know, who their academic “siblings” were, that might be actually difficult to discern, but we have the Math Genealogy site where we can get that information easily.

KK: Yeah. All right. So this podcast, though, is called My Favorite Theorem, so we asked you on here for a reason. So, Ranthony, what is your favorite theorem?

RE: Yeah, so I am taking it back, back, back, back. So I actually would say that my favorite theorem is the fundamental theorem of arithmetic.

KK: Okay.

RE: It’s very classic. And the reason that I like it is because it's sort of the first introduction to really meeting math in disguise, because I think a lot of people are at least aware of the concept of it in grade school, even if maybe we don't get into the implications. And so, you know, the fundamental theorem of arithmetic, it states that, given an integer, so positive whole numbers greater than one. So yeah, greater than one, excluding zero, you know, it can be written uniquely as the product of prime numbers. And that this decomposition into primes is unique except for the order. So in practice, it means give me a number, like any number that's an integer, and I can factor it uniquely into small pieces called primes. And that’s it. That's the only way I can factor this number. And it gives it a unique signature. And it's telling us that in the same way that atoms are the building blocks of ordinary matter, these prime numbers build up the integers. And I love it, because there are a lot of implications in the work that I do in factorization theory that can all sort of be traced down to this fundamental idea. And I also love it because when I talk to younger students about ring theory or things like this, I always start with the fundamental theorem of arithmetic. And I tell them when they're drawing factor trees — at least that's how I learned, I'm not sure how you guys — is that what you remember? You had the number and then you do the branches?

KK: Yeah.

EL: I loved doing that when I was a kid. I don't know if you two were also like that. That it was kind of a soothing little exercise. Like, write down a big number — not too big; I wasn't super ambitious — but like, write write down a number and just do a little tree figuring out, you know, yeah, that kind of thing. I don't know. I thought it was fun.

RE: Yeah, I usually start off with when I’m talking with — I don't want to say little kids, right — with general audiences. I'll start off by asking people to pick their favorite three-digit number. So I guess maybe, do you guys have a phone or calculator handy?

KK: Sure.

RE: So this may not pack the same sort of punch, but I ask people to pick their favorite three-digit number. And then I ask them to create a six-digit number by taking that three-digit number and repeating it twice. So I usually use 314 because it's an approximation for pi. I was also married on Pi Day. And so 314 is is my number, and then I create a six-digit number, so that's going to be 314,314. Yeah. Okay.

KK: I chose 312.

RE: Okay, all right. And so I want you to take your six-digit number and divide it by 11.

KK: Okay.

RE: Okay. I have 28,574 right now. And then I want you to take that number and divide it by 13.

KK: It’s amazing that you're getting integers here.

RE: Yeah.

EL: Or is it?

RE: So now I have 2198. Okay, and so now I want you to take this number and divide it by your original three-digit number.

KK: Yep.

RE: And did everyone get seven?

KK: Yes.

EL: Yay!

RE: So yeah, so it's like this really cool thing where if you take a number and you multiply it by 1001, it has the effect of creating a new six-digit number. That's your original original number repeated twice. And so essentially, because we know that 1001 factors uniquely into primes, which is guaranteed to us by the fundamental theorem of arithmetic, you know, 1001 is 7×11×13. And so if you divide away 11, then divide away 13, if you divide away that original number, no matter what it was, you're going to be left with 7. And so it's really exploiting this property of the integers. This is really cool. And so I don't know, I just really love the theorem.

So why, I guess maybe, do I care about it? In terms of the mathematical sense, besides the fact that it's cool? It’s because there's a lot of deeper underlying mathematics here. So it's like, we have this statement that tells us, given any integer, we can decompose it uniquely into the product of prime numbers. And so like I mentioned before, these prime numbers are acting kind of like the atoms of the integers. And so in factorization theory, this is sort of the name of the game. We're really interested in, how do we decompose a mathematical object into its smallest pieces? And this is our very first introduction to this idea. It's in elementary when we're breaking numbers into primes. And then typically, when we kind of level up, the next thing we try to break down are polynomials, right? And so in algebra, whatever level in which you had it, you have a polynomial and you want to break it down too. And so it's like, okay, we want to factor it. And the question is, how do you know when you're done factoring? So you know, with a prime number, you circle it, and it's like, we have our, you know, but with polynomials, it’s a little bit more hazy. There's not a list of, well, there are, but a list of just all the irreducible polynomials that ever are. And so the question is, is there some sort of fundamental theorem that exists for the set of polynomials over the reals? So if we had something like x4−1, I remember in algebra that this was a difference of squares, and so there was a pattern. So I could break this into x2+1 and x2−1. And then this was always a tricky one, because it was like, aha, another another square, x2−1. So you can keep going. But then the question is, you know, do you circle x2+1 or not? Is it irreducible? And the question depends on the setting. Like, it depends on if we're working over the reals, or if we're allowing complex numbers, because if we allow complex numbers, then we can suddenly say that x2+1 is (x+i)(x−i). But if not, then, you know, maybe we're done.

So feasibly, it's like, well, we don't want to have to come up with a fundamental theorem for every single set of polynomials that exist. That's not very efficient. So we kind of generalize this idea of the integers into something called a commutative ring. And we generalize this idea of primes into irreducible elements. AndI think that living that abstraction is what I've spent most of my mathematical career looking at, like how things decompose, but I think tracing it back down to, you know, what we're really trying to do here is to come up with really nice notions that generalize the fundamental theorem of arithmetic. So this is probably why it's my favorite theorem, because I feel like if you keep going down to just the bare bones of what it is we're trying to do, the best example I think is there in that theorem, and also the best things are the things you can talk about.

KK: Yeah, and it's kind of like the first real theorem you learn.

RE: Yeah.

KK: Because, you know, I mean, you start learning mathematics in elementary school, and you learn how to add and subtract, but there aren’t really — well, there are theorems there, or definitions, maybe, but this one, you learn how to do it somewhere like fifth grade, maybe?

RE: Yeah, you’re really young, but I don't know that it was given the name.

EL: Yeah, I didn’t know the name of it, I think until I was probably in grad school, maybe college?

RE: Yeah.

EL: Still, but you learn it. Maybe you don't learn it as a theorem.

KK: Yeah. You learn an algorithm, right? Essentially.

EL: Yeah.

KK: Yeah. How do you do it? I mean, what do they teach you to do? Like, start dividing by primes, maybe?

RE: I guess I felt like at that point, well, this was when I was still just using a lot of memorized facts. And that was math to me. And so I guess I had my list of things that I thought were prime. And then maybe if they threw in a big number, I'd have to think about it. Like if they threw in like a 37. It's like, Oh, wait, what's happening? But 2, 11, 13? You know, we were pretty good to know. But yeah, I think I had no idea that it was a theorem. I do remember learning it, though. And so my favorite things are when I'm learning something, especially in a more advanced mathematical setting, and it takes me back to a very young me who just didn't know that there was a lot more to this when I was first exposed to it.

KK: Mm hmm. Yep. And hopefully didn't fall to the Grothendieck trap of thinking that 57 was prime, right?

RE: No. So basically, I've done a lot of work looking at unique factorization. And so because I work with zero divisors, which I don't know that I need to get into the nuts and bolts, but I thought a lot about what makes a unique factorization domain tick. Because I think a lot about settings where we don't have those nice properties. And so a unique factorization domain is the is the exact generalization of the fundamental theorem of arithmetic. So the fundamental theorem of arithmetic, you know, we've got a setting, the integers, where everything factors uniquely into primes, and in a unique factorization domain, it’s commutative ring, which is a generalization of the integers and the nice properties that they have. And it's a commutative ring, where everything factors uniquely into atoms, so we're generalizing primes now into atoms. And so there are some really nice results related to polynomial rings, where if you have a ring that has unique factorization, then the polynomial ring extension also has that same property, and vice versa, too. And so in the world that I live in, there are a lot of times where these factorization properties don't extend. And so I spend time thinking about what can we do to try to make them extend. So yeah, I think a lot about factorization theory and commutative ring theory. And so a lot of this is sort of based on this very gold-star standard of a factorization setting, which is a unique factorization domain. It's the nicest place that you can live, where factorization is just really well-behaved. You don't have to distinguish between primes and irreducibles, it's just a beautiful place to be. And so I call this a utopia and it's really mimicking or generalizing the fundamental theorem of arithmetic and the results there.

KK: So another thing we like to do on our podcast is ask our guests to pair their theorem with something. So we hear you might have multiple pairings. You’re only obligated for one.

RE: Okay, so originally, my first thought with a pairing was was alcohol, and I don't really drink that much, but I do love mead. So there is a meadery here. Okay, so mead is like it's like when they fermented grapes to make the wine, they ferment honey. So it's sweeter. And so there's a meadery here in Columbus called Brothers Drake. And I believe that they have a cousin, or a brother? You know, another brother meadery that's in California. But that's really broad. I'm not sure which part in California, but they have an apple pie mead. And it's my happy place when I do you know, partake in a little bit of something. So, I think the apple pie mead, and just any mead in general, if you would like to try it, especially when we get into history and stuff. I feel like this is like a very historical drink. Yeah.

EL: Yeah. Like, I don't know, I think of, like, dank castles and that kind of thing. Probably a lot of like, Disney fantasy, you know, people coming from battles and drinking their mead or something.

KK: Right.

RE: Yeah. I think a lot about Thor because I just am a big Marvel Universe person. And so I feel like Thor and Loki would just be having some mead, you know, catching up. But okay, so I was trying to think of what else would go with my theorem that wasn't alcohol. And so because I feel like the fundamental theorem of arithmetic is a very classic thing. So I would just like to pair my theorem with two things. One, sleep. So this is like a shameless plug for everyone to attempt to get eight hours of sleep. This is something that I tried really hard to do last year. And it was really crazy how much I fought against. Like, “I don't have time for this because XYZ,” but it took me maybe a whole semester, and I finally am now sleeping eight hours a night no matter what. And so this is a nice pairing with math, is sleep because I think that it's really good to do math when you're rested and your head is clear. So that's one. And then the second would just be like nice long walks. I love nature. I love cycling and strength training, but you just can't beat a good walk. And so for those who are able and mobile, I just think taking the time to go on quick walks during the day, even if it's just, like, 10 minutes, in between a meeting or something it’s a really great thing to do. So those are, I guess, like, my classic pairings. So what I say is apple pie mead, eight hours of sleep, and a walk.

EL: A nice long walk. This sounds like a great day.

KK: This is amazing. And you’ve got your bike there in the background.

RE: Yeah. Oh, my gosh, we're getting to know each other.

KK: Yeah. Well, when I was a postdoc, I was a very serious cyclist. I mean, I spent a lot of time, it was good therapy for me to get on the bike. Like if I was stuck on my math, I went for a ride.

RE: Yeah.

KK: But I lived in Chicago at the time, so you can't really ride in the winter.

RE: Yeah. That's an interesting city. I guess what like, did you go maybe to like suburbs and ride out there?

KK: Yeah, I was in Evanston. So I was at Northwestern. So that was good, because you could just head north, and then you're out in the country pretty quick. But you know, I had a group I rode with and all that, but just very good therapy all the way around. And then I had a kid and moved to Detroit. And those two things will just kill your cycling.

RE: What about what about you, Evelyn? Do you have — because you were talking about birds in the beginning, and I see that your background is very scenic.

EL: Yes, this is a cold day at Bryce Canyon National Park down in southern Utah. It's extremely hot in all of Utah right now. So this was kind of nostalgic, like, bringing some cold weather into it. But yeah, I love biking and taking walks and stuff. I'm really lucky in the neighborhood I live in. Basically, if you go north from my house, which is also uphill, you end up in less than a mile going into this extensive trail network that can get you all over the place if you're willing to go for a long walk. And it's like, I live less than two miles away from the state capitol building in downtown Salt Lake, but the fact that you can get up into nature so quickly is amazing.

KK: Well, it's right up against it.

EL: We’re built into the foothills here. And it's great. So yeah, I love taking walks in nature and I I've never tried that kind of mead, but there's actually a local like fruit wine place here that has a whole mead series in addition to fruit wines. And it's really cool because they have some that are sweeter and then some that are less, where they fermented, like, all of the sugar and it's amazing some of these, they almost taste like a Chardonnay or something. Yeah. Because you think of mead, honey wine, it's going to be super goopy and sweet, and depending on how much you ferment it and stuff, it actually has all sorts of different flavors so yeah, it's a cool place. I think they've got some like apple and honey cider mix things so I should check those out. Yeah,

RE: You definitely should, but I do agree that there are so many different like flavor profiles. The meadery here, you can go and do samples and they have like, music nights, pre-COVID, I think they're starting to resume this, and, like, empanada nights, which was a very personal weakness of mine. But yeah, I love — like, some of it just is too strong for me, right? Because I was I was leaning towards mead because I was like, okay, I don't know if I'm a hard alcohol drinker. But it's not all just sweet. It's not all just like juice with alcohol. I really like it. And so the last time that we tried was called Purple Rain. And I believe that the guy said that he, and it depends also on the barrel on which you're aging the the the mead, but he did something, like it was like using some sort of like blackberry something, and they accidentally like made too much and it was like overflowing the barrel when they came to check in on it. And so he called it Purple Rain. So I thought that was was pretty cool. Back to the cycling comment, my bike was actually stolen out of my garage at the beginning of COVID, and I was so upset. So at the beginning of COVID and work from home, when I realized that we'd actually be here for a while, I started nesting. I did a lot of things to my office space. So you see this black peel-and-stick wallpaper that I put up and actually turned out really nice.

EL: And a beautiful — I was hoping to see the rest of that picture that you’re tilting up now because yeah, I was thinking that looks really cool.

RE: Yeah, I got art. So this is um, I Gosh, I want to say it's just an Israeli painter named Itay Magen, and I just really love. It does a lot of really vital prints, colorful art. And so this came as a canvas. And then I realized when you buy canvas prints, you actually have to go get them mounted, which can be a little pricey. And then I ordered this Blackboard that you see, but the point that relates to the bike is that I also put together some shells, there's a landing gear, so you're kind of blinded, but I was painting and staining the shelves in my garage, and I left it open for a little bit, just to let the air sort of, you know, vent because of the paint fumes, and my bike was stolen. I got a little too trusting living downtown, you know, moving here from Iowa, you know, I kind of learned my lesson in the big city, I guess. So I got a new one. And I'm still getting to know this one a little bit better. But I took it out for the first time last week, and I have new clipped-in pedals. I got a different pedal than I had last time. I've been practicing just getting clicked in and out just at home because it's a little bit more challenging. So yeah, but I love doing outdoor things. And I think it's really nice to get fresh air, just for balance. And then also it does help, I think, with math. There's a tendency, I think, to try to double down, like, “no, I got to get this result. And then like sleep can happen or life can happen.” But I found that, you know, actually taking the breaks is really helpful.

EL: Yeah, the number of times where, you know, you're stuck on something, and then you actually let yourself sleep and wake up and realize, Oh, I can approach this in some different way — I wish I learned better from that rather than continuing to torture myself sometimes.

KK: Yeah, yeah. All right. Well, this has been great fun. So where can our listeners find you online?

RE: Yeah, so you can find me online on Twitter. My handle is @RanthonyEdmonds, let's see, with regards to the OSU Black math history project I mentioned, we will have a website, probably by September. But in the meantime, you can contact us at blackmathstory@osu.edu if you're interested in telling a story related to your time, you know, at Ohio State or affiliated, or just you just want to tell a math story. You know, that's the place to go. And then I think lastly, I'll start posting a lot soon on Twitter about another project that I'm working on just by the end of the summer related to redistricting and communities of interest, and sort of synthesizing community input so that when the redistricting process happens at the end of this year, we are taking into account communities of interest, which is this sort of traditional redistricting principle that says that communities with shared interests should be kept together in the mapping process. But what are those communities look like? Where are their boundaries? What are their key characteristics? We're working with the MGGG Redistricting Lab along with Ohio Organizing Collaborative and their independent citizens redistricting commission to really collect a lot of public input related to communities of interest. And so I'm focused on what's happening here in Ohio. But this is an effort happening over 10 states this summer, as we prepare for redistricting in the fall and all that's going to happen with the release of the census data. So I guess just stay tuned. Some good places that aren't my Twitter profile will be Common Cause, Ballotpedia, and of course, here in Ohio, the let's see, don't let me lie, ohredistrict.org. And so this is where you can find the Ohio citizens redistricting commission information. And so this is an independent commission that is sort of focusing on modeling good redistricting practices. And we're working closely with them this summer. But like I said, this is definitely happening in over 10 states. And so I'll start posting about this soon. But in terms of not me, specifically, just, you know, look some things up about what's happening in redistricting. Try to get involved and make your voice heard, because it affects all of us and it's really important, but I don't want to go on a separate tangent. This is supposed to be like a closing plug. So follow me on Twitter @RanthonyEdmonds. Email me if you're interested in telling your story related to Black math history at Blackmathstory@osu.edu. And then just you know, ohredistrict.org and Common Cause are really great resources for learning more about redistricting that's happening this year.

EL: Excellent. That is a fantastic set of resources. Thanks so much for joining us. This was a lot of fun.

KK: It’s really great.

RE: Thanks for having me.

[outro]

On this episode of My Favorite Theorem, we had the pleasure of talking to Ranthony Edmonds from The Ohio State University about the fundamental theorem of arithmetic. Here are some links you might enjoy after you listen to the episode:
Edmonds' website and Twitter account
An interview with NPR about her Hidden Figures-based course about mathematics and society
Math Alliance, a program that supports mentorship for early-career mathematicians from underrepresented groups
Ohio History Connection and the National Afro-American Museum and Cultural Center
An article by Evelyn about why 1 isn't a prime number, which mentions the distinction between prime and irreducible
The Metric Geometry and Gerrymandering Group (MGGG)
Ohio Organizing Collaborative
Ohio Citizens Redistricting Commission
Common Cause

Ballotpedia

Episode 68 - Rekha Thomas

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Evelyn Lamb, one of your hosts. I'm a freelance math and science writer in Salt Lake City, Utah, and this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. How are you doing, Evelyn?

EL: I’m doing okay. I was trying to think of something interesting to talk to you about at the beginning of this and life isn't very interesting, but in a good way. So that's good.

KK: Yeah. Yeah. You know, monotony is underrated, isn't it?

EL: Yeah.

KK: We just had our 29th wedding anniversary on Sunday.

EL: Congratulations.

KK: Thank you. That's our big news. We went out and sat at a restaurant for the first time in nearly a year. So we were very excited

EL: Excellent. I hope that, you know, this podcast is going to go on for decades and decades, and someone is going to be catching up on old episodes, you know, in 15 years and say, “Why do they keep talking about all these very boring things that they're doing for the first time in a year?” So that'll be great. You know, last time it was haircuts, this time it’s restaurants.

KK: That’s right. That's right. Yeah.

EL: But whether they are saying that or not, they will be very excited that we are talking today to Rekha Thomas, and Rehka, would you like to introduce yourself?

Rekha Thomas: Hi, thanks for having me on the show. So I am a professor of math at the University of Washington in Seattle. And I come originally from optimization. My PhD is actually in operations research. But I have worked in a math department ever since I graduated. And my work lies somewhere at the intersection of optimization, applied algebraic geometry and combinatorics. And I very much like problems that have an applied background, or a motivation. Not necessarily because I work in applied things, but because I very much like the problem to be motivated by something real. And in the last 10 years or so I've been working quite a bit in the mathematics behind questions that come from computer vision. And this has been especially fun, but optimization in general does have that applied side to it. So I like problems like that.

KK: Right. You know, I talked to my colleagues over an industrial and systems engineering, where they do a lot of that work all the time. And I think, “You guys are basically mathematicians. Why don't you come over here?” Heavy overlap there. Yeah.

EL: Well, and I just want to say, how we came to invite Rehka onto the podcast.

KK: Yes.

EL: I think it's pretty cool. So one of her students sent in an email to our little submission form on Kevin's website, saying that he thought that his linear algebra teacher was great and would be a great guest for My Favorite Theorem. And this is especially remarkable because this is in the semester, we're all teaching has been online and stuff. And I just think to be connecting with students enough when you're doing Zoom teaching, that they actually reach out to a podcast to get you as a guest on a podcast, just you must have really made an impression. So I think that's very cool.

RT: Thank you. I was very honored by that piece of news. I thought, “Okay. We finally made contact.” That was great. But especially for an undergraduate to take that initiative to write to you was very touching.

EL: Yeah. So you know, not to build things up too much, but I'm sure that this is going to be a great episode.

KK: Yeah.

EL: So with that introduction done, what is your favorite theorem?

RT: Yeah, so I announced my favorite theorem in that class and told them about your podcast. So when you invited me, I thought, “Oops, is that really my favorite theorem? Is that what I really want to speak about?” And I think yes, so I'm going to stick with that theorem. And this is a theorem from linear algebra. It was a linear algebra class. And I think it's one that's not very well known to pure mathematicians. So it goes by the name of Eckart-Young theorem or the Eckart-Young-Mirsky theorem, but apparently the history is more complicated. I've been reading about this so I can tell you about it if we get to that. So the theorem is basically says the following. So if you have a matrix, say a real matrix of size N by N, and we know its singular value decomposition [SVD] — which is a very special decomposition of the matrix as a sum of rank-one matrices — then the closest rank-K matrix to the given matrix is just made up of summing the first K rank-ones in that decomposition. So if I wanted the closest rank-one matrix, I just take the first rank-one in the singular value decomposition of A. If I wanted the closest rank-two matrix, I take the first plus the second rank-one matrices in the singular value decomposition, and so on. So there is this neatly arranged set of rank-one matrices that add up to give you A, and if you truncate that sum anywhere at the K-th spot, you get precisely the closest rank-K matrix to the given matrix. This is the theorem.

KK: Closest in what sense? What’s the metric?

RT: Yeah, so that's a very good question. So closest in either Frobenius norm or spectral norm. So these are standard norms on matrices. Frobenius norm is just, you think of your matrix as a long vector, where you just take every row, let's say, and concatenate it to make make a long vector, and then take the usual Euclidean norm. Okay. So the sum of Ai,j squared, square root. [Maybe a little easier to read in math notation: (Σ Ai,j2 )1/2.] Spectral norm is the largest singular value, so it is the biggest stretch that the matrix can make on a unit vector.

KK: Okay.

RT: So in either norm this works.

KK: That’s pretty remarkable.

EL: Yeah. So just doing a little bit of mathematical free association, is this, can this in any way be thought of as like a Taylor's Theorem, like, you're kind of saying like, Okay, if you want your approximation to be this good, you go out this far, if you want it to be this good, you go out this far, maybe that’s — I don't I don't know if that's a good analogy.

RT: No, I think it is. So I think there are many constructs in mathematics like this, right? Like Fourier series is another one, where you have a break down into pieces, and then you take whichever part you want. Taylor series is similar. The SVD is special in the sense that the breakdown, this breakdown into rank-one matrices is actually tailored to the matrix itself. Like, for example, as opposed to, say, in Fourier series, where the the basic functions that we are trying to write the function as a combination of, they are fixed, right? It is always cos(θ) + i sin(θ), or cos(Nθ) + i sin(Nθ). So it's not particularly tailored to the function. It's just a fixed set of bases, and you're trying to write any function as a combination of those basis functions. But in the SVD, the basis that you construct, the factorization that you get, is tailored to the actual data that's sitting inside the matrix. So it's very, very nice and is incredibly powerful. So it's similar, and yet, I think, slightly different.

KK: Right. In other words, yeah, that's a good explanation. Because, as you say, with Taylor series, you're choosing a basis for a subspace of the space of all smooth functions or whatever. Whereas here, you're taking a particular matrix. Does it matter what — so if you change the basis of your vector space, do you get a different SVD?

RT: No. So what the SVD is — there is a very nice geometric way to think of the SVD, which may answer that question better. So the SVD is sort of a reflection of how the matrix works as an operator. So what it's telling you is if you take the unit sphere in the domain, so let's say we have an N by M matrix, so the domain is Rn, take the unit sphere, under the map A, the sphere goes to an ellipsoid.

KK: Right.

RT: In general, a hyperellipsoid, right? And this ellipsoid has semi-axes. The singular values are the lengths of those semi-axes. Oh, so it tells you, yeah, the length of the semi axes, and the unit vectors in the directions of those semi-axes are one set of basis vectors. So that's one set of singular vectors, and the pre-images in the domain are the other set of singular vectors. So if you're willing to change basis to the special bases, one in the domain, one in the codomain, then your matrix essentially behaves just as diagonal matrix. It just scales coordinates by these singular values. So it's a generalization of diagonalization in some ways, but one that works for any matrix. You don't need any special condition. So they're very canonical bases that are tailored to the action of the matrix.

KK: I’m learning a lot today. So I have to say, so before we started recording, I was mentioning that I think students need to take more linear algebra. But the truth is, I need to take more linear algebra.

RT: I think we all do.

EL: Yeah, I mean, I am really realizing how long it's been since I thought seriously about linear algebra. So this is fun. And it goes really well with our episode from a few months ago. I believe that was Tai-Danae Bradley who chose a singular value decomposition as her favorite theorem. So we've got, you know, a little chaser that you can have after this episode, if you want to catch up on that. So Rekha, you said that there was a bit of a complicated history with this theorem. So do you want to talk a little bit about that?

RT: Yeah, sure, I'd be happy to. So I always knew this theorem as the Eckart-Young theorem. I only recently learned that it had Mirsky attached to it. But then in trying to prepare for this podcast, I started looking at some of the history of the singular value decomposition. And there's a very nice SIAM review article by Stewart, written in 1993, about the history of the singular value decomposition. And according to him, singular value decomposition should be attributed, or the first version of it, is due to Beltrami and Jordan from 1873 — so the Eckart Yang theorem is from 1936, so almost 60 years before — and they were looking at more special cases, they were looking at square real matrices that are non singular, perhaps. So they, you know, people were interested in that special case. Then there were several other people, like Sylvester walked on it. Schmidt from Gram-Schmidt, he worked on it, Hermann Weyl worked on it. So from 1873 to 1912, this went on. And this article says that this approximation theorem that I mentioned, the Eckart-Young theorem, is really due to Schmidt from Gram-Schmidt fame. And he was interested not so much in matrices, but he was studying integral equations, where you have both symmetric and asymmetric kernels, non-symmetric kernels. And he wrote down this approximation theorem. So really, Stuart claims that that this theorem should actually be attributed to Schmidt. And then in 1936, Eckart and Young actually wrote down the SVD for general rectangular matrices. So that is in that paper, for sure. And they seem to have rediscovered this approximation theorem. So, that is my understanding of the history of how it is, but I did not know this till two days ago. And I'm not really a bonafide historian in any way. So, but this is what I've understood. It's an interesting story.

EL: Yeah, that sounds like an interesting article. I mean, I guess in the history of math, there are an uncountable number of places where unravelling back to where the idea first appeared is more complicated than you think.

RT: Right.

KK: And also, this sort of gets at your interests more generally, how you like things to sort of come from an actual application. Well, if this really came from integral equations, right, that's really an application.

RT: Absolutely.

KK: So it's working on lots of levels for you.

RT: That’s right. So he, of course, did this in infinite-dimensional vector spaces. And approximation will allow you to approximate an operator as opposed to a matrix, right? And apparently, that really elevated this whole theory from just a theoretical tool to something that's actually widely used. It became much more of a practical tool. And I guess, in modern day, the SVD, and versions of the SVD in exists in all kinds of mathematical sciences. So in signal processing, and fluid dynamics, all sorts of places. So it's, in some sense, one of our biggest exports from the math world, and yet we don't quite teach it normally, to math people. So yeah.

KK: Right. So was this like a love at first sight thereom? Was this the sort of thing that came up a lot in your work, and that's why you're now so enamored?

RT: So I did learn of at first in the process of writing a paper. I did not know about this theorem before, maybe about 10 years ago. But I think this theorem sort of perfectly fits me, which is why I love this theorem. So for different reasons, right? So first of all, it's an optimization problem, it's about minimizing distance from a matrix to the set of rank-whatever matrices. So it's an optimization problem. The space that you're trying to minimize to, which is the space of rank at most k matrices, that is an algebraic variety. So it can be written as the set of solutions to polynomial equations. So there's the applied algebraic geometry side, or at least the algebraic geometry side. And it's not a very simple variety. It's actually a complicated variety. So it's an interesting one. And lastly, this problem is sort of a prototypical problem in many, many applications. So a lot of statistical estimation problems are of this flavor. You have a model, which, let's say is our rank-K variety, so the rank being some measure of complexity. And then you have an observation that you have in the field with instruments, and it tends to be noisy, so it's not on the model. So that's your observed matrix. And now you're trying to find the maximum likelihood estimate, or the closest true object, that fits the observation. So this is a very standard problem that comes up in many, many applications. So in some sense, I feel it really lives at this intersection of optimization, algebraic geometry and applications, which is sort of what I do.

KK: That’s you.

RT: Yeah. So that's one reason that I think this theorem is so cool. And the other thing is, I think it's a very rare instance of an optimization problem where the object, the observed matrix, knows the answer in its DNA. It doesn't need to know the the landscape that it's trying to get to, which is your space of matrices of rank at most K. it doesn't need to know anything about that variety. Just inside its own DNA, it has this SVD. And from the SVD, it knows exactly where to go. So this is completely unusual in an optimization problem, right? Like even if you're minimizing, say, a univariate function over the interval [0,1], I really need to know the interval [0,1], to figure out what the minimum value is. The function doesn't know it. But this is, I think, kind of a gem in that sense. You don't need to know the constraint set. And then lastly, it appears all over the place in applications. So in things like image compression, you know, the Netflix problem is sort of a version of this, distance realization, you know, things that would come up in areas like molecular modeling, or protein folding, and so on. It's all — many of these problems can be thought of as low rank approximation to a given matrix.

KK: Very cool. Yeah, I'm now I'm thinking about that variety. It's not a Grassmannian. But it's sort of like, is it stratified by Grassmannians? Let's not go down this path.

RT: It’s just — yeah, it's a set of solutions to the equations that you get by setting all the rank, whatever minus to zero.

KK: Right.

RT: So the matrix cannot be rank more than K. So you set all the K+1 by K+1-minus to zero.

KK: That’s complicated.

RT: Yeah, yeah. It's a complicated variety with singularities and so on. Right.

KK: So another thing we like to do on this podcast is ask our guests to pair their theorem with something. So what pairs well with this theorem?

RT: So I thought about this. This, to me was a very interesting challenge that you posed. So one thing that I always think about when I teach this theorem or teach the SVD in general, is there’s sort of a layer cake analogy. Okay, so I have always drawn this layer cake picture in my class. But when I started thinking about what I should tell you on the podcast, I thought, “Okay, it's not quite a layer cake, like you would buy in a store.” But it's sort of like a layer cake. So there is a layer cake analogy going on here. And that is simply we can think of a matrix. So if you have, say, an M by N matrix, let's start by just thinking of it as a rectangular chessboard lying on the floor. And then every entry of the matrix is creating, let's say, a building on each square. So you have, you know, the buildings have different heights, depending on the entry there. And then in that sense, what we're doing is, if you think of what a rank-one matrix is in picture that I'm trying to draw, then what it is, would be this sort of city block with skyscrapers that we've constructed, where when you look from East Avenue or West Avenue, you see one skyline. And then you see, like, various up and down versions of the same skyline as you look across. And then similarly, if you stand on North Avenue or South Avenue, you see one skyline. And then you see these up and down versions of that skyline as you go in the other direction. So that's a rank-one matrix. And then the whole matrix is built of these puzzle pieces, if you like. They’re all rank-one matrices. And what we're doing is sort of, take different puzzle pieces, you know, the first puzzle piece captures the most amount of energy in the matrix, then the first and second, the next amount, and so on. So that's sort of one geometric thing. And so thinking of a pairing, that’s just a geometric thing, not not exactly a pairing. But in my mind, another way to think about the whole thing is you could think of your matrices as, say, living in a universe. We have this M by N, universe. And then each of these landscapes, these, you know, matrices of rank, at most K, they form landscapes inside this universe. They're nested one inside the other. And you could almost think of your matrix as sort of a flying object. And if it needs to make an emergency landing on one of these landscapes, it knows exactly where to land. It doesn't need any, you know, radio control from the ground, right? There's no air traffic control on the ground who needs to tell this matrix where to land. So that's sort of my geometric vision of what is happening. I love geometry. So I always try to make pictures like this. But the closest physical phenomenon that I was thinking that maybe we could match with this is with the way migratory birds work, right? Like these migratory birds, they have sort of an inherent genetic compass in their head that tells you where they should land. So Florida being one of the biggest places.

KK: Yes.

RT: And that's exactly before they fly over the Gulf, right, where there's a long stretch of water so they know exactly where the end of the land is, or where the beginning of the land is when they fly in the other direction. So I think that's some amount of this sort of DNA information that's in their head. So there’s either genetic information — of course, they also use celestial signals like the sun and the stars and so on. But yeah, so that that, that to me was the best pairing I could come up with, just thinking of matrices as having this inbuilt computer inside them.

EL: I love that!

KK: I do too. As an avid birdwatcher, I'm really into this pairing a lot. This is really nice. Yeah, and luckily for the matrices, it doesn't get messed up. You know, I mean, I get various — I subscribe to Audubon and things like that. And I just read an article recently about how light pollution is really a problem for migratory birds. Especially, you know, they fly over New York City. You think they're not there, but they are, and you can catch it on radar data and all of that. And it's really become a problem for them. And especially with climate change, they're getting messed up on all these things — not to get off into birdwatching, but this is a really excellent pairing. I love it.

EL: Yeah, well, I know Kevin is quite the birdwatcher. I have only recently been getting a little more into it. And so I will think about flying matrices the next time I go and look at some birds. I recently discovered a new birdwatching place not too far away from where we live. And there have been some great blue herons nesting there. They're probably leaving soon, but they were there for the spring. And so that was cool as a very beginning birdwatcher to suddenly have your first serious like, “I'm going to watch birds at this place” have these nesting great blue herons at them. It really raises the bar for subsequent birding outings.

KK: They’re impressive birds, too. I mean, I've seen them. We have lots of them here, of course, I mean, they're walking around campus sometimes. But yeah, they'll catch up like a big fish swallow it whole. And it's really pretty remarkable. So yeah. All right. Well, Rekha, so we always like to give our guests a chance to plug anything they're working on. Where can we find you online?

RT: Oh, so I have a basically just by webpage. I'm not a social media person at all.

KK: Good for you.

RT: That’s basically where you can find me.

KK: You're part of a wonderful department. I've visited there several times.

RT: Okay.

KK: Great department, great city. And we'll be sure to like the your homepage. Anyway, thanks for joining us. I learned a lot today. This has been great.

EL: Yeah. And can I just, I don't do a little bit of tooting our own horn was saying that I love this podcast because we get to do things like now every time I decompose these matrices I’m going to think about migratory birds. And, you know, it's just, like, building all these little connections. I love it. Thanks for joining us.

RT: Thank you.

[outro]

On this episode of the podcast, we were excited to talk to Rekha Thomas, a mathematician at the University of Washington, about the Eckart-Young-Mirsky theorem from linear algebra. Here are some links you might find interesting after you listen to the show:
Thomas's website
Our episode with Tai-Danae Bradley, whose favorite theorem is related to Thomas's
Stewart's article about the history of singular value decomposition

Episode 67 - Liz Munch

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Kevin Knutson, professor of mathematics at the University of Florida. I am joined today by your fabulous and glorious other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah. And I assume the “fabulous and glorious” means my new haircut, which I actually got professionally cut for the first time in quite a while recently.

KK: Good for you.

EL: I had been playing around with, you know, scissors — not official hair cutting scissors, just you know, the scissors that are lying in the kitchen drawer — and, like, my clippers and stuff for a while, but I went in and I look very sleek and chic right now.

KK: You do. You look very sleek. I’m getting a haircut tomorrow, I look less sleek.

EL: Not too bad, though.

KK: You know, I still have the plague beard though. I don't know, I can't decide what to do about that. And my hair has gotten longer, but you know, thinning, and I don't know what to do anymore.

EL: My husband actually decided to try for the ponytail. So he had been very excited about, like, two weeks after the second shot going and like getting a real haircut for the first time in a while. And then by that time, it was long enough that it's almost ponytailable. So he's like, “I've never had a ponytail. I think I want to try this.” So so now we're like at opposite, you know, going in opposite directions.

KK: I have a ponytail and my wedding pictures and then that’s the last time.

EL: Okay.

KK: Anyway, today we are pleased to welcome our good friend Liz Munch. Liz, you want introduce yourself?

Liz Munch: Hi. Yep, so I'm Liz Munch. So I am an assistant professor at Michigan State University. I'm in the departments of Computational Mathematics, Science, and Engineering, which is a mouthful, but we just say CMSE. And I'm also in the Department of Mathematics.

EL: All right, and what is your recent hair story?

LM: It’s the kind of, like, what can I do rolling out of the shower to like, make it do anything? And it's kind of like, wherever it lands. I did get a haircut, though, recently. It was very exciting. I had — it was getting so long that it was going out of control. And usually what happens is I I go get a haircut and I do like extreme versions. I'm like, “I hate my hair!” And then I chop it off to my ears. And then I grow it out again. And then “I hate my hair!” and I chop it off to my ears. This time, she convinced me not to chop it off all the way, but I'm due for one. So we're gonna work on getting that one soon.

KK: Looks good. Looks good. Yeah. So CMSE, that's interesting. You have this joint appointment. How's that? Yeah.

LM: I like it. It's different. So at least for me. So I do very interdisciplinary math, very applied math, but not your usual applied math. And so for me, it's a it's a really nice setting to be in. So CMSE is — so the faculty in CMSE are all sorts of different backgrounds. So there are mathematicians, but there are also statisticians and biostatisticians and biologists and plant biologists and geologists and physicists and engineers, and I'm sure I'm forgetting people. But it's basically rigged to be interdisciplinary. And so it's made a lot of fun, because you can find these interesting projects that are sort of in the intersection of complicated math and interesting stuff to do there with applied projects I would never have thought of before. So at least for me, that's been a really good fit for trying to do interesting applied math research explicitly outside of the usual academic silos.

KK: Yeah. Well, that's nice, because I think a lot of us in math departments often want to engage in these activities, but you know, the physics department’s in another building, and it's more effort, right? You know, it's true, there’s this idea of these collision spaces. So you know, if you have colleagues in all these different disciplines just right down the hall, it might lead to more interesting stuff. So that's a nice model. I wish we could do more of that. Anyway. So you have a favorite theorem, I hear.

LM: I do have a favorite theorem.

KK: You going to tell us what it is?

LM: I was going to talk about max flow min cut.

KK: Okay.

LM: Which is sort of my, I don't know — so the reason I like max flow min cut, is partially because, again, I do interdisciplinary applied mathematics, and I tend to fall into the theoretical computer science land a lot, just based on a lot of things we want to do. Because I work in TDA, topological data analysis research, and so a lot of things come down to here's a complicated thing I want to compute, and I’ve got to go figure out how to do it in a computer in a reasonable amount of time. So this is an example of that.

So max flow min cut. So here's the game. So you get yourself a directed graph. And this graph has a source and a sink. So you've got your S vertex source and your T vertex sink. And you essentially have capacities on each of the edges. So my directed edges, I now have an amount of stuff I can push across the edge. So I like to think about this with, like, water tube flows kind of thing, right? So I've got how much capacity each of the edges can take. And so the game is, try to see how much stuff you can push along from the source to the sink, basically playing nice with those capacities, so flowing from one side to the other. So if you're at any vertex, all of your inward flow amounts should be equal to all of your outward flow amounts, because you can't have anything hanging out at the vertex. And so your other restriction is your flow values — this is essentially like a second choice of weighting on all your edges — your flow values have to be less than your capacity values, right? So you can't flow more across an edge than the capacity allows.

KK: Sure.

LM: And so the trick is, what's the most amount of flow you can get across? I hand you one of these graph setup problems, what's the most flow you can get across these things? So the max flow min cut, as the title would suggest, is that you also need to know something about the minimum cut. So what's what's a minimum cut? So same starting input information. What you can do is you can try to divide your collection of vertices into two piles. And the rule is that you have to have your source in your sink in different piles. And the cost of whatever choice of piles has to do with, essentially the cost of the capacities that go from one pile to the other, right? So you sort of add up those values, there's some negatives if things are backwards, etc. But that's the cost of a cut.

KK: Okay.

LM: And so the game with that part of the problem is you want to make that as small as possible, right? Can I reorganize these vertices in such a way to make that cut cost low? And so the max flow min cut theorem says the capacity of the highest possible flow value is equal to the cut, the minimum possible cut value you could get. And so this is cool, because this means that you could answer your problem either by going and hunting for a maximum flow value or by hunting for a minimum cut value. And so it gives you two very different ways of looking at this problem to try to solve something. And these show up in all sorts of different application settings, right? So you can imagine, like, railway networks, where you're trying to move stuff around, you could try to do things with electric grids, and you can do things with water flow. There are lots of places where this thing would show up. And so having access to two different ways of looking at the problem is super useful.

EL: Yeah. So I have never thought about this before. So I'll just kind of — I’m not even sure if I can formulate the right question. But I'm trying to think, my mind immediately goes to extremes. So like, if you put everything except your source in one pile, or everything except your sink in one pile, is it obvious that usually that's not going to be good?

LM: I guess it kind of depends on the sort of setup you have, right? So I guess the simplest version would be like, okay, let's start with a graph that's just a path, right? And so in that case, I can stick a bunch of random numbers as capacities on my path, and so the maximum flow I can get across this path is going to be something like the minimum weight capacity that I've got, right?

KK: Yes.

LM: That minimum weight capacity is also going to be if I chop my path graph in half at that particular edge, that's as low as my cost for my path can go.

EL: Sorry, just to slow down a little bit for me. So that capacity is like, this one edge can only hold, you know, whatever amount of whatever stuff?

LM: Yeah.

EL: Okay. Great.

LM: This is kind of like the bottleneck in the system, right? Like I could have pushed, like, 30 gallons back here. But if this particular tube only allows for five gallons, it doesn't matter how much I can push in the backend, right? It's got to be, it's limited by this one, right?

EL: Yeah.

LM: And so then you basically take that, that path problem and scale it up. So then you've got all these other possible paths where you could push stuff through. You could imagine a vertex where you have, like, a capacity of 10 coming in, and maybe five edges coming out. And so your flow could be 10 in and then it got split up and there's all sorts of things. And so the way you prove this, so this is the Ford–Fulkerson algorithm from the 1950s, I think? It basically comes from looking for paths like that. So the whole problem comes down to basically that path example I was giving you. So you essentially rig up this modified version of your graph, and then look for paths from your start to your sink, where you try to see places where I could push some amount of flow across that line, across that path. And what you do is you essentially update some sort of flow that you're trying to build as big as possible. So if I've got a path where the bottleneck value was five, I now push value of five across each of the edges in that path. And then I update this other graph I can use to keep track of things. This is called a residual graph. Basically, then I update this flow, and then I go look for a new path. And then I update this flow, and then I look for a new path. And it all comes down to just finding these paths that have some sort of bottleneck in them and updating accordingly.

KK: That seems slow.

LM: Oh, yeah. Oh, yeah. This is not nice.

KK: So there must be better algorithms for doing this. But that’s the easiest to explain algorithm?

LM: Yeah, right, easiest to explain, and the one you get the proof off of. So that's the the Ford–Fulkerson one. And of course, I did not check this beforehand, but I believe the running time has something to do with the value of your maximum flow. And there are some caveats in here. So if you have integer-valued flows, so I'm only allowed to put integers on each of my edges, that process will terminate, which is a good sign, right? It will terminate at the best possible flow, which is a good thing, right? And worst-case scenario, you basically updated by one every time, and so the the order of the number of times you have to do that update procedure has to do with essentially the maximum flow value.

KK: Okay.

LM: So yeah, it's not pretty, but it works. This is another one of these examples where in practice, it tends to not be that bad. But of course, you can construct the nasty examples.

KK: Sure. Right. So yeah, the theory is bad, but the practice is good. So that's fine.

LM: Yeah.

KK: So do you use this much? Does it come up for you?

LM: I haven’t in a long time. The reason I was thinking of this was a while ago, I was teaching this summer program for bright high school kids, where we were doing a topology class for high school kids that basically have seen calculus. And so one of the reasons I like this was I was remembering I had rigged up this thing where I created a graph on the ground. And I put capacities on it involved in the — sort of like, I don't know, I think it was squares of paper or something like that — and messing around with getting them to actually manually push stuff around on this graph and start thinking about flows and how you can update information. Yeah, it hasn't come up in my research recently. But I think in general, just the messing around with problems where you can have multiple ways to find a solution, that's been really interesting lately, because trying to rig, trying to look at a problem from a different angle where you might have an easier ability to go solve something than something else, right? So like, if you told me go find the minimum cut in this particular graph, the knee jerk reaction is like, “Okay, let's test all possible ways of splitting the two piles!” which you really don't want to do, right? Bad idea. Never do that, right? And I don't have any good intuition for how — if you had to be that problem, I would start going at any sort of algorithm that would get me anywhere. But because we've got this duality idea, right, I can go back and do this pushing flows through a network thing that gives you access to a way to solve a problem that wasn't there before.

EL: Nice. And so do you remember, like really loving this theorem when you first saw it? Or is it something where your appreciation has grown over time?

LM: Ah, there's probably — now I;ve got to remember where I saw it. I'm going to guess I saw it in an algorithms class in my PhD. Yeah, I don't know. I don't know that I appreciated it at the time. I feel like a lot of the beginning of my math career was like, “Okay, this theorem exists, but I'm not sure why I care.” Right? It's taken me much longer to start getting to the like, “Oh, the point is the duality. The point is accessing a problem from different sorts of viewpoints.” And having the ability to sort of question the vantage that you're looking at something from because there might be a better way of doing something. And “better” here is now a subjective term, right? Like, mathematically, the theorem says, I don't care. But practically, right, there is a better and a worse way of doing something. And so you're going to end up having choices, as you go about doing math research, that are going to be better or worse in a practical sense,

KK: So it's clear listening to you that you work in an interdisciplinary environment, right? Because, you know, the mathematician loves the duality, but you know, practically, the engineer wants to find the most efficient way to do it. Yeah, very cool.

EL: So I wanted to come back to something you said a little while ago, and I think it just percolated through my brain. So you said that, like, if your your capacities on these are all integers, then the process terminates. And the fact that you needed that “if” actually just made me really nervous. So is it the case that if you don't have integers, and possibly even rational numbers or something like this? So like, maybe if you had like a bunch of irrational or transcendental values on these, that it might not terminate? And then you’d be sad.

LM: Yep, yep, yep. Yeah, then you'd be sad.

EL: Or maybe you would be sad. I don't know.

LM: Well, yeah. Okay. So I actually, I'm not sure of the answer if it's rational. I know that if you allow irrational values on the edges, you can definitely have examples that don't terminate. And basically what ends up happening — okay, so the way it is pushing the flow through your residual graph, there's a setup that allows you to essentially undo pushing through edges. So these residual graphs get this sort of backwards edge that's added to them. There’s the flow that you've already pushed through that you could undo and the flow that's available, which is the capacity minus the flow that you've done to this point. And so finding a path in this graph could push new stuff through, or essentially undo pushing things through and move it to some other path in your graph. So what ends up happening if you — again, if you create incredibly contrived examples — is that finding these paths in this graph can keep updating and then undoing the update. And you end up with this vicious cycle where you kind of like spin stuff around in circles, and you just never get there. And it's got all sorts of weird behavior, like not only it, like, doesn't decrease, it doesn't convert, like, it's, it's all kinds of nasty.

KK: Right? I would imagine with rational, though, you can do the standard trick of just scaling everything to be an integer. Like, that's probably okay.

LM: Oh, yeah. Because especially if you've got a finite starting graph, yeah. Yeah. Right.

EL: Oh, “if you've got a finite…” you're putting all these “ifs” that I wouldn’t even think you’d need!

LM: Let's have everybody be finite before I say something false.

EL: That’s fine with me.

LM: I am going to bet somebody thought about that.

EL: Oh, I'm sure. But, you know, since you started saying this with water pipes, I just immediately thought of, like, okay, a city water system. And I just know that that is finite. So yeah, in my mind, this is always going to be finite.

KK Right. And you'd sort of create one source by having it all come in your case, Evelyn from the Wasatch, right?

EL: Nice job.

KK: And then it all goes to the water treatment plant. That's it. That's your whole, you’ve got this crazy network.

EL: Or that eventually to the the Great — the Lessening — Salt Lake as it dries out.

KK: Yeah. All right. So also on this podcast is, we ask our guests to pair their theorem with something. And you mentioned what it was going to be ahead of time and I'm really intrigued. So let's look. What pairs well with max flow, min cut?

LM: Okay, so we're gonna pair this with the cross-strung harp, which I promise is going to make sense. So give me a moment. So just for some backstory, so my undergraduate, I was actually a harp performance major. That was my previous life. And I sort of backdoored my way into the math department.

EL: So I have to interrupt say that my sister is a harpist, was a harp performance major, is now working on a PhD in music theory, but teaches harp and has a harp studio and yeah, that’s such a cool coincidence.

LM: You probably might already know some of the stuff I'm about to say. So that's super exciting. Because, yeah, most people you talk to have no idea what I'm talking about. This is great. Yeah. So I spent the first part of my professional life, I was going to be a professional harpist. I got about halfway through and was like, “No, that's not for me.” I got into, started taking math classes. Ended up in a math PhD program, unbeknownst to me. And here we all are, right? Yeah, so the thing with harps is if you look at a regular harp, what you're actually looking at are, like, the white keys on the piano. So no chromatics, no sharps and flats. It's basically in one key, right? So there was an issue in, sometime around the 1800s, where they were trying to figure out how to make a chromatic scale possible on a harp. So prior to that, there was sort of a trick where you could put things like levers on each of the strings and they would shorten each string by a little bit. And so you could basically get, if you tuned your string in natural, you could get the sharp. If you tuned your string flat, you could get the natural, but that was it. So you could do the key of B — the key of F and the key of D, and then things got hard. And so in the 1800s, they were trying to figure out how to deal with this problem, and so there were two solutions that showed up at about the same time. So there was the cross-strung harp, which is not what you've seen before, and the double-strung harp which is what you've probably seen before. So if you go to an orchestra concert, somebody most likely has the cross-strong harp, or sorry, a double action harp, which is, you have two sets of essentially these twisty peg things that go on top of the string. And so now you engage a pedal, and the pedal, if you push it down once it engages one of these twisty bits, and if you engage it a second time, you get two of them. And so now you have three notes on every string. And so you tune it in such a way that now you've got flat, natural and sharp on every string.

EL: And there was much rejoicing.

LM: And there was much rejoicing, right. So this was invented by a watchmaker who was trying to solve this problem, because the mechanics of this are absolutely nuts. Like, I don't even know how the inside of my harp works. At the same time, so there was another option that was starting to show up, which was the the cross-strung harp, which was literally — you're going to have to Google this later — but it looks like two harps with two sets of strings that meet at an X in the middle sort of interlaced.

EL: Whoa!

LM: Yeah. I've seen these things. My brain goes fuzzy. Like, you can't focus on any of the strings, right? But the idea is that now you can move your fingers up or down to get sharps and flats. And so now you have access to all the strings. And it involves different techniques and things like that. And so there were these two companies. So it was Erard at the Erard company. So Erard was the watchmaker that invented the double action harp. And the Pleyel company had invented the cross-strung harp, and they were fighting about who was basically going to win the business.

KK: Harp domination.

LM: Yeah. There's some there's some catty stuff in the harp world, let me tell you. And so each of the companies, so there are actually two very famous pieces for harp that were commissioned by each company to prove that their harp was better. And so the Pleyel company, commissioned the Debussy dances, and the Erard company commissioned the Ravel Introduction and Allegro. And these two pieces are basically made to be — I'm not going to say easy, but like, manageable on your harp and not good on the other one, right? Right, so fast-forward 100 years, and the double-action clearly won out, right? So that's what everybody has now. But we still play both pieces. So if you go to college for harp performance, you're going to play both. And so the Ravel isn't, you know, again, it's not easy, but it is manageable, because you are playing it on the instrument it was written for. And then you get to the dances. And it is a combinatorial nightmare, because basically, what you're trying to do is figure out these fingerings, and how to play these different chromatic notes at the same time, when you don't have access to the other set of strings. You are now limited to trying to play the note on another string that you might have needed a half a second ago to do whatever.

So I promised that this was going to come back to something that made sense. So the whole point of this is that I was thinking of the cross-strung harp because it's one of these two solutions for the same problem. And not necessarily that one of them is better than the other, but in certain cases, there is definitely a priority. One of them really, truly did better.

EL: Yeah. That's so cool. And I have to ask one more question about the cross-strung harp. I don't want to totally derail this, but I'm going to slightly. So you know, when I watch my sister play like her right hand, you know, as on one side of the strings that are left hand is on the other. So with the cross-strung harp, then, your right hand and left hand — if you were playing, like on the diatonic, would have one would have to be up and one would have to be down, right? Because the strings go…

LM: Yeah. I will admit I've never played one.

KK: You’d switch them, right?

EL: Yeah, like if you move to flats, you move like this.

LM: You kind of want to be able to do both at once. So your technique would change entirely. I don't actually know how you would deal with this. If you want to Google even more things, there’s also a thing called a Welsh triple, and the triple — so what this is is now, this will make your brain hurt if you try to look at this thing — it’s now three rows of perpendicular vertical strings. So instead of being crossed through the middle, they’re, like, up, straight up. And so now you've got white notes on the outside, that are spaced wider than a normal harp, with sharps and flats down the middle. And so now you have to reach through the strings to try to grab.

EL: That seems bad.

LM: Oh, and I’ve played one of those too. And it's like your eyes just can't focus. Like I can't decide which layer of strings to be looking at. It was it was rough,

KK: But I'm sure that's mostly a function of having grown up playing the one that you play.

LM: That’s true.

KK: If you grew up playing the cross-strung or the West triple, that would just feel absolutely, totally natural.

EL: Yeah, but I guess you'd have to do something for visual markers, like so the harp has, is it the F and C strings are a different color to help you? Because on the piano, you can look at where the black keys are, and that tells you what notes you're playing. On the harp if they're all the same color, it's just a nightmare. And I guess on the Welsh triple, maybe you'd have to…

LM: I don’t remember what the coloring was on the Welsh triple. Yeah, on a modern harp, C’s are red and F’s are black. And then if you get older strung harps, they used to be switched, and that also makes my brain hurt.

EL: Oh no!

LM: It’s very much what you're used to.

EL: Yeah. Well, that's so cool. And what a funny coincidence that you’re harpist, just like my wonderful sister.

LM: Hi to your sister.

EL: Hi, Rachel!

KK: Do you still play much?

LM: Oh, not much anymore, unfortunately. I've moved my harp around to multiple cities and houses. Yeah, so I basically, I kind of put myself through grad school playing weddings. So that was nice. And then after, you know, kids and everything else, it's just kind of turned into less so.

KK: Well, you’ll come back.

LM: It might come back.

KK: Yeah. As someone who's now an empty-nester, I can tell you it does happen. So the other thing is, you know, my kid’s about to go off to grad school for music composition. So apparently just like Evelyn’s sister, he’s going to go be a theoretical musician, I guess. He's a percussionist, though, you know, and I can't understand how he plays drumset right. That’s, like, two arms and two legs, doing four different things. But harp sounds pretty bad too.

LM: Yeah, I thought I had to move around a bunch of stuff. They are so much worse. They got, yeah, that was that was always the trick. So when I was in college, the harpists saw got our own practice rooms, so we didn't have to move the harps around. Everybody else had to, like, fight for it. But the the percussionists also got their own practice rooms where they got, like, nobody wants to move around, you know, the xylophone.

KK: Yeah, although I've certainly moved Gus’s drum kit plenty of times.

LM: Oh, yeah. No, if you've gotten to the point that you've bought vehicles based on your instrument…

EL: Yep. I was going to say that's Rachel’s, I think Rachel has taken a harp to a car place to, like, test in the lot to see how easy it is to load.

LM: I have done that. I've purchased vehicles entirely because it could fit my harp in the back and I've confused — oh my goodness, I love showing up to car dealerships like this. It's my new favorite thing.

EL: Yeah. So before we end this episode, I just had — I didn't get a chance to say this earlier. But Max and Min can both be names, and I just feel like max flow and min cut just sound really snappy and sound like either a superhero duo or maybe, like, superhero antagonists or something. I’ve just got to put this idea out there for like, you know, a graphic novel about Max Flo and Min Cut.

LM: I want to read that book. Absolutely.

KK: Or Max Flow sounds like a news reporter from the ‘20s. You know, like His Girl Friday, right? You know, Cary Grant and Rosalind Russell or Max Flow and Min Cut in there. Yeah. So we also give our guests a chance to plug anything, or where can we find you on the line or …

LM: Oh, I definitely should have thought of that in advance. I don't know. Come see MSU. I like her. I like our new department. That's probably the first thing I should plug. I'm on Twitter more than I really should be. So you can always find me there. [Her handle: @elizabethmunch] That's — Yeah. I love Math Twitter. It's a happy place.

KK: Mostly.

LM: Yeah, I think that's mostly it. The other thing I guess I should plug is, so I'm very involved in the women in computational topology network. So if anybody is interested in anything in that general direction, so this is a group for not just women. So we're aiming for a broader community of gender minorities and a place for people to do math in a supportive environment because that, in my opinion, is the reason I am still here, and so I want to create that for other people.

EL: Cool, yeah.

KK: I can verify Liz is well-known for her no assholes rule.

LM: I yes — oh good, we can swear on this podcast.

KK: Sure. Why not?

LM: You just did. I do have a tendency to curse like a sailor, and so my husband before I started this was like, “Remember, Liz, you need to not swear on that.” I was like, “Okay.” I have a very strict “does not collaborate with assholes” rule. And it is gotten me very far in life and made for much better research.

KK: Life’s too short not to have that.

LM: Exactly. Yeah, exactly.

EL: So now are we going to go back and rerecord, like, the swear version of max flow min cut? Just kidding.

LM: Now that I'm allowed to curse for the whole thing?

KK: It’s pretty light swearing.

LM: You’ve got your R rating, don’t worry.

KK: No, no, no, R rating requires f bombs.

LM: See? We’re doing okay.

KK: So PG— so movie ratings for PG-13, you’re allowed one f-bomb. If there's more than one it gets an automatic R.

LM: Good to know. So we’ll hold off on the f-bombs, just in case this turns into a movie.

KK: That’s why you don't hear it as much in PG-13 movies because they they deliberately leave them out to get a PG-13 rating so the teenagers can come. But you can use any other swear you want. Anything. But just one f-bomb.

EL: All right. Well, with that important knowledge…

LM: We’ve gone on so many tangents at this point. I probably got math in here somewhere, right? Somebody did math.

EL: Yes. Our math, hair, harp, and movie ratings podcast can now conclude.

KK: That’s right.

EL: Great to have you, Liz.

LM: Thank you. Thank you both for having me. I really appreciate it.

KK: Take care.

[outro]

On this episode, we were happy to talk with Liz Munch, an applied mathematician at Michigan State University, about the max flow, min cut theorem. Here are some links you might enjoy after you listen to the episode.
Munch's website and Twitter account
The Women in Computational Topology Network
The Max-flow Min-cut theorem at Brilliant.org
The Ford-Fulkerson algorithm on Wikipedia
The cross-strung harp on Wikipedia
Harp.com's history of the harp

Episode 66 - Érika Roldán

Evelyn Lamb: Hello and welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb. I am a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. How are you, Evelyn?

EL: I’m all right. It's been raining a little here, which is very good, because we are in a perhaps somewhat historic drought and every bit of moisture we can get is fantastic. Probably not very close to your experience in Florida right now.

KK: It’s been pretty dry. But yeah, it's not really an issue for us. I mean, it’s actually really lovely right now, and I'm looking forward to kayaking some this week.

EL: Oh, fun.

KK: And my son graduates college in two weeks. And yeah, all kinds of fun stuff on the horizon for us. So anyway, let's talk math, though.

EL: That is exciting. Yes. We are very happy today to have Érika Roldan joining us. So yes, Érika, would you like to introduce yourself?

Érika Roldán: Yeah, thank you. Thank you very much for the invitation. I'm super happy to be here. And, well, I'm a postdoc right now. I finished my PhD thesis in 2018. And then I started jumping here and there, from Mexico to the states and now Europe. I’m at Munich, Technical University of Munich, and École polytechnique fédérale de Lausanne in Switzerland is my co-host. And yeah, I have this fellowship, the Marie Curie fellowship, for 11 more months, and then jumping again. Yes.

EL: Yeah. Well, that's very exciting. And what is your field of research?

ÉR: Well, it's stochastic topology and topological and geometric data analysis. I think most of the time, most of the brain time, goes to that. But also, there is something that is related because it gives extremal examples — you will never see them typically when you're using kind of random processes — but these extremal examples allow is to contrast with random ones, so I also do extremal topological combinatorics a bit.

EL: Okay, and I also am familiar with some work that you've done in recreational mathematics, which I guess might have to do with this extremal combinatorics too. And so if I can self-promote and Érika-promote a little bit, a couple of the puzzles that Érika has worked on appeared in the mathematics-themed calendar that I put together a couple years ago, which is still available for purchase through the bookstore of the American Mathematical Society and which is not specific to a year, so you can still enjoy this calendar whatever year it is when you are listening to this episode. So anyway, yeah, you did a couple of fun puzzles. I don't know if you want to talk about any of those. I am actually blanking right now on there's one with like, polyomino things, right?

ÉR: Yes. So there are two. One of them actually has a very special place in my heart because it was my first paper, and I wrote it for the Gathering for Gardner, this meeting that is every two years in Atlanta. It is a wonderful meeting. It is the first mathematical community, research community, that I got in contact with. And yeah, I did a complete analysis and characterization of a type of puzzles with colored cubes, and you have to stack them, and you have to do a tower and have some interesting coloring properties. And the most famous one is called Instant Insanity, just to give you the name. The name has a little bit of a clue of how interesting it is to try to solve them by trial and error. And yeah, I guess I did some computations and everything to characterize all possible different kinds of puzzles like this. And, yes, that's one of the of the entries of the calendar, this this wonderful calendar. Thank you for sending it to me. I enjoy it very much. And yes, and the other one was about maximally many holes with polynomials.

EL: Yes, that’s right.

ÉR: And for sure, we're going to go back to polynomials. Because it's related with the story that I want to talk about today about my favorite theorem.

EL: Yes. And you've provided the perfect segue now. What is your favorite theorem?

ÉR: Yes. Well, first of all, I want to say that I was thinking, and I changed my mind different times during the past two weeks. And I decided that I wanted to talk today about my favorite theorem, choosing it in particular for my mother to be able to follow the podcast. So today's the 10th of May. And, yes, because of the corona crisis, I think a lot of people have not been able to travel, and I haven't seen my mother for more than, almost a year and a half. And this is a way of sending her all my love and appreciation. So my favorite theorem, okay, so one of the things that my mom used to do, or does still very, very well, is shuffling cards. So every time that we gather together with my grandma, and in Mexico, this is very common that you gather with your family very often. It doesn't matter what, you always want to be in a huge crowd, just eating and playing. And yeah, she's a very good shuffler of cards. And my favorite theorem is a theorem by Perci Diaconis, the theorem for today is by Persi Diaconis, where he proves, and actually there is like a set of different papers that have different ways of modeling this shuffling, the usual shuffling that we know, and they prove how many shuffles you need to do to be a sampling from a fairly uniform distribution from all possible ways of having 52 cards of a deck in a certain order.

KK: This is a very famous result, and it's a surprising number, you're gonna tell us the number, right?

ÉR: Actually, it depends. It depends on the distribution that we use. And then this depends on the algorithm. And but yes, it's a very famous theorem. And it's very well-appreciated by by mathemagicians. Persi Diaconis is one of the most well-known mathemagicians.

KK: I have to say, I'm not surprised this is your favorite theorems. I've been at various TDA [topological data analysis] conferences with you. And topologists like to play fast and loose with distributions, and you are always quizzing us about which distribution we're trying to use. It's always “Wait a minute, wait a minute, what's the distribution?”

EL: So,

KK: Go ahead, Evelyn.

EL: Yeah. Um, so for someone who has not thought about card shuffling a whole lot, what do you mean by different distributions?

ÉR: So let me withdraw the word “distributions” for the first part.

EL: Okay.

ÉR: So you want to play poker or any other game, and you want to shuffle the cards in such a way that you feel comfortable betting $100. So, I have the deck of cards. And let's say that I decide to shuffle like this: I take the uppermost card, I take it, and then I put it in one of the possible positions that is not on top, or could be on top actually. And I choose where to put it uniformly randomly. That means I select one of the possible 51 spots, 52 spots, and I put it there. And let's say that I do this five times. Question for both of you: Will you bet $100 with me in this poker game?

KK: Uh, no.

EL: Probably not.

KK: It doesn’t feel very well-shuffled.

ÉR: Exactly. And I think no one will do it. Like, it doesn't matter if you know or not probability. We have a sense of when things are shuffled, even if we haven't heard the word “probability.” And so, let me just tell you that actually, for doing this, if I do this 10 minutes, you will start getting more and more comfortable, right?

EL: Yeah.

ÉR: At some point, you can say, “Okay, let's play.” That's enough after an hour, right? And the solution, or roughly the number from starting from which you feel comfortable and theoretically can be proven that is the kind of right stopping time, is around 200 shuffles.

EL: So this is not a very efficient way to shuffle, in case that wasn't already clear to everyone. One card at a time. Not so good.

ÉR: Yeah, well, in a Mexican family it will be okay, because everyone will be talking everything. No one will get, just, bored. But perhaps if you are there just for playing, it's not the right way to go. Now, in general, I want to say here, the right rate. In general, if you have n cards, with this shuffling you have to wait nlogn time. So n times logn is the rate that with this kind of shuffle will give you a uniform distribution as a convergence, and when you start to feeling that yeah, it is uniformly sampled. Now let's compare these to the actual way that my mom shuffles the cards. And this is commonly referred to as a riffle shuffle. So you take the deck of cards, and then you kind of try to cut it in into two packs, half perhaps. And then what you do is you hold each one of the packets with each one of your hands. And then you make — I’m trying to generate a mental image that everyone knows — and then you start with your hands, like, trying to put one after the other one, alternating from the packets from one and the other one. And there are different things to do here to convince you that this is for sure a different way of shuffling than the first one. That is one thing, and the other thing is, so if I do it one time, perhaps you will not be very happy. But whenever we're playing with friends usually is what? Five times, six times.

EL: Yeah.

KK: It depends on how many times I drop them. You know, when you do it, you do it the down and then you do that little flip where you reverse the cards to get that little “shhhh” sound. It took me for—I was an adult before I could finally ever do that. But then a few cards always go flying, right?

ÉR: And this is one of the things that I love seeing my mom doing because I think she has some secrets that she doesn't tell us because she does it so well. Like amazingly well.

KK: My mom was good at that too, actually. That's interesting. Okay.

ÉR: We have these two different ways of shuffling. And we we kind of feel that, or we're comfortable with saying a riffle shuffle shuffles faster than the other way. And so now I want to use to use this shuffling perspective to go and shuffle other mathematical structures. Because what happens is that I ended up using this way of shuffling objects, or mathematical entities, in my research. And that was a very pleasant outcome of my PhD thesis. Yeah.

EL: So you're saying that like with these polyominoes with holes and stuff, there's some way to think about shuffling polyominoes, or do you shuffle entire configurations of them?

ÉR: Yes, exactly. Yes. It's super fun. So okay, let me start with Tetris.

EL: Okay.

ÉR: I think if we start with Tetris, everyone who's listening to us will go directly to the picture, the image of what is a polyomino.

EL: Oh, that's right. Yeah, we didn't actually say what a polyomino is. So in Tetris, you know that each picture is made up of four blocks. And so that's a tetromino is. So it's like a polyomino, I guess, is something that's made up of any number of these blocks.

ÉR: Yes, yes. Exactly. So. I think my, my favorite definition of a polyomino was the first mathematical definition given by Solomon Golomb when he was a PhD student. He gave a talk, and he said, well, a polyomino is a rook-wise, connected subset of squares of the infinite checkerboard.

EL: Okay.

KK: Rook-wise.

ÉR: So you can imagine you select a finite number of squares, and then you place a rook in any one of the squares, and you have to be able, with only rows and column moves, to go and visit any other square of the structure. So rook-wise connected.

EL: Yup.

ÉR: Yeah. And with Tetris, what happens is that we — let's now come back to to the mindset of sampling. And when we're playing Tetris, there is some entity that is giving us one polyomino after another one. Question: I don't know if people have thought about this. But it's a fun thing to do. I did one morning, and I was like, “I don't know.” Yes. So how is this done? How does the game decide how to give you the next polyomino?

KK: It’s not just random?

EL: Yeah, I guess I assumed it was — there was a however many Tetris pieces there are. I actually can't think off the top of my head, but six or seven, maybe different pieces?

ÉR: Seven.

EL: Okay, then just so a 1/7 chance of each one, is that not the case?

ÉR: Well, you soon will be able to play the game that you're describing, because I'm programming a version with uniform distribution, just to make people really mad and upset.

EL: Okay.

KK: I can see, actually, now that I think about it, if it were just random, uniform, that would be bad. You’d lose pretty quickly, right? I mean, now that I think about it, often you'll get that long, the straight four in a row right when you need it, right? And and if it were uniform, you wouldn't necessarily expect that to happen.

ÉR: It could, you could have bad luck. Yeah. You could have had as long as you want bad luck.

KK: Right? Sure. That's right.

ÉR: Right. So the algorithm is very interesting. What they do is, they take the seven, the possible seven polyominoes with four squares that they give, and then they shuffle these seven in a way that you have a one over seven factorial, each one of the possibilities of the order has one over seven factorial, that means it’s a uniformly distributed sample. Okay, so what what can happen? Well, the worst that you can wait for that “I” shape is that if you have at the beginning of a set of seven, and then in the next set of seven, you have it at the end. That’s the worst that can happen in terms of these wonderful pieces that allow you to kill four rows.

KK: Interesting. Okay, so they're just choosing a random shuffle. Oh, wow. Well, that is better.

ÉR: It is for playing.

EL: I guess with, like, computer games like this. We, as you're saying earlier, Kevin, we don't actually want it completely random because we don't want to get, you know, five squares in a row or something like this, which can definitely happen when you play it randomly. Especially when you're me and my brother when we were, like seven and nine years old, always going in and playing for two hours to try to erase the other ones’ high scores.

KK: I still remember, so when I was in graduate school, I had a Nintendo Gameboy. My mother gave me one of these, and I played Tetris. And I can still hear that song. The music that went along with it, and I would go to sleep at night picturing these damn pieces just falling. I can hear this song over and over and over my head.

ÉR: That is a thing I think that happened when we were doing a PhD because I was always with my Nintendo playing Tetris and procrastinating, beautifully procrastinating. And it turns out.

EL: I have a question. How do you — like, did you find how Tetris is programmed? Or did you just do some — Like, how did you figure out that this is how Tetris works?

KK: Yeah.

ÉR: Well, so I started reading about it. But first of all, I started playing, and I started noticing this pattern, right? This pattern, because you never see — I think, I think what what tells you immediately that it cannot be a random distribution, uniform random period, is that you will never see three times the the same shape.

EL: Okay, you could see two in a row, if it happened, right at the end and then the beginning, but you can't see three in a row.

ÉR: You can’t see three in a row! So this is one of the immediate signs. So whenever you don't see three in a row, and you have a decently small set — and we're going to go back on the size of the set, because it plays a role. So it plays a role, the distribution that you have in the set, and it plays a role also the size of the set. And so in this case, because it's such a small set, not seeing three in a row is like, uh-uh, something fishy is happening here. This cannot be the uniform distribution, period. Yes.

EL: Okay.

ÉR: Yeah, and this is with four squares. And now imagine that we three get a very nice contract by a company that tells us, I want you to build a very nice Tetris, but it has to be for any number of tiles. And then we start discussing, okay, so let's see, one of the things that we need to do is to extend, or just to apply perhaps this algorithm, but it has to work for any n, any number of squares, let's say 57 squares, right? So, first step, go and find all polyforms with 57 squares.

KK: That’s a lot.

EL: Yeah.

ÉR: That’s a lot. Yes. And I kind of, yeah, I kind of, in purpose, choose 57. Because 57 is the first amount of squares that we as humans, don't know how many polyominoes there are with 57 squares.

EL: Okay.

KK: All right.

EL: It’s just too too many for for us to have sorted them out somehow.

ÉR: It’s too many for the patience that people and computers and computer time have. Like, 56, I think was two years.

EL: Wow.

KK: So there's some algorithm but it is not super-efficient.

EL: It hasn't stopped yet.

ÉR: It’s a little bit worse. Because even even if we wanted to say, Okay, let's cut at 56 because we know how many there are, or up to 56 and we deliver the game like this, right? So the problem is that this algorithm can count them. But this algorithm cannot really store the pieces or actually look at them, which is something that is possible to do. This is one of the magics of mathematics and computer science, that we can say something about something that we cannot see. And then we know up to 28 tiles, Tomás Oliveira e Silva in in 2015, was able to look at them and, for example, say the symmetries that they have because it's unknown, really, how many of them are going to have which kind of symmetries and so on and so forth, all the geometric and topological properties that you can have with polyominoes. And yes, so I guess we have to talk to the manager and say to the manager, we have a problem here. I don't think we can create Tetris for any amount of tiles.

EL: Yeah. Plus, I mean, it would be — I bet a lot of these tiles, as they get more pieces in them, have holes in the middle that you can just never plug. Because you can you can have rook-wise moves that will sort of surround this hole. Which I am thinking of because of your puzzle from the calendar. So I guess, do you happen to know like, at what size Tetris stops being fun?

KK: Eight.

ÉR: There is the answer. Well, with eight, perhaps not yet. Because almost all of them will not have holes.

KK: Right. But you’re still going to get that one with a hole every once in a while. And then you're done. Well, not yet. But it's, it's gonna cause problems.

ÉR: Super difficult.

EL: Yeah.

ÉR: Yes. And that is a very, very good observation, because actually, one of the things that is a mathematical truth is that most polyominoes have holds.

KK: Sure.

EL: Okay.

ÉR: Meaning with probability one, as n goes to infinity, if you take a polynomial at random, it will have most likely a linear amount of holes with respect to the number of tiles.

KK: Okay.

ÉR: And this is one of the topics that I started studying in my PhD, is how does the number of holes grow with respect to the number of tiles? And because we're asking a question upon a set that we don't know how many there are, and we don't know how to actually look at all of them. Right?

EL: Yeah.

ÉR: So the only way that we can say something is, for example, to have actual numbers, is sample. Sample and tell me. Sample them. Yeah, but we have another problem. How do you sample from a set that you don’t know?

EL: Yeah, that doesn't seem easy.

ÉR: Yeah. And that is huge. That is humongous. Because it’s — the number of polyforms is growing exponentially fast with respect to the number of tiles.

KK: Right? That's what I was going to say. I mean, even, you know, if you're up to, like, 10, the number is probably so large that you can't construct a reasonable game out of it, right? Yeah. So I mean, so Tetris is — so I have this game with blocks called Katamino, where they’re pentominoes, right? And that's fun, and challenging enough. Yeah. Okay. So now I understand why you're interested in stochastic topology, this feels very, if you're looking for holes in these things, you've set up some rules. And now the question is when do the Betti numbers change and things like that.

ÉR: Yes, yes. And now that you mention polyominoes, I also like like putting here and there one puzzle, so that people, perhaps even build it in their homes or something. So when Matthew Kahle was in Mexico visiting for my candidacy exam, we exchanged puzzles, and one of the puzzles that we exchanged is you take an eight by eight, a chess board, a usual eight by eight chess board. And then you have and then you break it on the head — no. And then the story in Diedonne’s book, it goes like this, that there were two royal persons playing chessboard, one loses, and then breaks the chessboard on the head of the other one, and then you get pieces. And these pieces are the 12 pieces that you were referring to, Kevin, the pentominoes, plus the two by two square. And then one of my favorite puzzles, and now so because Matt did this, built this puzzle with bamboo, with a very, very thin bamboo material, and he gave it to me as an exchange of the puzzles that we had. We actually had a night of puzzles and games back there.

KK: That’s fun.

ÉR: A lot of polynomials in my mathematical life.

KK: Very cool.

EL: Yeah. So, another thing we like to do on this podcast is asks our mathematicians to pair their theorem or example or their mathematics with something else. And so, what would you like to pair with this theorem about shuffles? And it may be even with the applications to all of these polyomino puzzles.

ÉR: I will say that is drawing and 3D modeling.

EL: Okay.

ÉR: One of the things that I like is to run these run some of these algorithms for generating poly-cubes in 3D, and different algorithms. And then to get these 3D crazy structures, and I like doing the modeling. So I guess, is the is a way of I, I think, it’s random art?

KK: Sure. Sure.

EL: And what programs do you use to actually draw those? ÉR: Oh, well, here, and this is one of the things that I have been really into it ,is there is a program called Maya. And it’s for doing 3D modeling and rendering. But it can be combined with Python.

KK: Okay.

EL: Okay. So do some other math with it, and then put it into Maya?

ÉR: Yes. And so I found someone to give me some classes on Maya. And then this person was always like, “But how do you manage to do 1000 cubes and decide where to put them?” And no, like, I just click and play, and I have an algorithm that decides to go all the way to 1000. So I guess it was a very interesting combination, knowing the kind of fast-track things that we do as mathematicians, and then mixing that with art. I like digital art. Yes.

EL: Nice.

KK: Very cool. Okay, so so we also like to give our guests a chance to plug anything, or where can we find you on the internet, or anything else you want to tell us let people know.

EL: Yeah, if people want to play anything that you made, or where they can find this frustrating Tetris game that you'll be coding once you make it.

ÉR: Yes. That is going to be available very soon. But also, because I’ve been developing these apps with a computer science collaborator, and one of the apps that I have, and that I will provide a link for people to play, is for finding polyforms formed by squares and triangles. So you can also play this in the infinite the triangle tessellation of the plane, or you can play this with any tessellation. You could play it even in hyperbolic spaces, you could play it in higher-dimensional spaces, but the app that I’m going to share stays with the square and the triangular grid. So, you will be able to find polyforms with maximally many holes and it will tell you if you are actually reaching the maximum number of holes that you can for that amount of tiles, so it has that property. And I guess another another random way that I have for for producing these random polygons is like a cell growth model. And what you do is you start with a cell, and then you add uniformly random one of the cells that are neighboring, so you can imagine this is a cell colony growing. And there is a huge area of research called first passage percolation that has been analyzing these kind of models. But what I do is I study the topology of this other way of randomly generating polyforms. And I have an app that I will also provide here so that people can think about it and perhaps come up with some conjectures that later, they can see in one of my papers with Benjamin Schweinhart and Fedor Manin.

EL: Okay, great.

KK: That’s fun.

EL: So, you’ll get homework after this episode, but it's not a quiz. So it's still okay.

KK: All right. Well, this has been a lot of fun, Érika. I always like to hear about these sorts of questions, even though I'm no good at them. I always like to talk to people who are good at counting and these kinds of things. So it's been it's been great fun talking to you. Thanks for joining us.

ÉR: Yeah. Thank you so much for the invitation. And yes, looking forward for seeing you in the USA soon.

EL: Yes. Bye.

ÉR: Bye.

[outro]

On this episode of My Favorite Theorem, we had the pleasure of talking with Érika Roldán, a Marie Curie fellow at Technical University Munich and École Polytechnique Fédérale de Lausanne, about shuffling cards and Tetris pieces.

To read about the mathematics of riffle shuffles, this article by Persi Diaconis is a good place to start. To get a copy of the American Mathematical Society page-a-day calendar, click here. (If you already have a calendar, check out Dr. Roldan's puzzles on August 28 and October 12.)

Dr. Roldán shared some other links to and explanations of some of the apps and videos she mentioned in the episode:

The COVID crisis has allowed me to start developing digital material for my research, teaching, and outreach on mathematics and its applications. It has also allowed me to collaborate as a developer of digital material (in Germany) with artists whose projects promote gender equality, and diversity & inclusiveness awareness. Here, I share some of the links to explore this digital playground (new digital material created will be posted soonish at my website: http://www.erikaroldan.net/):
1) https://000612693.deployed.codepen.website
Follow the link above to find Extremal Animals, that is, polyforms with maximally many holes. A polyform is built by gluing together squares or triangles (in the case of this app) by their edges. And a hole in a polyform (that mathematicians call the first Betti number in this 2D case) is a finite connected component of the complement of the polyform. To get some intuition, just build a polyform with squares with 7 tiles and one hole, or a polyiamond with 9 triangles and one hole. Could you create one hole with less tiles in any of these two cases?
Have a look at these papers for the maths behind this (Extremal Topological Combinatorics) puzzle of finding polyforms with maximally many holes:
https://arxiv.org/pdf/1807.10231.pdf
https://www.combinatorics.org/ojs/index.php/eljc/article/view/v27i2p56/pdf
https://arxiv.org/abs/1906.08447v1
2) https://000612976.deployed.codepen.website
Here a link to an app that has a model to generate a random polyform by a cell growth process called The Eden Model. Pay attention to how the holes are created and destroyed as time (the number of tiles) evolves. Do you have any conjectures about how the number of holes is changing concerning time? Have a look at this link to see if your conjectures are stated and/or proved in this paper:
https://arxiv.org/abs/2005.12349
3) My first proto-game with Unity was developed for the Film “Broken Brecht” directed and produced by Caroline Kapp and Manon Haase, for the Brechtfestival Augsburg, Germany (Mar 2021). This is a project that will be extended during 2021! Here some links to the festival, the proto-game, and an extract from the film that happens within the proto-game.
Brecht Festival
https://brechtfestival.de/brokenbrecht/
Extract from Broken Brecht
https://vimeo.com/542287814
Link to the 3 min Archive Video Game
https://simmer.io/@ErikaRoldanRoa/~56f30f68-048c-c027-7aa0-aeaca82508fc
4) Some 3D models created with Python & Maya to explore (random) cubical complexes.
https://sketchfab.com/erikaroldan

Episode 65 - Howard Masur

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, a math podcast where there's no quiz at the end. I remember we did that tagline, like, I don't know, probably two years ago or something. And I forgot that I wanted to keep doing it. But I did it today. I'm Evelyn Lamb, one of your hosts. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. I forgot that tagline, too, and it's a pretty good one. Let's, let’s—look.

EL: We’ll see if we remember later.

KK: After our last recording session, we agreed we needed we needed a real tagline. So yeah. We're recording this on February 18, which means that Texas is largely without power and frozen.

EL: Yeah.

KK: And it's 82 degrees in Florida today.

EL: Oh wow. Yeah. Most of my family is in Texas, and it is not great.

KK: Do they have power? Or? No?

EL: Most of them do. All of them do sometimes.

KK: Right. Actually, now I think water is getting to be a problem now. Right?

EL: Yeah. I haven't heard about any problems with that from my family. But yeah, it's not great. I hope that it warms up there soon and everything can come back online. But yeah, today, we're very happy to be talking with Howard Masur, who is in a place that is very used to being cold and snowy. So yeah, Howard, do you want to introduce yourself? Tell us a little bit about yourself?

Howard Masur: Okay, thank you. First of all, thank you very much for inviting me to do this. I’ve been very excited thinking about about it. Yes, I'm on the math faculty at the University of Chicago. And I've, I guess, been working in mathematics for quite a long time and still enjoy it a great deal. It’s a major part, a very big part of my life. And your invitation to talk about my favorite theorem led me to, you know, think about what that would be and why I chose what I did. And and it made me think that, yes, what I really like the most in mathematics, or one of the things, is mathematics that connects different fields of mathematics. And maybe unexpectedly connects different fields. And I personally, have worked on and off in complex analysis and geometry and dynamical systems, another field. And I love the part of mathematics that sort of connects them.

EL: Well that's perfect. Because I mean, you you're a frequent collaborator with my husband, Jon Chaika. But also with my advisor, Mike Wolf, who, you know, isn't quite in the same area of math generally. So yeah, you have worked in a lot of a lot of different fields that I feel like your name pops up, you know, in a very wide range of things related to geometry, analysis, dynamics, but yeah, you’ve got your finger in a lot of pots.

KK: Right. Well, okay, so what is it? What's your favorite theorem?

HM: Okay. It's called the Riemann mapping theorem.

KK: Yes.

HM: So, let me let me give a little bit of background. The first thing, it involves subsets of the plane which are called simply connected. And this is a notion from topology. And let me just say I looked at one of your podcasts and someone else talked about the Jordan curve theorem, where if you have a simple curve in the plane — it could be very, very complicated — a simple closed curve, then it has an inside and an outside, then the inside is simply connected. And a way of thinking about what simply connected means is heuristically it doesn't have any, it has no holes. But as also has been pointed out, they can be very complicated, Jordan curves. Certainly they can be simple looking like a circle. The inside of a circle is simply connected, the inside of a rectangle. But on the other hand, the Jordan curve can be very complicated like a snowflake, a Koch — I never remember how to pronounce that; is it “coke” snowflake?

KK: Let’s go with Koch [pronounced “coke”].

HM: Pardon me?

KK: Let’s go with that.

HM: Okay. And so that's very complicated. It's the boundary — the curve is a fractal. So already simply connected domains can be very complicated, but they don't even have to be just the inside of a Jordan curve. You could take the plane itself, there’s a very simple example. You could take all the positive real numbers, include zero, and take it away from the complex plane. So the plane minus the positive real axis and also subtract the origin, that’s simply connected, it doesn't have any holes. And it's not the inside of a curve. You could also, on the other hand, here's something that isn't simply connected: you could remove the interval [0,1], including zero and one from your plane, just that interval on the real axis. And that is not simply connected because the complement, or the plain minus that, has a hole, which is that interval [0,1], it can be thought of as a hole. So that's the notion of simply connected. I don't know whether I should say more. I mean, that's what I thought to say about what simply connected means.

KK: That’s great. Yeah, yeah, that's a good explanation.

HM: Okay, and so that's a topological notion. And then the other thing that goes into this theorem is a notion from geometry, well, actually a notion from geometry and a notion from complex analysis. But let me let take a basic notion from geometry, which is called conformal. And the idea is that if you suppose you have two domains in the plane, and you have a transformation from one to the other, you say it's conformal if it's angle-preserving. So that means that if in the first domain, you have a pair of arcs — or maybe you prefer to think of them as straight lines, but it's better to think of a couple of arcs — that meet at a point, and then you apply the transformation, and you get a pair of arcs that meet in the image under the transformation. And you could measure the angle that you started with between the pair of arcs and the angle of the images of the pairs of arcs, and if the angles are equal at every point for every pair of arcs at those points, then you say the transformation is conformal, angle-preserving. Now, in some ways, the nicest — so let me give some examples that are and are not. The nicest transformations, certainly of the plane, are linear transformations.

KK: Sure.

HM: Given by two by two matrices, and they turn out not to be typically conformal. There are some that are, for example, a rotation about the origin is conformal. You know, if you have two lines and you rotate them, the angle they make after rotation is the same as the angle they started with. If you — this isn't strictly a linear transformation, it’s called affine — if you take a translation of the plane, if you take every point and you add the same vector, think of them as vectors, that's angle-preserving, that's a conformal transformation. Here's another one that's back to linear. If you take, for example, every point, which has, say, coordinates (x,y), and you multiply x by 2 and y by 2, so you multiply the coordinates by the same number, 2. That's called a scaling. And that's angle-preserving. One can sort of check that out. What that transformation does is, for example, it takes a square with one vertex at the origin, a unit square, and then another vertex on the x-axis at the point (1,0) and another at the point (0,1), last point at (1,1), and it takes a unit square to a two by two square, and that's angle preserving. But that's it — well, and the composition — but typical linear transformations are not angle-preserving. So, for example, if you took (x,y) and the transformation took (x, y) to (2x, y/2), so it multiplies in the x direction by 2 and multiplies in the y direction by a half, it takes a unit square into a rectangle, and that's not angle-preserving. It preserves the right angles, but it doesn't preserve other angles.

EL: Yeah, you can imagine the diagonal is, you know, [demonstrates with arm gestures that are very helpful to podcast listeners].

HM: The diagonal is closer to the x-axis, so the diagonal which made an angle of 45 degrees will be moved with the x-axis. The x axis goes to itself, and the image of the diagonal is moved closer to the x axis. Yeah, exactly.

So there aren’t maybe, there aren't so many linear transformations of the plane to itself, and so let me tell you what the theorem is, and this is a beautiful, beautiful theorem, I think, and it was really a cornerstone of, in the 19th century, of the beginnings of complex analysis. Oh yes, I’m sorry. Before I do that, let me also connect conformal, as I had mentioned, to complex analysis. One also can think of the euclidean plane as the complex plane, where (x,y) becomes x+iy, becomes a complex number z, and then conformal, another way of saying it, is that the map, the transformation from some region in the plane to some other region in the plane has a complex derivative. It’s what you call complex analytic. It has a derivative and the derivative is not zero. Again I looked at your podcast. Someone talked about the Cauchy-Riemann equations, and that's exactly what complex analytic means is that the Cauchy-Riemann equations hold. Where where w is u+iv and z is x+iy, then it's complex analytic if ux=vy and −uy=vx. That’s the Cauchy-Riemann equations, and that's from complex analysis. It has the names Cauchy and Riemann, who where in some sense the founders of complex analysis. And that's equivalent to conformal, so even there just in this, there's already kind of an amazing theorem that relates — I think obviously you had somebody on your podcast maybe talk about this — that relates complex analysis to geometry, conformal meaning angle-preserving and complex analytic meaning, let's say, the Cauchy-Riemann equations hold.

KK: Right.

HM: Okay, so the theorem is that if I take any simply connected set’s domain in the complex plane, other than the complex plane itself, okay? And I take the unit disc — so that's inside the circle of radius one, so that's simply connected — I can find a conformal transformation from the unit disc to this simply connected domain, and maybe thinking about the inverse, it's a conformal transformation from that (maybe crazy) simply connected domain to the unit disc, and so that's the Riemann mapping theorem

EL: Yeah, and it's just amazing. I mean I think there's part of me that still doesn't believe that it's true. I've actually just, I don't know when it was, maybe a month or two ago, I think I was brushing my teeth or something and just thinking, why hasn't someone pick the Riemann mapping theorem yet for My Favorite Theorem?

HM: Okay, all right.

KK: It's a really mind-blowing theorem. So when I teach the undergraduate complex analysis course that we have, I don't get to it until the very end.

HM: Yeah.

KK: And it's kind of hard. You can't even really prove it especially at that level, but students just look at me like, there's no way this is true. This just can’t be true. So it's really remarkable that anything — I mean, you're right. I mean, these simply connected domains can be bizarre. But they're conformally equivalent to the unit desk. That's just blows my mind still. Yeah,

EL: Yeah. It's just hard to imagine, like, this fractal snowflake, you know, how can you straighten that out enough to just be like a circle?

HM: Let me contrast it — and this also goes back kind of to the founding mathematicians of the subject. If I take what's called an annulus, let's say I take the circle of radius 1. And I take the circle of radius R, where R is bigger than 1. And I take the region between them. So the region between two concentric circles, that's not simply connected because it has a hole, namely, the inside of the unit circle is the hole. And so if I take one of radius, the inner is radius 1, the outer radius is R, and I take another one, inner radius 1 and outer radius R’. And let's say R’ is not equal to R. So it's a different outer radius. They are not conformally equivalent, even though they are very simple boundaries, their circles. So there was something very, very special about simply connected. And that's also kind of what makes the theorem amazing. And then the fact that it doesn't work for something not simply connected started a whole field of mathematics that has been going on for close to 200 years.

EL: And so was this kind of a love at first sight theorem for you the first time you saw it?

HM: You know, I guess I'm not 100% sure. I was in college a little while ago, and I don't don't think I had complex analysis in college. And so I may not have run into it then. But certainly, as a first-year graduate student at University of Minnesota, and my professor, who then became my thesis advisor within a year, you know, for my PhD advisor, that was somehow his field. And so I certainly learned it as a graduate student. And that led me — again, I can't exactly say it led me to what I do — but, you know, it certainly had a big influence, and things that I do sort of have grown out of this whole history of this, from the from from the Riemann mapping theorem.

KK: So, is this one of those theorems is actually named correctly? Did Riemann actually prove it?

HM. I don't know, I'm not a historian. You know, I mean, I could ask. For that matter, are the Cauchy-Riemann equations named after the right people? Yeah. I mean, I know the modern proof that one sees in books on the Riemann mapping theorem is not due to Riemann. It’s I think, early 20th century.

EL: Is it Poincaré maybe?

HM: You know, my mind is going blank here for a second.

EL: It’s someone.

HM: I don't know. I'm not a historian, and I did not look it up to say “Does Riemann really deserve credit?”

KK: But wait, I looked at Wikipedia. I’m cheating. The first rigorous proof of the theorem was given by William Fogg Osgood in 1900.

HM: Oh, okay. Okay. Yeah.

KK: So apparently Riemann, this is in his thesis, actually. But there were some issues, it depended on the Dirichlet principle. And Hilbert sort of fixed it enough that it was okay. But Osgood is credited with the first rigorous proof.

HM: Well, isn’t it also somehow the case again, that mathematicians 200 years ago did not quite have the rigor that we have now?

KK: That’s true. Cauchy sort of put limits on the right footing more or less, but I think it still took a little while to get it cleaned up, right? So are there any really interesting applications of this theorem that you like? Or is it just beauty for its own sake?

HM: Gosh, you know, I'm not sure. I think beauty for its own sake, I mean, but also to my mind, it opened up a whole branch of mathematics where you study, well, for example, you study surfaces. Or maybe it's the difference between topologists and geometers. A topologist says that a doughnut, famously, a doughnut is the same as a coffee cup with a, you know, with a handle and so forth. And geometers say, well, we could put different ways of measuring angles, different metrics on a torus that are not conformally equivalent, that there's no transformation from one to the other that preserves angles.

KK: Right.

HM: And this Riemann mapping theorem says, No, you can't do that for simply connected. They are conformally the same. But as soon as we move to topologically more complicated things like a torus or even these annuli, or surfaces with more holes, genus, then there are different ways of putting metrics and measuring angles and so forth. And so it opened up, and again, this actually also has Riemann’s name to it, it’s the Riemann moduli space, is the study of all metrics on a space. And so, yeah, again, I haven't thought of an application so much to other fields, but something that is a beautiful and unexpected theorem that opened up whole vistas of mathematics, I think, in the last whatever. I don't remember when Riemann stated this problem. When did he live, in the 1840s?

KK: Yeah, middle 1800s.

HM: So it’s been 175 years or something that people studying, have been studying? consequences in some sense of, of this or analogs of this?

EL: Yeah. Well, and so this is something: I never wonder it at the right time to check and see, but is there a place where you can go and say, like, this is my domain 1 — and maybe it's a square or maybe it's the flag of Nepal or something — and this is my domain 2, or just the unit circle, and here is the conformal map between them. Is that something that exists?

HM: Typically not. There are certainly examples where you can, but it's very, very rare that you can write down an explicit formula for the map. That's again, maybe why it's such a beautiful theorem, but you cannot, I don't let’s see, I hope I'm not — maybe you can do it for a circle to an ellipse. Um, maybe. I'm not 100% sure. There are people who know much, obviously know much, much more about finding something explicit. But in general, no, if you take some crazy Jordan curve, no way, do you know an explicit formula.

EL: You just know it's there.

HM: You know it's there.

KK: Well, that's important, though, right? If you're going to go looking for a needle in the haystack, you do, in fact, want to know there's a needle in it.

EL: Yeah.

KK: All right. So another fun part of our podcast is we ask our guests to pair their theorem with something. So we're dying to know what pairs well with the Riemann mapping theorem?

HM: Well, I thought about that a lot.

KK: This is the harder part.

HM: I tried desperately to find food, but I couldn't think of really the right thing. So I do love music. And this is maybe crazy far-fetched, but I paired it with Stravinsky's Rite of Spring only because to me, this Riemann mapping theorem revolutionized geometry and complex analysis. And I think of the Rite of Spring of Stravinsky, which was premiered in the early 20th century, revolutionized modern music, contemporary music. That's the best I can do.

EL: I like that.

KK: Well, I do too. And for all we know, there were riots after Riemann published his theorem.

HM: Could have been.

KK: You know, “there’s no way this is true!” Mathematicians stormed out.

HM: Maybe he gave a lecture and people threw tomatoes at him.

EL: Yeah, well, I must say when I was thinking about asking you to be on the podcast, I did think about the many wonderful meals that we have shared together, and I know that Howie is a great appreciator of the finer things in life, including music, too. I think we've gone to a concert together. And yeah, so I thought that this would be an excellent thing. You know, I was I was talking with Jon earlier about, what is Howie going to pair with it? And my first thought was actually pancakes, which I think are a little pedestrian, but that you can make them into so many different shapes. And there's even, there are people who will do these things where, you know, if you pour the batter on, in a certain way, you know, you can get these beautiful things. I mean, part of it is part of the batter cooks longer than the rest of it. And so you've got shading based you know, how they do I've seen, I think, you know, Yoda and like, I don't know, all sorts of different things. There’s this Instagram account. But that was one, all these different shapes you can do. I'll say that Jon actually suggested jigsaw puzzles. Oh, no, sorry, he first suggested jigsaw puzzles, but there’s only one right way to do that. But then he said tangrams, you know those things with all the shapes, you know, there's a square and triangles and stuff. And then you can rearrange them to make all these different shapes, although those are non-continuous maps. So it wouldn't be quite as good. But, I do like the Rite of Spring. And it means that Stravinsky is doing really great on My Favorite Theorem because Eriko Hironaka actually picked Stravinsky also.

KK: Firebird.

HM: She picked the Firebird. I’ll have to look at her podcast. Maybe I'll give her a zoom meeting and we can compare music and the math.

EL: Yes. But I like that. And I am now going to ret-con in some riots following Riemann declaring that you can make these conformal maps.

KK: Well, this has been great fun. I do love the Riemann mapping theorem and Howie, thanks for joining us this

HM: Well, thank you for having me. It was a pleasure.

On this episode of My Favorite Theorem, we were happy to talk with Howard Masur, a math professor at the University of Chicago, about the Riemann mapping theorem. Here are some links you might find interesting as you listen.
Masur's website
Evelyn's article about the Koch snowflake
Jeremy Gray's article about the history of the Riemann mapping theorem (pdf)
A recording of Stravinsky conducting the Rite of Spring
Did the Rite of Spring really cause a riot at its premiere?

Episode 64 - Pamela Harris and Aris Winger

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast. We need a better tagline, but I'm not going to come up with one today. I'm Kevin Knudson, professor of mathematics at the University of Florida. Here is your other host.

Evelyn Lamb: Hi, I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City. And I think that our guests might be able to help us with that tagline. But we'll get to that in a moment because I have to share with you a big kitchen win I had recently.

KK: Okay.

EL: Which is that that I successfully worked with phyllo dough! It was really exciting. I made these little pie pocket things with a potato and olive filling. It was so good. And the phyllo dough didn't make me want to tear out my hair. It was just like, best day ever.

KK: Did you make it from scratch?

EL: No, I mean, I bought frozen phyllo dough.

KK: Okay, all right.

EL: Yeah, yeah, I’m not at that level.

KK: I’ve never worked with that stuff. Although my son and I made made gyoza last month, which, again, you know, that that's a lot of work to because you start folding up these dumplings, and you know. They’re fantastic. It's much better. So, yeah, enough. Now I'm getting hungry. Okay. It's mid afternoon. It's not time for supper yet. So today we have we have a twofer today. This is this is going to be great, great fun. It's like a battle royale going here. This will be so much fun. So today we are joined by Pamela Harris and Aris winger. And why don't you guys introduce yourself? Let's start with Pamela.

Pamela Harris: Hi, everyone. I like how we're on Zoom, and so I get to wave. But that’s really only to the people on the call. So for those listening, imagine that I waved at you. So I am super excited to be here with you all today. I'm an associate professor of mathematics at Williams College. And I have gotten the pleasure to work with Dr. Aris Winger on a variety of projects, but I'll let him introduce himself too.

Aris Winger: Hey everybody, I’m Aris Winger. I'm assistant professor at Georgia Gwinnett College. I've been here for a few years now. Yeah, no, we, Pamela and I have been all over the place together. I've been the honored one, to just be her sidekick on a lot of things.

PH: Ha, ha, stop that!

EL: So we're very excited to have you here. So you've worked on several things together. The reason that I thought it would be great to have you on is that one of the things is a podcast called Mathematically Uncensored. And it's a really nice podcast. And I think it has a fantastic tagline. I was telling Aris earlier that it just made me very jealous. So we've we've never quite gotten, like, this snappy tagline. So tell us what your podcast tagline is. And a little bit about the podcast.

PH: Maybe I can do the tagline. So our tagline is “Where our talk is real and complex, but never discrete.”

AW: Yeah, that's right. That is the tagline. And yeah, it's a good one. And sometimes I have to come back to it time and again to remember, so that we live up to that during the podcast. We're taping the podcast later today, actually. And so it should be out on Wednesday. So yeah, the show is about really creating a space for people of color in the mathematical sciences and in mathematics in general, I think. And so one of the ways—I think for us the only way that can happen—is we have to start having hard conversations. Right. And so a realization that comfort and staying on the surface level of our discussions doesn't allow for us to have the true visibility that all people in mathematics should have. And so for too long, we've been talking surface-level and saying, “Oh, we have diversity issues. Oh, we should work harder on inclusion.” No, actually, people are suffering. No, actually, here's our opinion. And stop talking about us; start talking to us. So it really is a space where we're just like, you know what, screw it. Let us say what we think needs to be said. Listen to us. Listen to people who look like us. And yeah it’s hard. It's hard to do the podcast sometimes because when you go deeper and start to talk about harder topics, then there are risks that come with that. Pamela and I, week after week, say, “Oh, I don't know if I really should have said that.” But ,you know, it's what needs to be said, because we're not doing it just for us. We're doing it to model what what needs to happen from everybody in this discipline, to really say the things that need to be said.

KK: Have you gotten negative feedback? I hope not.

AW: Yeah, that’s a good question. So I mean, I think that the emails we've gotten are have been great and supportive. But I think, so for me, I'm expecting no one to say — I’m expecting the usual game as it is, right, that people aren't going to say anything, but of course there's going to be backlash when you start saying things that go against white privilege and go against the current power structures. You know, I'm expecting to be fired this year.

PH: Yeah, those are the conversations that we have constantly — that we’re having on the podcast are things that Aris and I are having conversations about privately. And so part of what's been really eye-opening for me in terms of doing a podcast is that I forget people are listening. There are times Aris and I are having just a conversation, and I forget we're recording. And I say things that I normally would censor. If I were in a mixed crowd, if I were in a department meeting, if I were at a committee meeting for, you know, X organization. And I think it's not so much that we would receive an email that says, “Hey, you shouldn't have said X, Y, and Z,” it’s that we are actually getting targeted. For example, I was just virtually visiting Purdue University giving a talk about a book that Aris and I wrote, supporting students of color. And accidentally, the link got shared to the wrong people. And all of a sudden, I'm getting Zoom-bombed at a conversation. That's targeted, right? So those are the kinds of things that we are experiencing as people of color, and we have to have conversations about how are we ensuring that this isn't the experience when you bring a Black or brown mathematician to talk virtually at your colloquium. And if we're not talking about that, then no one is talking about that, because people are trying to hide their dirty laundry. Purdue University is not putting out an email to their alumni saying, “By the way, we invited Pamela Harris to show up and talk about how we best support students of color. And then we got Zoom-bombed, and somebody was writing the N-word and saying f BLM.” Right? Like, that's not happening.

AW: Yes. Wait, they didn't say anything about it?

PH: Well, they're actively doing things about it. But you know they're not putting out the message.

AW: Right. So then it gets sanitized, right? So a traumatic attack gets sanitized to be something else. There are two things about the podcast that Pamela and I, and the Center for Minorities in the Mathematical Sciences, really are trying to work with is making sure that we call out these things, but then not to center it, right, because the the podcast itself is supposed to be about our experiences. But a lot of ways there's a significant part of our experiences that is tied to having to continuously fight against this type of oppression against us.

EL: Yeah. And I think it's really important to have that. And it's so important that it decenters— I think I was listening to an episode recently where you talked about the white gaze and what you have to deal with all the time in trying to present things to a majority white audience. And I think it's really important for us white people to listen to this and realize that not everything is about and for us. And I mean, there are so many things in life where this is true: movies, TV shows, books and stuff. And yeah, I think it's great that that your voices are there and having these conversations, and I think that people should listen to your podcast.

AW: I appreciate that. Yeah. Because it requires a deep interrogation, a self-interrogation by white people to really deal with the feelings. Let me just step back and give the usual disclaimer. Everybody's nice. Everybody's good. Nobody's mean. Nobody is a bad person. Let me just say all that to get that out of the way, right? But what we're talking about is that when I say something on the show, when Pamela says something on the show and you get this feeling like “Wow, that doesn't feel good to me,” then you need to take some time and figure out why it is that you're feeling this way. And it's tied to your privilege, something that you need to interrogate, and it will make you a better person and for everybody.

KK: I don't know. I can't wrap my head around people, like, Zoom-bombing. This is nothing that would ever come to my mind. “You know what, I'm going to go Zoom-bomb this person.” I just…

EL: Well, I mean, it’s just a bad way to spend your time, but not everyone has the same time priorities.

AW: Well, no. So I think that's a great question. And let me just say that it that's how deep and pervasive it is in people, right, that people grow up and have this experience of being raised by other people who have ingrained within them that it is fundamentally, and in some sense, it just burns their soul to have somebody who does not look like them, have someone who is “lesser” than them take the center stage, be deemed the expert. And so again, I'm not calling these people bad. But there is something within some of us that says — and it’s called white supremacy, by the way — that we all have, that we all have to fight, that is so ingrained in some people that they feel compelled to do it. And so they, again, no one's going to fix that for them. And the person who did this to Pamela has it in spades, right? And so when we say that, so I think too often we make it an intellectual exercise, right? We say that it just makes no sense. Right? It doesn't make any sense because white supremacy makes absolutely no sense. But it is a thing. And it's there. And that's what it is, right? So I've been working a lot on calling, naming things so that we don't get confused, because as long as we don't name it, then it just gets to be out there. Like, “Oh, I don't understand.” We understand this exactly. It's called white supremacy. And we need to fight it in our discipline, and across the board.

PH: And it doesn't always just show its face via Zoom-bombing with the N-word in the chat, right? It shows up with who you invite to your podcast. It shows up with who's winning awards from our big national organizations. It shows up with who gets tenure, who even lands into a tenure track position, who even gets to go into graduate school, who actually majors as a mathematician, who actually goes to college, who actually graduates high school, who actually gets told that they're a mathematician. Right? So this is showing its ugly head in very visual ways that we all feel a huge sense of, “Oh, no, this is terrible. I'm sorry, this happened to you.” But the truth is that white supremacy is in everything within the mathematical sciences. And so you know, we got to pull it at its root, my friends. At its root!

AW: Yes.

PH: So this was just one way in which it showed itself, but I want to make it clear that it is pervasive.

KK: Sure. Right.

EL: So what I love about hosting this podcast is that we get to know both people and their math and their relationship to their math. And so we're gonna pivot a little bit now, maybe pivot a lot now, and say, Okay, what are your favorite theorems? And, yeah, I don't know who wants to go first. But, yeah, what's your favorite theorem?

KK: Yeah, let’s hear it.

PH: I’ll do it. I’ll go first. I always like hearing Aris talk. So I'm just like “Aris, go,” right? But no, I’m going to take the lead today. Alright, so I wanted to tell you about this theorem called Zeckendorf’s theorem. I don't know if you know about it.

KK: I do not.

PH: And it goes like this. So start with the Fibonacci numbers without the repeated 1. So 1, 2, and then start adding the previous two, so 3-5-8, and so on. Alright. So if you start with that sequence, his theorem says the following, if you give me any positive integer N, I can write it uniquely as a sum of non-consecutive Fibonacci numbers.

AW: Oh, wow!

EL: Uniquely?

PH: Yes. And this is why you need to get rid of the 1, 1. Because otherwise you have a choice. But yeah. So it's hard to do off the top of my head, because I'm not someone who, like, holds numbers. But say, for example, we wanted to do 20. Maybe we wanted to write the number 20 as a sum of Fibonacci numbers that are not consecutive. So what would you do? You would find the largest Fibonacci number that fits inside of 20. So in this case, it would be 13.

AW: Yeah.

PH: 13 fits in there. Okay, so we subtract 13. We're left with 7. Repeat the pattern.

KK: Ah, five and two.

KK: Five and two! They're non consecutive.

KK: Okay.

PH: Yeah.

AW: Wow!

PH: Three is in between them, and eight is in between the others. And so you can do this uniquely. And so this is using what's known as the greedy algorithm because you just do that process that I just said, and it terminates because you started with a finite number.

KK: Sure.

PH: And so the the proof, of course, there's the, you know, “Can you do it?” but then “Can you do it uniquely?” So the thing that you would do there is assume that you have two different ways of writing it, each of which uses non-consecutive, and then you would argue that they end up being exactly the same thing. So that, in fact, they use the same number of Fibonacci numbers and that those numbers are actually the exact same.

KK: Sure, okay.

EL: Yeah. Like I'm trying to figure out — and I don't, I also am not super great at working with numbers in my head just on the fly. But yeah, I'm trying to figure out, like, what would have gone wrong if I had picked eight instead of 13 to start with, or something? And I feel like that will help me understand, but I probably need to go sit quietly by myself and think about it. Because there’s a little pressure.

PH: Yeah, it's a little subtle. And it might be that you don't get big enough, you end up having to repeat something.

EL: Yeah, I feel like there's not enough left below eight to get me there without being consecutive.

PH: Yeah. Right.

AW: Right. Because you’ve got to get 12. Yeah, yes. Yeah.

KK: Yeah, it makes sense, right? Like, I guess, you know, if you pick the largest one less than your number, then it's more than halfway there. That's sort of the point, right? So that's how you prove it terminates, but also the the non-uniqueness, the non-uniqueness seems like the hard part to me somehow, but also the non-consecutive. Wait a minute, I don't know, which is.

AW: Well it sounds hard, period.

KK: Yeah. I like this theorem. This is good. What attracts you about this theorem? What gets you there?

PH: So I, in part of my dissertation, I found a new place where the Fibonacci numbers showed up. And so once you find Fibonacci numbers somewhere new, I was like, what else is known about these beautiful numbers? And so this was one of those results that I found, you know, just kind of looking at the literature. And then I later on started doing some research generalizing this theorem. So meaning, in what other ways could you create a sequence of numbers that allows you to uniquely write any positive integer in this kind of flavor, right, that you don't use things consecutively, and consecutively, really, in quotes, because you can define that differently. And so it led me to new avenues of research that then I got to do. It was the first few research projects with some of my undergraduate students at the Military Academy. And then I learned through them — they looked him up — that he actually came up with this theorem while he was a prisoner of war.

AW: Oh, wow!

PH: This is when Zeckendorf worked on this theorem. And to me, this was really surprising that, you know, my students found this out. And then I was like, “See, mathematics, you can just take it anywhere.” lLke this poor man was a prisoner of war, and he's proving a theorem in his cell.

KK: Well Jean Leray figured out spectral sequences in a German POW camp.

PH: I did not know that!

AW: Anything to pass the time.

KK: What else are you going to do?

EL: I mean, Messiaen composed the Quartet for the End of Time — I was about to say string quartet, but it's a quartet for a slightly different instrumentation, in a concentration camp, or a work camp. I'm not sure. But yeah, I'm always amazed that people who can do that kind of creative work in those environments, because I feel like, you know, I've been stuck in my house because of a pandemic, and I'm, like, falling apart. And my house is very comfortable. I have a comfortable life. I am not as resilient as people who are doing this. But yeah, that is such a cool theorem. I'm so glad that you said that. And I'm trying to think, like, Lucas numbers are another number sequence that are kind of built this way. And so is there anything that you can tell us about the the sequences that you were looking at, like, I don't know, does this work for Lucas numbers? I don't know if you've looked at that specifically, or did you look at ways to build sequences that would do this?

PH: Yeah. So we started from the construction point of view. So rather than give me a sequence, and then tell me how you can uniquely decompose a number into a sum of elements in that sequence, we worked backwards. So one of the research projects that we started with is what we called — there's a few of them — but one, it was a “Generacci” sequence. And so what we would do is, instead of thinking of the numbers themselves, imagine that you have buckets, an infinite number of buckets, you know, starting out the first bucket all the way to infinity, and you get to put numbers into the sequence in the following way. So you input the number 1 to begin with, because you need a number to start the sequence. And since you want to write all positive integers, well, you’ve got to start with 1 somewhere. So you stick the number 1 in the first bucket. And then you set up some system of rules for which buckets you can use to pull numbers from that then you add together to create new numbers. Well, you only have one bucket, and you only put the number 1 in it. So then you move to the next bucket. Well, okay, you want to build the number two, and you only have the number one, and as soon as you pull it from the bucket, you don't have any other numbers to use. So let's stick the number two in the second bucket. Oh, well, now I could maybe in my rule, grab a number from two buckets, and add them together to get the next number. Oh, that starts looking familiar. The third bucket will have not the number three, because you were able to build it. So what next number could you grab? Well, maybe you can stick in the 4 in there. And so by thinking of buckets, the numbers that you can fill the buckets with, that you couldn't create from grabbing numbers out of previous buckets under certain rules, you now start constructing a sequence. And provided that you very meticulously set up the rules under which you can grab numbers out of the buckets to add together to build new numbers, then you do not need to add that number into the buckets, because you've already built it.

EL: So what rules you have about the buckets will determine what goes in the buckets.

PH: Exactly, exactly. So you might say okay, maybe our buckets can contain three numbers. And you're not allowed to take numbers out of consecutive buckets, or neighboring buckets, or you must give five buckets in between. So what must go into the buckets to guarantee that you can create every single number and you can do so only uniquely? And so these are these bin decompositions of numbers. But you are working backwards. You start with all the numbers, and then you decide how you can place them in the buckets and how you can pull them from the buckets to add together. So I'm being vague on purpose, because it depends on the rules. And actually it's quite an open area of research, how do you build these sequences? You set up some some capacity to your bucket, some rules from where you can pull to add together. And the nice thing is that it's very accessible, and then it leads to really beautiful generalizations of these kinds of results like that of Zeckendorf.

EL: This is very cool.

AW: Fantastic.

EL: All right. So Aris, I feel like the gauntlet has been thrown.

KK: Yeah.

AW: Yeah, well mine is simple. This is not a competition. Yeah, no. I guess mine is influenced — I’ve been thinking about a bunch of different things, but I keep coming back to the same one, which I think is influenced by my identity as a teacher first and foremost when I think about the fundamental theorem of calculus. I just keep coming back to that one. And I don't know how many people have used this one with you on this podcast before, but for me, it hits so many of the check marks of my identity in terms of thinking about myself as a mathematician and a teacher, in the sense that for a lot of students who get to calculus, it's one of the first major, major theorems that will show up in their faces that we actually call out and say this is a theorem. And we call it fundamental, right? We don't often bring up the fundamental theorem of algebra in college algebra, right? Or in other places, or the fundamental theorem of arithmetic, right? But so it's one of these first fundamental theorems. And so it also helps to tell the story of a course, right? And so that really hits the teacher part of me where too often people in the calculus sequence spend all this time talking about derivatives, and all of a sudden, we just switch the anti-derivatives. And we don't really say why. You'll figure out in the next couple of sections, and then we start adding rectangles, and we don't say why. And so it really is, at least the way that the order of calculus has gone and in terms of how to teach it, in my experience, it really is this combination, like, oh, this is why we've been doing this. And this is the genius of relating two things. Sometimes I've gone in, and I've talked about, like I put up a sine curve, and a cosine curve, and we talk about how one of them measures the area under the curve. And then I pretend to bump my head and get amnesia. And then I'll come and say, “Oh, look, looks like we've been talking about derivatives. Right?” And they’re like, “Wait, what do you mean we're talking about derivatives?” “This is the derivative of this one.” And they’ll go, “What? We were measuring the area under the curve.” “Well, we’re also measuring the derivative, right?” This is the derivative, but this is measuring the area. And it's like, Oh, right, and so it's just one of these “aha” moments, where if people have been paying attention, it's like, oh, that's actually pretty cool, right? And then also in terms of the subject itself in relationship to high school, just really thinking about — because I get a lot of students who know all the rules, right? And they look at the anti-derivative with the integral sign and say “That's the integral.” Well, that's an anti-derivative, right?

EL: We’re not there yet.

AW: Yeah, that the anti-derivative and the integral are actually different. And so just having that conversation. And it also is a place to talk about the history of the subject and stuff like this.

EL: Yeah, I love it. And, at least for me, I feel like it's a slow burn kind of theorem. The first time you see it, you're like, “Okay, it's called the fundamental theorem of calculus. I guess some people think it's really important.” So that might be your Calculus I class. And then you see it again, maybe in an introductory real analysis class. And you're like, “Oh, there's more here.” And then you teach calculus, and you’re like, “Ohhhh!”

AW: Oh right, yes!

EL: Your brain explodes. You're like, “This is so cool!” And then your students are where you were several steps ago. And they’re like, “Okay, I guess it’s all right.”

AW: If I get the success rate of like, I've had three or four people go, “Whoa!” And it's like, okay, yeah, you're with me. And so this is out of hundreds of people.

EL: If you can get a few people that do that the first time they see it, that’s awesome.

AW: Yes. Yeah. No, it's been fulfilling for sure. And so then the proof itself, you know, it's also great, because then it culminates all of the theorems that you've been talking about beforehand. Depending on the proof, of course, but like, there's the intermediate value thereorem. There's the mean value theorem for integrals. There's unique continuity, at least in this version of it, in order for it to work. So yeah, it's great.

KK: So when you teach calculus, there's always two parts to the fundamental theorem. And so I like the one where the derivative of the integral is the function back, right? That's the fun, like for the mathematician in me, this is the fun part. Your students never remember that. Right? They always remember the other one, where we evaluate definite integrals by finding it the anti-derivative. So I was going to ask, if you had to pick one of the two, which one is your favorite?

AW: I mean, part of it is because at least the way that I've taught it, we're coming out of the mire of Riemann sums.

KK: Right.

AW: And so people have suffered through doing rectangles so much. And then I just get to say, “Oh, you don't have to do this anymore.” I mean, I've had a few students go, you know, now that we do — I always use the antiderivative of x squared on zero to 10, or the area under the curve of x squared from zero to 10. And like, sometimes I'll say, “Oh, that's 1000 over 3, right?” And then it was like, “Well, how did you do that so quickly?” We'll see. Right? But then, at the end, when I'll say, okay, and then we do another one again. And then I show how to apply the theorem, and people say, “Well, why didn't you just say that?” And then we have a great conversation there about how this isn't about the answer, that this is about a process and understanding the impact of mathematical ideas, that the theorem, as with all theorems, but this one is my favorite, is an expression of deep human intellect. And that if we reframe what theorems are, we get a chance to rehumanize mathematics. And so I think that too often in our math classes, and our math discourse, we remove the theorems from the humanity of the people who created them. And so people get deified, like Newton and Leibniz, but you know, these same people had to sit down and work hard at it and figure it out.

KK: No, it’s certainly a classic, but it is surprising how little it has come up on our podcast. It was the very first episode.

AW: Oh, okay.

KK: Yeah. Amie Wilkinson chose it. And then this will be episode 60-something.

AW: Okay. Yeah.

PH: Wow.

EL: We've talked, we've mentioned it in some other episodes. But it isn’t — I mean, there are just — I love this podcast, obviously, I keep doing it. And there are just so many types of theorems. And I love that you two picked different types. Yours, Aris, is one of these classics. Everyone who gets to a certain point in math has seen it, hopefully has appreciated it also. And Pamela, you picked one that none of us had ever heard of and made us say, “Whoa, that's so cool!” And people just have so many relationships with yours. And that's what this podcast is really about. Actually it's not about theorems. It's about human relationships with theorems and what makes humans enjoy these theorems. And so you picked two different ways that we enjoy theorems. And I just love that. So yeah.

KK: Yeah, that is what we're about here, actually. I mean, I mean, yeah, the theorem. But actually what I like most about our podcast, so let's toot our own horn here. We’re trying to humanize mathematics. I think everybody has this idea that mathematicians are a very monolithic bunch of weird people who just — well, in movies we’re always portrayed as either being insane, or just completely antisocial. And I mean, there’s some truth and every stereotype, I suppose, but we are people, and we love this thing. We think it's so cool. And sharing that with everyone is really what's so much fun.

AW: Yeah. And I think also that, for me, the theorem itself, and what it reveals, touches something that’s inside of us. There’s something about it, right? There’s the “Whoa” part that is that is indescribable and that I think really touches to our humanity. There is a eureka moment where you're just like, “Oh, I understand this now.” Or this connection is amazing, right? Yeah, it's indescribable.

KK: So we all agree these things are beautiful. So here's a question. Where do people lose this? I mean, I have a theory, but — because we've all had this experience, right? You're at a cocktail party and someone says, they find out you're a mathematician and like, oh, record scratch. I hate math. Okay.

AW: Yes, yes, yes.

PH: But I don’t think they hate math, though, Kevin.

KK: No, they don’t. Nobody hates math. Nobody hates math when they're a kid. That's exactly right. So I think when they say that they mean that the algebra caused them trouble. When x’s started showing up.

PH: I don't even think that's it.

KK: Okay. Good. Enlighten me because I want an answer to this that I can’t find.

PH: I don't think it's that people hate math or that they hate that the alphabet showed up all of a sudden in math that they hate how people have made them feel when they struggle with math. Math is an inanimate object. Math is not going out there and, like, punching people in the face. It's the way that people react to other people's math. Right? The second that you don't use the language in the way that somebody expects you to use it and you're trying to communicate properly and somebody says, “That’s not how you say it. It's not FOILing. It's called distributing!” Right? But you knew what I meant when I said FOIL the binomial!

KK: Of course I did.

PH: FOILing this gives you the middle term, blah, blah, right? So it’s again about human interactions. And if you make someone feel dumb, they'll never like what it is that they're trying to learn

AW: Amen to that. And they will conflate the two, which is what always happens.

PH: That’s exactly it!

AW: They will replace the experience with the subject itself, when in fact, they're talking about the experience. Yeah. So yeah, we've been working a lot about this in the last few years, Pamela and I and Dr. Michael Young, about when people say they hate mathematics, they’re really talking about their mathematical experience. So my immediate response to your question is just bad teaching. Let's just call it what it is.

PH: Right.

AW: I don't want to get on my podcast too early. We're recording later.

PH: We’re recording in a bit, yeah.

AW: But yeah, we're talking about people. And I say this as a loving critique of the greatest discipline in the history of people. I truly believe that, but I believe that the way we teach it, and the cultural norms we take with it, devalues people, and so I want every person who's listening to this now to then the next time they hear somebody say they hate it, look at them as an innocent person who had a bad mathematical experience. And then, because I see too often amongst my people in the community who say they hate having these conversations with people who say they hate it. And I think we need to return innocence back to that person. And say that this is not a person who hates you or even hates the subject. This is a hurt person. Yes, this is a person who has been damaged in our subject. And by the way, I go much farther than that. It's our responsibility to try and help repair that because this person is going to impact their cousin, their child, their relative, by bringing this hate of the subject, when in fact, it doesn't have anything to do with the subject.

EL: Yeah. It’s about the traumatic experiences. And actually, I think mathematicians often have a bit of a persecution complex and think this is the only place where people have this reaction. But one of my hobbies is singing, and in particular, singing with large groups of untrained people who are just singing because we love singing. And the baggage that people bring to singing is similar. I’m not saying it's entirely the same, but people have been made to feel like their voice isn't good enough.

AW: Yes.

EL: They have this trauma associated with trying to go out and do this sometimes. Obviously a lot of people love to sing and will do it in public. A lot of people love to sing at home and are scared of doing it in public because they're worried about, you know, their fourth grade music teacher, who told them to sing quieter, or whatever happened.

PH: Yes,

AW: That’s right. That's right. And the connection is similar, because what are we saying? We're saying that if you don't hit this right note, then it doesn't count. As opposed to if you don't get the answer seven, then we're not going to value you because the answer is seven, right? Because we have this obsession with the correct answer in mathematics. Right.

PH: And not only that, but also doing it fast.

AW: Yes.

PH: You and I have talked about this before, that — maybe in singing, this is different. I'm not sure. I definitely can relate to the trauma of never singing out loud in public. But is there this same sentiment that you must get it perfect the first time and pretend that it doesn't actually take you hours of training?

EL: I mean, it comes up. There’s definitely, people can feel more valued if they're quicker at picking things up than others, although, you know, it's not the same. There's no isomorphism between these two, I don't know, to bring a little silly math lingo in. But there definitely, there are a lot of similarities, and I think about this a lot, because two things I love in my life are math and singing with my friends. And, you know, I just see these relationships. But yeah, I could go on a whole rant, and I want to not do that.

AW: No, no, no, I appreciate you bringing it up.

EL: But I think it's a really interesting correspondence.

AW: And then the final one is that, you know, in the music space, what is it that we really should be trying to do, value everybody's voice? And in mathematics, we should be valuing everybody's contribution. Right? This is all we're saying. And what does each discipline look like when we value people's voices, no matter where they are on the keys? And we value everyone's contribution to trying to solve a problem.

EL: Yeah, yeah. And how can we help people, you know, grow in the way they want to? You can say, like, “Oh, I like I am not as good a sight reader as I want to be. How can I get better?” How can we help people grow in that way without feeling cut down?

AW: Yeah.

EL: Yeah, it is true for math, too. Yeah. It's just, everything is connected. Woo.

AW: Yes. But you know, we've been talking about, you know, these human relationships we all have with math. And so another part of our podcast that we love is forcing you to do make one more human connection between math and something else with the pairing. So what goes well, Pamela, with this theorem about uniquely writing the numbers in terms of the Fibonacci sequence?

PH: So I was trying to think about my favorite food, and when it was the epitome of perfection, and I came up with, okay, so if we're going to pair it with something to drink, I was like, I want to think about happy moments. Because this feels like a happy theorem. And so I want to go with some champagne.

KK: Okay.

PH: Okay, I was like, “We're gonna go fancy with it!” But then for food, I'm thinking about, oh, this is hilarious. So I went to a conference in Colombia, we visited Tayrona which is a beach in Colombia. And on the side of the beach, I paid to have ceviche, fresh ceviche. And I've never been happier eating anything in my life. And so I imagine myself learning Zeckendorf’s theorem at the beach in Tayrona in Colombia, with some champagne and the ceviche.

EL: Oh man.

AW: Wow.

PH: Beat that, Aris! Beat. That.

AW: There’s no way. So wait, so I want to make sure I understand. So is this while you're reading the proof? Or is this while you’re—

PH: This is like the gold standard. If I were to put all the, like, uniqueness of my favorite food, my favorite drink and my favorite theorem, I would put them in a location which is Tayrona in Colombia, at the beach, eating ceviche sipping on some champagne, learning Zeckendorf’s theorem.

AW: Okay.

KK: Is this the Pacific coast or the Caribbean?

PH: You’re asking questions I should know the answer to, and I believe it’s the Caribbean.

KK: Okay.

PH: Nobody Google that. [Editor’s note: I Googled that. It is the Caribbean.] I have no idea where they took me in Colombia. I just went.

KK: Sure.

EL: Yeah, that sounds so lovely as I look out of my window where there's snow and mud from some melted snow.

PH: Ditto.

AW: So I yeah, I think for the fundamental theorem of calculus, I think this is something that's just classic. Like you're just having a nice pizza and some ginger ale. You're just sitting down and you're enjoying something hopefully that everybody likes and that connects with everybody, that everybody hopefully sees that they get to get that far. So yeah, I mean, my daughter recently — I didn't realize this. She's 9. And we were talking. We visited my aunt in DC. My aunt raised me. And my daughter was much younger at that time, but then every time she thinks about going to visit, she thinks about the ginger ale that my aunt got her because that was the only time she ever got ginger ale. So she’s like, “Oh, I like your aunt, Daddy, because you know, I had ginger ale there.” And I was like, Oh, I should have ginger ale more often. So that made me think of that.

PH: That’s adorable.

EL: I can really relate to that feeling of, like, when you're a kid, something that is totally normal for someone else isn't what's normal for your family. So you think it's a super special thing.

AW: It’s amazing.

EL: I think I had this with, like, Rice-a-Roni or something at my aunt's house, and my mom didn't use Rice-a-Roni, and I was like, “Whoa, Mom, you should see if you can find Rice-a-Roni.”

PH: Amazing.

EL: She was like, “Yeah, they have Rice-a-Roni here.”

AW: Rice-a-Roni’s the best.

KK: I haven't had that in years. I should go get some.

AW: Me either. All right.

PH: That’s how you know you made it.

KK: You know what? You know, single mom and all that, and I lived on Kraft macaroni and cheese when I was a kid. And yeah, you would think I don't like it any more. But, aw man.

PH: Listen, that thing is delicious. So good.

AW: I was about to say.

EL: They know what they’re doing. Yeah. Well, that's great. And I mean, pizza is my favorite food. As great as ceviche on the beach sounds, pizza, just, when you come down to it, it's my favorite food. And so I love that you paired the fundamental theorem of calculus with my favorite food.

KK: So I'm curious, there must be a human who doesn't like pizza, but have you ever met one? I've never met one.

PH: No.

EL: I know people who don't like cheese. And cheese is not — I mean, to me cheese is essential to the pizza experience, but you can definitely do a pizza without cheese.

AW: Yeah. No, my wife also always says that for her it's about the sauce. So I think she might be a person who can get rid of the cheese if the sauce is right. Yeah.

KK: But the crust better be good too.

AW: Of course, of course. It's a full package here.

EL: But okay, so you say that, but on the other hand, I would say that bad pizza is still really good.

KK: Sure.

EL: I mean, you can have pizza that you're like, “I wish I didn't eat that.” But I have very rarely in my life encountered a slice of pizza that was like, “Oh, I wish I wish I had done something else other than eat that pizza.”

AW: It’s actually a pretty unbeatable combination, right? Tomato sauce, cheese and bread.

PH: Yeah. It kind of can't go wrong. Yeah.

KK: When I was when I was in college, there was a place in town. It was called Crusty’s Pizza, and I don't think it exists anymore. And it was decidedly awful. But we still got it because it was cheap. So we would occasionally splurge on the good pizza. But you could get a Crusty’s pie for like five bucks.

AW: Absolutely.

KK: This is dating myself. But yeah, absolutely. Always. All right, so we've got we've got theorems, we’ve got pairings. You've plugged your podcast pretty well, although you can talk about it more if you'd like. Anything else that either of you want to plug, websites, the Twitter?

EL: Yeah, but can you say a little more about the book that you mentioned?

AW: Yeah, the book is a series of dialogues that was an extension of an AMS webinar series that we gave about advocating for students of color mathematics. And so we had just decided, you know, there was so much momentum, we had hundreds of people coming every time to the four-part series. And so we were like, you know, we've gotten to a place where we've given all these talks, and then you give talks, create momentum, and then it just ends. And we're just like, you know, what, not this time. Let's create a product out of this. And so, we decided quickly to get the book together, just answering some of the unanswered questions from the webinar series. So we had the motivation, in terms of answering their questions. And yeah, we got it together. And it was an honor. So it really is just a list of our dialogues, a transcription of our dialogues, answering some of the unanswered questions from that webinar series. And so it's gotten some really good reviews, and people are using it in their departments. And so it's been fantastic so far.

PH: Yeah, I think that's that's the part that I'm really enjoying, getting the emails from people who have purchased the book. And so maybe I should say the full title, so it is Asked and Answered: Dialogues On Advocating For Students of Color in Mathematics. And the things that I hear from folks who have purchased the book — so thank you all so much for the support — is that they didn't expect that there is part of a workbook involved in the book. So it isn't just Aris and I going back and forth at telling you things. I mean, a lot of that there is, that is part of the content. But there's also a piece about doing some pre-reflection before we start hearing some of the dialogue that we have, and then also the post part of it. So how are you going to change? And how are you going to be a better advocate for students of color in mathematics? And so it leaves the reader with really a set of tools to come back to time and time again. That's really what I see as a benefit of the book. And people are purchasing it as a department to actually hold some kind of book club and really think about what of the things that we suggest that professors implement in their department, in their classrooms, in their institutions, what they can actually do. And so the reception has been really wonderful. And I'm just super thankful that people purchase the book, and we're supporting our future work.

EL: Yeah. And can you also mention, is it minoritymath.org, the website that hosts Mathematically Uncensored?

AW: That’s correct. That's right. So yeah, that's the home of the podcast. And that's a place where we're trying to create voices for underrepresented minorities in the mathematical sciences. And so you can go there not just for the podcast, but for other content as well that centers around that experience.

KK: Okay.

EL: Fantastic. Thank you so much for joining us.

KK: Yeah.

EL: I had a blast.

PH: Thank you.

KK: This was a really good time.

EL: Yeah. Over lunch today, I'm going to be writing down numbers and writing them in terms of Fibonacci numbers. It’s great.

AW: It will be fantastic.

PH: Awesome.

AW: Thanks.

PH: Bye, everyone.

KK: Thanks, guys.

On this very special episode, we had not one but two guests, Pamela Harris from Williams College and Aris Winger from Georgia Gwinnett College, to talk about their podcast, Mathematically Uncensored, and of course their favorite theorems. Here are some links you might be interested in as you listen to the episode.

Harris's website

Winger's profile on Mathematically Gifted and Black

Mathematically Uncensored, the podcast they cohost
Minoritymath.org, the Center for Minorities in the Mathematical Science, a website with information and resources for people of color in mathematics
Asked and Answered: Dialogues On Advocating For Students of Color in Mathematics, their book
Zeckendorf's theorem and a biography of Edouard Zeckendorf

Jean Leray, a French mathematician who worked on spectral sequences as a prisoner of war
Olivier Messiaen's Quartet for the End of Time, composed when he was a prisoner of war
A paper generalizing the Zeckendorf theorem by Harris and coauthors
Our episode with Amie Wilkinson, who also chose the Fundamental Theorem of Calculus, making it 2 for 2 among mathematicians with the initials AW.

Episode 63 - Lily Khadjavi

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, the podcast from 2021. I don't know why I said that, just, it's a math podcast, and it is currently being taped in 2021. I'm your host Evelyn Lamb. I'm a freelance math writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. No, look, it's important to say it's 2021 because 2020 lasted for about six years. It was—I couldn't wait for 2020 to be over. I don't think 2021 is much better yet. It's January 5. I'll leave our listeners to figure out what's going on right now that might be disturbing. And, and yeah, but anyway, no, happy new year. And I had a very nice holiday. My son has been home for nine months now. He's going to go back to school finally next month to finish up his senior year in college. And I did nothing for a week. I mean, like when I say nothing, I mean nothing. Just get up, watch some TV, like we’re watching old reruns of Frasier, like this is the nothing levels I saw. It was fantastic.

EL: Very nice.

KK: How about you guys? Did you have a nice holiday?

EL: Um, I had a bad bike accident right before Christmas. So I had some enforced rest. But I'm mostly better now. I have gotten on my bike a couple times, and nothing terrible has happened. So still a little more anxious than usual on the bike. We were taking a ride yesterday and I could tell I was just like, not angry, but just, you know, nervous and worried. And it's just like, Okay, I'm just at the scene of the trauma, which is my bike seat, and getting over it. But I hope I will continue to not fall off my bike and keep going.

KK: That’s the only thing to do. Back in my competitive cycling days when I was a postdoc, I had some pretty nasty crashes. But yeah, you just get back on. What else are you going to do? So anyway, enough of that. Let’s talk math.

EL: Yes. And today, we are very happy to welcome Lily Khadjavi to the show. Hi, will you introduce yourself and tell us a little bit about yourself?

Lily Khadjavi: Hi. Oh, thanks, Evelyn. It's so great to be here. I'm Lily Khadjavi, as you said. I'm a professor of mathematics at Loyola Marymount University, which is in Los Angeles, California. I'm a number theorist by training, but I'd say that I'm lucky to have taken some other mathematical journeys, especially since graduate school, and I don't know, for example, this past year, maybe my biggest excitement is I was lucky to be appointed to a state board in California. So by the Attorney General, Xavier Becerra, to be appointed to an advisory board looking at policing and law enforcement and the issue of profiling. And so that's an issue that's very important to me. And it was an unexpected mathematical journey.

EL: Yeah.

LK: If you’d asked me 20 years ago, what would I be up to, I might not have thought of that. And I've taken many a bike spill in my day, so I could feel some nice affinity being here today. You’ve just got to get back on and be careful, of course.

EL: Yeah. And that that must be an especially important issue in LA, because I know the LAPD has been the subject of some, I guess, investigations and inquiries into their practices and things like that.

LK: That's exactly right. And over the years, it was under a consent decree, so an agreement between the US Department of Justice and the City of Los Angeles, with many aspects monitoring police practice. And actually, some of that included data collection efforts looking at traffic stops. And that, combined with teaching a statistics course, is what really gave me a window more into policing practice, into problems that where I wanted authentic engagement for my students with the real world and took me on, maybe I'll say unexpected journeys to law conferences and elsewhere, as I started to learn more about the issues, the ways that as mathematicians, we can bring tools to bear on on these social questions too.

EL: Yeah, very cool.

KK: Yeah.

EL: So what is your favorite theorem? And I know that's an unfair question, but I will ask it anyway. And then, you know, you can run away with it.

LK: Yeah. I know this podcast is not visual, but I'm already kind of smiling in a terrified way because I found this question so difficult, really an impossible task, because I thought it's like asking me when my favorite song—I don't know, do you have a favorite song?

EL: That is hard to say. If you asked me, I would start listing things. I would not, probably, be able to tell you one thing.

LK: What do you think, Kevin?

KK: I, uh, Taxman?

LK. Okay, I thought you would name the opening the music for the podcast as a favorite too.

EL: Oh, yeah.

LK: You know, shout out to that.

KK: I do like that. But now, you know, maybe What Is Life by George Harrison? Single?

LK: Oh, yeah. Okay, well, maybe I'll count that as listing, which is what Evelyn started to do. Because it's very, difficult.

KK: It is.

LK: You know, I was really wrestling with this. And it got me kind of thinking about why do we like certain theorems. I think I pivoted to what Evelyn said. I started wanting to make lists. And of course, it's fun to talk about things that are new to everyone. And, you know, it's been a remarkable podcast, and lots of people have staked out, I mean, they've grabbed those beautiful favorite theorems. But I started thinking, could you have a taxonomy? I really saw a taxonomy of theorems. Not by discipline. So not a topological statement or an analytic proof, but by how mathematicians feel about them, or the aesthetic of them. And so my first you know, category had to be sort of the great workhorses, like those theorems that get so much done, but they also they never cease to amaze you. And I mean, it’s hard not to point right away to the fundamental theorem of calculus, and I think maybe in your very first episode. That's right, that might be what?

KK: Yeah, Amie Wilkinson.

EL: Yes, Amie Wilkinson just came in and snatched that one. Although as everyone knows, we do double theorems, you know, we don't have a rule that you can't use the same theorem again.

LK: No, because that's one we use again and again and again. You know, even this past semester, I was teaching multivariable calculus. And you know, we have this march through line integrals, double, triple integrals, and we build, of course, to Green’s theorem, Stokes’ theorem, the divergence theorem. So these main theorems in calculus that the machinery is heavy enough for the students that even if I'm trying to put them in a context where, “Oh, this is really all the fundamental theorem of calculus,” I think that gets obscured obscured for students first trying to get their head around these theorems. Even though you relate them, you say, Oh, but they've got the boundary of this—maybe endpoints of a curve or some other surface boundary, and you're relating it as the relationship between differentiation integration, and it's so it's beautiful stuff. But I think I'm not convinced my students thought of it as the same theorem, even if I tried to emphasize this perspective. But still, they, all of us can be blown away by how powerful the theorem is in all of its incarnations. And so that's a great workhorse. So we don't have to talk at length about that one. It's been here before, but you know, you just have to tip your hat to that one. But I was wondering, are there other great workhorses something you put in that in that category?

KK: So I argue—I mean, so you mentioned the fundamental theorem—the workhorse there is actually the mean value theorem.

LK: Hmm.

KK: Because the fundamental theorem, at least for one variable, is almost a trivial corollary of the mean value theorem. And I didn't appreciate that until I taught that sort of undergraduate analysis course for the first time. And I said, “Wait a minute.” And then I sort of came up with this joke, I'm actually going to write a book. It's like a “Where's Waldo” style thing: Where's the mean value theorem? Because in every proof, it seemed like, Well, wait a minute, by the mean value theorem, I can pull this point out. Or there is one, I don't know where it is, but it's in there somewhere. So I really like that one.

LK: That’s a really great perspective. I also will say that I did not happen on that feeling until teaching analysis for the first time, of course, versus, you know, for seeing these theorems or learning about them, and even learning them in analysis, not just using them in calculus. Know, that reminds me that it wasn't till grad school, maybe taking a differentiable manifolds class, and that's not really my area. But seeing, Oh, you can define a wedge product, you can define these things in a certain way. Oh, they really are literally all the same theorem. But I like this perspective, maybe that would have been a way to convince my students a little bit more, to kind of point to the mean value theorem, because it would put them on more familiar turf too. I really like that. Yeah. Are there other workhorses?

EL: So the first one that came to my mind was classification of surfaces, in topology, of like, you know, the fact that you can do that—I feel like I it's like so internalized to me now. And yeah, I don't know, that for some reason that came to mind, but it's been a long time since I did research and was keeping up with, you know, proving things. So yeah, it’s—but yeah, I think I would say that anyway.

KK: Yeah. And I would sort of think anything with fundamental in its name right.

LK: Yeah, I was thinking that.

KK: So the fundamental theorem of arithmetic, okay, so that you can factor integers as products of primes, or the fundamental theorem of algebra, that every polynomial with complex coefficients has a root. But then more obscure things like the fundamental theorem of algebraic K-theory. You guys know that one?

LK: That one, I'm afraid does not trip off my tongue.

KK: All it is, is it's a little bit weird. It just says that the K-theory of if you have a ring, maybe it needs to be regular, that if you look at the K-theory of the ring, and the K-theory of a polynomial ring in one variable over it, they're the same. And the topological idea of that is that, you know, it's a contractibility argument somehow. And so it's fundamental in that way.

LK: These are great workhorses. Yeah. And also, Evelyn, you mentioned the classification, like these results are just so fundamental. So in whether they have fundamental in the name or not, they are.

EL: Like, naming it fundamental, it's almost like cheating that point. Or, maybe not cheating, maybe stealing everyone else's thunder. It’s like, “No, I already told you that this is the fundamental theorem of this.”

LK: My poor students, whenever I want them to conjure up the name and think of something that way, I make the same corny joke. I'm like, “It's time to put the fun back into…” and they’re like, “Ugh, now she's saying fundamental again.” So yeah, I was thinking, too, that in different fields, we reach back, even as we're doing different things in our own work, back to those disciplines that we were sort of steeped in. And I think for topologists, there are so many great theorems to reach to.

KK: Sure.

LK: But I was thinking even like the central limit theorem in statistics and probability, so this idea that you could have any kind of probability distribution—start with any distribution at all—but then when you start to look at samples, when the samples are large enough, that the mean is approximated by a normal distribution. That somehow never ceases to amaze me in the way that the fundamental theorem of calculus, too. Like, “Oh, this is a really beautiful result!” But it's also a workhorse. There are so many questions in statistics and probability that you can get at by gleaning information from the standard normal distribution. So maybe I’d put that into a workhorse category.

KK: Sure.

EL: Actually, Heine-Borel theorem, maybe could be kind of a workhorse, although I'm sort of waiting for for you to say that it's actually the mean value theorem too.

KK: No, it's just, it's just that, you know, compact sets are closed and bounded. That's it. Right?

EL: Yeah. Yeah, actually, yeah, that, once again, is such a workhorse that it's often the definition that people learn of compactness.

LK: That’s right.

EL: Like the first time they see it. Or, like such an important theorem that it it almost becomes a definition. Actually the Pythagorean theorem, in that case, is almost a definition.

KK: Sure.

EL: Slash how to measure distance in the Euclidean plane.

LK: Yeah, that's a good example. So maybe now we have so many workhorses, well, another category I was thinking of — it's beautiful stuff. I was thinking of those theorems where the subtlety of the situation kind of sneaks up on you. So maybe you hear the statement, and you kind of even think, “Oh yeah, I believe that,” like the Jordan curve theorem, I think you had a guest speak about this, too. So this, you know, idea of a simple closed curve. So you just draw it in the plane, there's an inside, and it divides the plane into an inside and outside. And I kind of really remember—I can't tell you what day of the week it was—but I remember the first time this came up in a class, and I thought, “Yeah.” But then we started thinking about how would you go about proving something like this, or even just being shown, someone drawing, a wild enough crazy curve, where suddenly you can't just eyeball it and immediately see what's inside and what's outside. So I don't know what this category or set of theorems should be, but the subtlety sneaks up on you even though statement seems reasonable.

EL: “I can't believe I have to prove this.” Maybe that’s slightly different. Well, what I mean is like, I can't believe this is a—It seems so intuitive that understanding that there is something to prove is a challenge, in addition to then proving it.

LK: Yeah. And maybe you can't even prove it—Well, how about the four color theorem? So this map coloring theorem, this idea that the four colors suffice, so if you have states or counties or whatever regions, you want to make your map of, that if they share a common edge boundary, then use different colors, that four colors is enough. I don’t know, has a human being ever proven that? My understanding is that it took computing power.

KK: It’s been verified.

EL: I think they’ve reduced the number of cases, also, that have to be done from the initial proof, but I still think it's not a human-producible proof.

KK: That’s right. But I think Tom Hales actually verified the proof using one of these proving software things. So I mean, yeah, but that was controversial.

LK: That brings up a neat question about what constitutes proof in this day and age. I've seen interesting talks about statements where, or journals where something's given as this: “Okay, here's a theorem. And here's the paper that's been refereed.” And then later, oh, here's something that contradicts it. And people are left in a sort of limbo. Well, that's another discussion, things unproven, un-theorems, I don't know. Well, anyway, in this category, that's going to help the subtlety of the situation sneaks up on you. If I start coloring maps, testing things out, after a while, I’d say, “Oh, there's a lot to this.” But the statement itself has an elegant simplicity.

KK: Well, it's not easy. So I curated a math and art exhibition at our local art museum, in the Before Times, and one of the pieces I chose was by a Mexican artist, and it's called Figuras Constructivas. And it was just two people standing there talking to each other, but it was sort of done in this—we’ve all done, you probably when you were a kid—you took a black crayon and scribbled all over a page, and then you fill in the various regions with different colors, right? It reminded me of that. And the artist used five colors. And so when I was talking about this to the to the docents, I said, “Well, why don't we create an activity for patrons to four-color this map?” So they did, they created it, because it was just a map. And they did it, and the docents were just blown away by how difficult it was to do a four-coloring. You know, five colors is fairly easy. But four was a real challenge.

LK: That sounds really fun. And what a great example of math and art coming coming together. And my understanding of the history of this, too, is that the five-color theorem was proved not just before four colors, but was kind of doable in the sense that

EL: I think it’s just not that hard.

LK: Certainly not that hard in the sense of firing up the computers and whatever else has done.

KK: Needing a supercomputer in 1976.

LK: Which is basically my phone, maybe. Well, I had another category mind, which is, theorems where the proofs are just so darn cute.

KK: Okay.

LK: And so what I was thinking of—I tried to have an example for each of these—which was the reals being uncountable.

EL: Yeah.

LK: And I think you've had guests talk about this. And you know, like a diagonalization argument, like say, just look at the reals only from 0 to 1. And suppose you claim that that is a countable set. Okay, go ahead and list them in order, in whatever ordering you've got for countability. And then you can construct a new element by whatever was in the first place of your first element, do something different in your first place, whatever was in the second place of the second element, do something different in your second place of your new element, and so on down the line. So you go along the diagonal, if you had listed these and so this, I don't know my crude description of a diagonalization argument, that you can construct a new element that wasn't in your original set and so contradict the countability. I don't know, I thought that's really cute.

EL: Yeah. And that was probably the first theorem that really knocked my socks off.

KK: Mm hmm. It's definitely a greatest hit on our show.

EL: Yeah.

LK: So I guess that’s right. We've had a Greatest Hits show, so I don't know, this taxonomies kind of disintegrating, like “Workhorses,” “Just so darn cute,” “Situation sneaks up on you.” But yeah, I don't know if there are others that fit into the “Just so darn cute.” That was the one that came to mind because I kind of wanted it on my favorites, and then I was like, “Oh, someone's already talked about this on the show.”

KK: Well, I really like—so I'm a topologist. And I really like the theorem that there are only four division algebras over the reals. So the reals, the complexes, the quaternions and the octonians. And it's a topological proof. Well, I mean, there's probably an algebraic proof. But my favorite proof is topological. So I don't know if it's cute.

EL: That isn't what you'd expect the proof of that to be, for sure.

KK: No. And it's it's sort of—I'm looking through it. So I taught this course last year, and I'm trying to remember the exact way the proof goes, not that our listeners really want to hear it. But it involves cohomology. And it's really pretty remarkable how this actually works. Oh, here it is. Oh, yeah. So it involves, it involves the cohomology rings of real projective spaces. And so if you had one of these division algebras, you look at some certain maps on cohomology, and you sort of realize that things can't happen. So I think that's very, well, I don’t know if it’s cute, but it's a pretty awesome application of something that we spend a lot of time on.

LK: Yeah, it’s so neat when a different field. So you know, we have these silos, historically: algebra, topology, and so on. So the idea that a topological proof gives you this algebraic result is already a delight, but then that's heavy machinery. That's sounds like a really neat.

KK: Or fundamental theorem of algebra, right?

LK: Well, that's when I was thinking when you started saying saying, “Oh, there's a topological proof.” I started thinking, “Oh, fundamental theorem of algebra.” You know, fire up your complex analysis. And yeah, neat stuff. Yeah.

EL: Well, and there's this proof of the Pythagorean theorem that I have seen attributed to Albert Einstein, I think, that has to do—Steve Strogatz wrote, I think, an article for The New Yorker about it. So Oh, yeah, listening to my bad explanation of it semi-remembered from several years ago, you can go read it. But it has to do basically with scaling. And it's a kind of a surprising way to approach that statement.

KK: I think it was in the New York Times [editor’s note: Evelyn was right, it’s the New Yorker! [note to the editor’s note: Evelyn is the editor of this transcript]], or it's also in his book, The Joy of X, I think it's in there too. And yeah, I do sort of vaguely remember this, it is very clever.

it's a nice one to record.

LK: Yeah, this makes me want to swing back to many things. It's also reminding me, so here we are in pandemic times. And so at the university I'm at, we're not spending time in the department, but you reminded me that when I wander around the department, sometimes we have students’ projects, or work from previous semesters, up here and there, along with other posters. And I'll look at something and say, “Oh, I haven’t thought about Pythagorean Theorem from that context, or in that way.” So just different representations of these. So maybe there should be a category where there are so many proofs that you can reach to, and they're each delightful in their own way, or people could you could start to ask people what's your favorite proof instead of a favorite theorem, maybe.

KK: I think we did that with Ken Ribet because he did the infinitude of primes. He gave us at least three proofs.

LK: And I think three pairings to boot. Yeah. Nice. I'm wondering if another, so there was the “so darn cute,” how about something where the simplicity of the statement draws you in, but then the method of the proof may just open up all kinds of other problems or techniques. So in other words, I guess what I'm saying is some theorems, we really love the result of the theorem. Maybe the Fundamental Theorem of Calculus. That result itself is so useful. But on the other hand, Fermat’s Last Theorem, I don't know if anyone's even pointed to that on the show, but something in number theory where the statement was—I mean, this is how I got suckered into number theory. That's what I would say. So you have this statement. You mentioned the Pythagorean theorem, so this idea that, that you could find numbers where the sum of two squares is itself a square, like three squared plus four squared equals five squared, but what if you had cubes instead, could you find a cubed plus b cubed equals c cubed, or any a to the n plus b to the n equals c to the n. And, you know, that's a statement that, although the machinery of number theory that's developed to ultimately prove this is so technical, and involves elliptic curves and modularity, all kinds of neat stuff, but that the statement was very simple. And of course, at some level, then it wasn't even just proving that statement. It was the tools and techniques we can develop from that. But I remember telling a roommate in college about, “Oh, there's this theorem, it's not even proven.” So that was a question too. Why are we calling this a theorem? So back in the day, that was not a theorem, but it was still called Fermat’s Last Theorem. And in telling, you know, relating the story that Fermat was writing in the margin of his I don't know Arithmetica or something in the 1600s. And that he said, “I had the most delightful proof for this, but the margin is too small to contain it.” And my roommate’s first reaction actually was “Has anyone looked through all of his papers to find the proof?” And that was nice, because, you know, coming from a different discipline, studying English and history and so on. Because to me that wasn't the first reaction. It was like, oh, if Fermat had a proof, can we figure it out too? Or can we figure out what he—maybe he had something, but what mistake might he have made? Because there's more to this one perhaps. But anyway, the category was “statements that draw you in with their simplicity.” Maybe the four-color theorem should have landed here.

EL: Yeah.

LK: I don’t know.

EL: Yeah, draw you in. It's kind of—I don't know if this is maybe a bad analogy to draw, but kind of catfishing. Yeah. There’s just this nice, well-behaved statement. And oh, yeah, now it's a giant mess to prove. Actually, maybe like the Jordan curve theorem.

LK: Yeah, maybe a lot of these end up there. Then there's that way, though, if something's finally— sometimes when you finally prove something, you're like, “Oh, why didn't I think of that earlier?” I don't know that Fermat will ever land there for me, but maybe the Jordan curve, maybe there are aspects of some of these that you just come to a different understanding on the other side of the hill.

EL: Yeah. So I think if I were doing this taxonomy, one of my categories—which is probably not a good category, but I think I would have a sentimental attachment to it and be unable to get rid of it—would be like, theorems with weird numbers in them or, or really big numbers in them, like the one that we talked about with Laura Taalman, where there’s this absurd bound for the number of Reidemeister moves you have to do for knots. Like there are some theorems where like, you've got some weirdness, it's like, oh, yeah, this theorem is, works for everything except the number 128. And it's just like, theorems with weird numbers in them, or weird numbers in their proofs, I think would be one of mine. Or, like the proof of the ternary Goldbach conjecture several years ago, which I only remember because I wrote about it, is basically proving that it works up through a certain very large number of just individual cases, and then having some argument that works above 10 to the some large number, and like, that's just a little funny. It's like, “Oh, yeah, we checked the first 12 quadrillion. And then once we did that, we were made in the shade.” And I don't know, I think I think that goes a long way with me.

KK: How about theorems with silly names? Like, like the ham sandwich theorem.

LK: I think the topologists corner the market on this, right? Yeah? No? Maybe?

KK: We really do.

LK: Yeah, the ham sandwich. No, I like so we need to find one that's like, unusual cases, or a funny number comes up and it has a funny name to boot. I love these categories. Well, how about how about something where the statement might surprise the casual listener. So in other words, like, the Brouwer fixed-point theorem, so when I’m I chatting with my students, I say, “Oh, you toss a map of California onto the table (because I'm in California) and there's some point on the map that's lying above its point in the real world.” And then oh, I can do it all over again, toss it again, it doesn't land the same way. And then, and they start to realize, oh, there's something going on here. But I don't know if that's surprising. Maybe my students are a captive audience. I say surprising to the casual listener. Maybe it's surprising to the captive audience. I don't know.

EL: Yeah, well, that's definitely like a one where the theorem doesn't seem surprising, or, you know, the theorem doesn't seem that strange. And then it has these applications or examples that it gives you that you're like, oh, wow, like that makes you think like, for me, it's always the weather. What is it? That there are two antipodal points on the earth with the same, you know, wind speed, or at any given time or temperature, whatever the thing is you want to measure?

KK: The Borsuk-Ulam theorem.

EL: Maybe the same of both? I don't remember how many dimensions you get.

KK: Well, you could do it in every dimension. So yeah, it's the Borsuk-Ulam theorem, which is that a map from the n-sphere into R^n has to send a pair of antipodal points at the same point. Right.

EL: So the theorem, when you read it, it doesn’t seem like it has anything weird going on. And then when you actually do it, you're like, “Whoa, that's a little weird.”

LK: Oh yeah, I like that. Maybe that's true, so many of the things we we look at. So I guess I realized, as I was thinking about these, I was tipping towards theorems where there's also some kind of analogy or way to convey it without the technical details. Certainly, if the category is to draw in the casual listener, or to sucker someone in without the technical machinery. Yeah, so I don't know what would be next in the taxonomy of theorems. Do you have other ideas?

EL: I’m not sure. Yeah, I feel like I’d need to sit down for a little bit. Actually first go through our archives and like look at the theorems that people have picked, and see where I think they would land.

LK: I had a funny taxonomy category that's very narrow, but it could be “guess that theorem.” But I was thinking theorems with cute names or interesting funny names that have also been proven in popular films.

KK: Oh, the snake lemma.

LK: Ding-ding-ding, we have a winner.

KK: You know, don’t pin me down on what the movie is. I can't remember.

EL: I think t's called It’s My Turn.

KK: That’s it.

LK: Wow, the dynamic duo here has exactly. And I have to admit, when I was thinking of it, I was like, “I don’t remember the movie.” And I had to look it up. But anyway, algebra comes to the rescue.

EL: Yeah, I’ve seen that scene from it, but I've never seen the rest of the movie for sure.

KK: Has anybody?

LK: As mathematicians, maybe we should.

EL: I don’t even know if it’s on DVD. It might might never have been popular enough to get to the new format.

KK: And isn’t that the last time that there's any math in the movie? Like it's this opening scene, and she proves the theorem, and then that's it? Never any more?

LK: So it's really a tragedy, that film. But no, they say this is the year that people said, Oh, they watched all of Netflix. I don't know if that's possible. So this is the year, then, to reach out to expand. Or maybe if we rise up and request more streaming options for the movie. I would like to show my students students that. Yeah, but I also admit, I haven’t seen the film.

Maybe a big core category we're missing is those theorems that really bridge different areas or topics. So Kevin, you give an example of a statement that could be algebraic, but it's proven topologically. But then I was thinking, are there theorems that kind of point to a dictionary between areas? And I only had one little example in mind, but maybe I'll call it my little unsung hero, a theorem that won't be as familiar to folks, but I was thinking of something called Belyi’s theorem, so not as well known as the others, perhaps, but that number theorists and arithmetic geometers are really interested in. And then actually, I went ahead and printed out ahead of time, these quotes of Grothendieck, who was so struck when this theorem was announced or proven because he'd been thinking along these lines, but was surprised at the simplicity of their proof. But my French is not very good, so I'm not going to read anything in French. But I don't know if you want to take a moment to talk about this theorem.

KK: Sure.

EL: Yeah.

KK: So what's the statement?

LK: Yeah, so maybe I'll say en route to the statement that number theorists and arithmetic geometers are interested in ramification, but I'm maybe I'm going to describe things in terms of covering maps, and whether you have branching over a covering so. So like, if you had a Riemann surface, you're mapping to Riemann surface, and you had a covering map, you might expect, okay, for every point down below, you'd expect the same number of preimages, or for every neighborhood down below, the same number of neighborhoods, if it's a degree D map, maybe a D-fold cover. And in fact, I remember my advisor first describing this to me by saying, if you had a pancake down below, you'd have D pancakes up above. And it really stuck in my head, frankly, because he was so precise and mathematical in his language at every moment, this was one of the most informal things I ever heard him say. Maybe he was hungry at the moment, he was thinking about pancakes. So as a concrete example where something different could happen, suppose I was mapping to the Riemann sphere, and I suppose I had a map, like I don't know, take a number and cube it, like x cubed, and started asking what kind of preimages points have. For example, x cubed equals 1, there are three roots of unity that map to 1, but something different is happening at zero, so only zero maps to zero. There's no other value that when you cube it, gives you zero. So now we no longer have, instead of a cover, maybe I'll say we have a cover, except at finitely many points. So somehow zero, and in that case, infinity, there's some point at infinity that behaves differently, but everything else has three distinct preimages. And maybe just to make a picture, let's take the interval from 0 to 1. So a little line segment, the real interval, and we could ask what its preimage looks like. And so above 1, there are three points up above. There are three roots of unity that map to 1, and on the other hand 0 was the only point that mapped to zero. And for the rest of the interval, all of those points have three preimages. So you could draw, maybe I'm picturing now a little graph on my original surface that's got a single vertex, say, at zero, and then three segments going out for each of the preimages of the real line, and ending at these three roots of unity, ending at the preimages of 1. And so now I'm not even thinking very precisely about what it looks like. I'm just picturing a graph. So I’m not worrying about how beautiful my drawing is. I just have one vertex over zero and then three branches. So what number theorists describe in terms of ramification, in this setting we might think of as branching. So these branch points. So I'm interested in saying when I have a map, say to the Riemann sphere, or number theorists might say to the projective line, I'm interested in what kind of branching is happening. And it turns out that — so now Belyi’s theorem — he realized that in the situation where you're branched over at most three points, so in the picture, we had over 0 and also infinity. I was kind of vague about what's happening at infinity. So that was two points. But if there are at most three points where branching happens, something very special is going on. So he was looking at maps from curves to the projective line. So in a nutshell, really what he proved was that a curve is algebraic if and only if there's one of these coverings that's branched at at most three points. So what is that saying? So saying a curve is algebraic? That's an algebraic statement. You're kind of saying, Well, if you had an equation for the curve — suppose I could write down an equation and then the solutions to that equation are the points of the curve — he’s saying that the coefficients have to be algebraic numbers. So they don't just have to be integers. I could have coefficients, like the square root of two could be a coefficient, or i, or your favorite algebraic number, but not pi, or e or any non-algebraic number. So that's an algebraic statement. But saying that that can happen if and only if, and now he has a map actually, from the curve, well I'm going to say from some Riemann surface to the Riemann sphere, that's branched over at most three points, that second statement is very topological. And it's actually sort of combinatorial too, because that graph I was describing earlier, people use those to kind of describe what's happening with these maps. And so the number of edges, the number of vertices, there's a lot of combinatorial information embedded in that picture. And so I don't know how much of the theorem really comes through in this oral description. But the point is, people were really surprised, including Grothendieck was surprised. He was so surprised and agitated, but excited, that he wrote a letter to the editor, and it's been published. Leila Schneps has done these amazing volumes about a topic called dessins d’enfants, or children's drawings, but I have to read a piece of this because he wrote something like “Oh, Belyi announced this very result.” So this idea, he says actually, “Deligne when consulted found it crazy indeed, but without having a counterexample at hand. Less than a year later, at the International Congress in Helsinki, the Soviet mathematician Belyi announced this very result, with a proof of disconcerting simplicity contained in two little pages of a letter of Deligne. Never was such a profound and disconcerting result proved in so few lines.” So Belyi had actually figured out not only a way to show that these maps exist, but he had a construction. And it reminds me of something you were saying earlier, Evelyn, where the construction exists, maybe it's an unwieldy construction, in the sense that if you really wanted to work with these maps, you might want to do better, and if you try to bound, something I tried to do earlier, you get these really huge degree bounds on maps that are not so practical, in a sense, but the fact that you could do it, so it was the fact not only of the existence, but also there was a constructive proof, opened the door to lots of other work that folks have done.

And maybe I just want to say I was looking — so my French is not good enough to read and translate on the fly. But this “disconcerting result” the word that was used déroutant, can also mean strange and mysterious and unsettling. So even our taxonomy could include unsettling proofs or unsettling results. But I really wanted to put this in the category of something that that bridges different areas, because this picture I was describing earlier really was just a graph with three edges and four vertices. It’s an example of what Grothendieck called, he nicknamed them dessins d’enfants, or children's drawings, the preimagesof this interval. And yeah, so this is really a topic that's caught people's imagination, and Frothendieck was thinking “Are there ways to get at the absolute Galois group?” Because these curves I mentioned were algebraic, so something behind the scenes here is purely algebraic. You can look at Galois actions on the coefficients, for example. But meanwhile, you have this topological combinatorial object. And when you apply this action, we preserve features of the graph, we preserve the number of vertices and edges and so on. Can you start to look at conjugate drawings? And so these doors opened up to these fanciful routes, but it also pointed to these bridges between areas. Maybe algebraic topology is full of these, where you have some algebraic tools, but you're looking at something topological, just things that bridge or create dictionaries between between areas of mathematics, I think are really neat. Yeah. So in the end, you could even bring a stick figure to life this way. So I described this funny-looking graph with just three edges, but you could actually draw a stick figure in this setting, labeling vertices and edges. So I'm picturing, I don't know, literally a little stick figure.

EL: Yeah.

LK: And give some mathematical meaning to it. And then through these through Belyi’s theorem, and through this dictionary, is actually related to curves and so on. And then you can do all kinds of fun things. Like I mentioned some Galois action, although I wasn't specific about it. You could start to ask, are there little mutant figures in the same family as a stick figure? Maybe there's a stick figure with both arms on one side? And is that conjugate somehow to your original, and so somehow there was something elusive about this. The proof had eluded Grothendieck. But it opened this door to very fanciful mathematics. And there's really been kind of an explosion of work over the years looking at these dessins d’enfants. It's a podcast, but I saw you nodding when I mentioned these children’s drawings.

EL: Well, that's a term I've definitely seen. And then not really learned anything about it. Because I must admit, algebraic geometry is not something that my mind naturally wants to go and think about a whole lot.

LK: There’s a lot of machinery, and actually one direction of Belyi — I said this theorem as and if and only if — but one direction was sort of known and takes much more machinery. And it was this disconcerting direction, as Grothendieck said, that actually took less somehow. Some composition of maps and keeping track of ramification, or using calculus to see where you have multiple images of points, or preimages. Yeah, in fact, Grothendieck, there was one last sentence I found, I culled from this great translation by Leila Schneps, who said, “This deep result, together with the algebraic geometric interpretation of maps, opens the door to a new unexplored world, within reach of all, who pass without seeing it.” And you know, we really don't usually see mathematicians speaking in these terms about their work. So that's something I loved. I also loved that Belyi’s proof was constructive too, because even if it creates bounds, I might not want to use, it becomes a lynchpin in other work that connects — the fact that it could be made effective, like not just that this map exists, but you can actually have some degree bound on a certain map, is a lynchpin. And maybe the funniest example takes me to a last category, which is how about theorems that may not be theorems? Like what counts as a theorem? And there's this statement called ABC conjecture. Which is—

EL: A can of worms.

LK: Yeah, so is it proven or not?

KK: It depends on who you ask.

LK: Yeah, so there’s this volume of work by Shinichi Mochizuki, it’s 500-plus pages, and he's created this, I think it was called inter-universal Teichmüller theory. And I, you know, I can't speak to it, but experts are chipping away, chipping away. And maybe it's — I don't know if it's too political to say it's in kind of a limbo. There may be stuff there. There's a lot of machinery there. And yet, do lots of people understand and sort of verify this proof? I'm not sure we're there.

KK: I mean, he’s certainly a respected mathematician. So that's what people taking it seriously. But that's right. But didn't Scholze point to one particular lemma that he thought wasn't true? And the explanations from Kyoto have not been satisfying?

LK: Yeah, I don't have my finger on the pulse. But it’s this funny thing where if you unravel a thread, does the whole thing come apart? And on the other hand, when Wiles proved Fermat’s last theorem, well, some people realized that it would need to do a little something more here. But then it happened. And it kind of was consistent with the theory to be able to sure to fill that in. Yeah. So this is — I don't know, it's exciting to me, but it's also daunting. But this ABC conjecture, so I mentioned Belyi’s theorem. So there's a paper that assuming the ABC conjecture — we don't know if we have a proof, but going back when we've still just called it a conjecture — you can imply or from that, you get so many other results in number theory that people believe to be true. And Noam Elkies has this paper ABC implies Mordell, so Faltings’ theorem, so this theorem about numbers of points on curves. And there's this, I thought this is funny. So I’ll mention this last thing, but this paper has been nicknamed by Don Zagier: Mordell is as easy as ABC. And it's kind of funny, because they're quite difficult no matter how you slice it. You've got something that's still an open problem. And then something that had a very difficult proof. So to say one thing is as easy as the other is sort of perfect. Yeah, there's much more to say about the ABC conjecture, but maybe that's a topic for My Favorite Conjecture.

EL: Yeah. Or My Favorite Mathematical Can of Worms.

KK: Yeah, yeah. Okay, so.

EL: I like this.

KK: Yeah. Well, I was going to say it might be time for the pairing.

EL: I think it is.

KK: So I think I think maybe you're going to pair something with Belyi’s theorem, but maybe not. Maybe there’s something else.

LK: Yeah, I wanted to. I feel like I didn't do justice to Belyi’s theorem, and originally, I'll admit it, I was going to say a gingerbread man because I mentioned stick figures. And so I was like, okay, pairing, well, I love food, made me think of food, made me think of a gingerbread man because of this theory of dessins, or drawings, of Grothendieck. So you can attach a meaning to this little stick figure. And maybe when you're baking, you start making funny-looking figures and those are your Galois conjugates, I don't know. But actually, you know, I was so long on this list of theorems, I'll be short. I think I just have to go with coffee too. Maybe a gingerbread man and coffee because, you know, I wanted to be clever and delicious. But instead I’m just going with coffee because, well, I drink a lot of coffee. They say mathematicians turn coffee into theorems. So can't go wrong. And during the pandemic working at home, I would say I've consumed a lot of coffee in all its incarnations. And maybe it takes me back, too. When I was first hearing about Belyi’s theorem and elsewhere, I was very lucky to have the chance to spend some time in the Netherlands because my advisor Hendrik Lenstra was spending time there, and so as students, we got to go for periods of time. It was very influential to me to be there. But there's a coffee you can get in the Netherlands, which is probably sort of cafe au lait meets latte. But it's called something like koffie verkeerd, and I'm going to mispronounce it, but it basically means messed up coffee. And that's one of my favorite coffees, coffee with, it has too much milk in it. I guess that's what messes it up. So maybe that will be my pairing, just to stick with coffee.

KK: All right. Yeah.

EL: Well, I thought you might go like a pairing for this whole taxonomy and just go with, like, the taxonomy of animals, which, you know, I feel like we didn't do a great job of like, getting theorems exactly into one category or another. And historically, that has also been true for our understanding of biology and like, how many kingdoms there are, you know, in terms of, like, animals, plants, and then a bunch of other stuff.

LK: That’s right, I'm counting on someone to hopefully listen enough to this sprawling, fanciful discussion and say, “Oh, no, no, no, here's how we should do it,” and actually come up with a decent but entertaining, I hope, taxonomy.

EL: Well, we also like to give our guests a chance to plug anything. You know, if you have a website, books or projects that you're working on, that you want people to be able to find online, feel free to share those.

LK: Yeah, that's such a gracious door that you open to everyone. And I mean, maybe I do want to say, in honor of work with collaborators, that math sent me on sort of an unusual journey, as I mentioned in the beginning. So now, for example, looking at the issue of racial profiling and statistics and policy and law. And I do think that there are ways that mathematicians are very creative and can carry that creativity to all of their endeavors, including many of us are spending a lot of time in the classroom. And so that interest has led to a collaboration with Gizem Karaali. She's at Pomona College. And so we do have some books that we've been lucky to co-edit, so many creative people have contributed to. So these are books around mathematics for social justice. There are some essays. There are contributed materials of all sorts. The first volume came out in 2019, in the Before Times. The second volume is due out in 2021. But these represent the work of so many people. And actually, many of the theorems that have come up in your beautiful podcast have come up there, like Arrow’s impossibility theorem around voting theory. Kevin, I think you've been in talks about gerrymandering. And that’s, you can imagine, a topic of great interest. And these materials are more introductory, for folks to bring into the classroom. But as I said, I think mathematicians are very creative, and so it's neat to see what other people have done. And so I hope others will be inspired by those examples as they're creating authentic engagement and cultivating critical thinking for ourselves and all the students we work with.

EL: Yeah, well we’ll make sure to put links to that in the show notes.

KK: Sure.

LK: Yeah. Well, thank you for a sprawling conversation today.

KK: This has been a sprawl, but it has been a lot of fun, actually. I kind of felt like you were interviewing us a little more.

LK: Oh, I that sounds fun to me.

KK: Yeah. This is a great one. I'm going to look forward to editing this one. This will be a good time.

LK: Well, maybe a lot will end up on the editing floor.

KK: I hardly ever cut anything out. I really don't.

LK: There’s always a first time.

EL: You’re on the hot seat!

KK: Lily, thanks so much for joining us. It's been a lot of fun.

LK Thank you for your time.

On this episode, we talked with Lily Khadjavi, a mathematician at Loyola Marymount University in Los Angeles. Instead of choosing one favorite theorem, she led us through a parade of mathematical greatest hits and talked through a taxonomy of great theorems. Here are some links you might enjoy as you listen.

Khadjavi's academic website
Her website about mathematics and social justice, which includes the books she mentioned with co-editor Gizem Karaali
Leila Shneps's book The Grothendieck Theory of Dessins d'Enfants
Steve Strogatz's article about Einstein's proof of the Pythagorean theorem

Try your hand at four-coloring Joaquin Torres-Garcia’s Figuras Constructivas
And some past episodes of My Favorite Theorem about some of the theorems in this episodes:
Adriana Salerno and Yoon Ha Lee on Cantor's diagonalization argument
Henry Fowler and Fawn Nguyen on the Pythagorean theorem
Susan D'Agostino on the Jordan curve theorem
Belin Tsinnajinnie on Arrow's impossibility theorem
Ruthi Hortsch on Faltings' theorem
Ken Ribet on the infinitude of primes
Francis Su and Holly Krieger on Brouwer's fixed point theorem

Episode 62 - Tai-Danae Bradley

Evelyn Lamb: Welcome to my favorite theorem, a math podcast. I'm Evelyn Lamb, one of your hosts. And here's your other host.

Kevin: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. It's been a while. I haven't seen your smiling face in a while.

EL: Yeah. I've started experimenting more with home haircuts. I don't know if you can see.

KK: I can. It's a little a little longer on top.

EL: Yeah.

KK: And it's more of more of a high and tight thing going here. This is Yeah. All right. It looks good.

EL: Yeah, it's been kind of fun. And, like, depending on how long ago between washing it, it has different properties. So it's very, it's like materials science over here, too. So a lot of fun.

KK: Well, you probably can't tell, but I've gone from a goatee to a plague beard. And also, I've let my hair grow a good bit longer. I mean, now that I'm in my 50s, there's less of it than there used to be. But I am letting it grow longer, you know, because it's winter, right?

EL: Oh yeah. Your Florida winter. It's probably like, what? 73 degrees there?

KK: It is 66 today. It's chilly.

EL: Oh, wow. Yeah, gosh! Well, today we are very happy to invite Tai-Danae Bradley to the podcast. Hi, Tai-Danae. Will you tell us a little bit about yourself?

Tai-Danae Bradley: Yeah. Hi, Evelyn. Hi, Kevin. Thank you so much for having me here. So I am currently a postdoc at X. People may be more familiar with his former name, Google X. Prior to that, I recently finished my PhD at the CUNY Graduate Center earlier this year. And I also enjoy writing about math on a website called math3ma.

EL: Yes, and the E of that is a 3 if you're trying to spell it.

TDB: Yeah, m-a-t-h-3-m-a. That's right. I pronounce it mathema. Some people say math-three-ma, but you know.

EL: Yeah, I kind of like saying math-three-ma my head. So, I guess, not to not to sound rude. But what does X want with a category theorist?

TDB: Oh, that's a great question. So yeah, first, I might say for all of the real category theorists listening, I may humbly not refer to myself as a category theorist. I'm more of, like, an avid fan of category theory.

KK: But you wrote a book!

TDB: Yeah, I did. I did. No, I really enjoy category theory, I guess I'll say. So at X, I work on a team of folks who are using ideas from—now this may sound left field—but they're using ideas from physics to tackle problems in machine learning. And when I was in graduate school at CUNY, my research was using ideas in mathematics, including category theory, to sort of tackle similar problems. And so you can see how those could kind of go hand in hand. And so now that I'm at X, I'm really just kind of continuing the same research interest I had, but, you know, in this new environment.

EL: Okay, cool.

KK: Very cool.

EL: Yeah, mostly, we've had academics on the podcast. We’ve had a few people who work in other industries, but it's nice to see what's out there, like, even a very abstract field can get you an applied job somewhere.

TDB: Yeah, that's right.

EL: Yeah, well, of course, we did invite you here to talk about your job. But we also invited you here to ask what your favorite theorem is.

TDB: Okay. Thank you for this question. I'm so excited to talk about this. But I will say, I tend to be very enthusiastic about lots of ideas in mathematics at lots of different times. And so my favorite theorem or result usually depends on the hour of the day. Like, whatever I’m reading at the time, like, this is so awesome! But today, I thought it'd be really fun to talk about the singular value decomposition in linear algebra.

KK: Awesome!

TDB: Yeah. So I will say, when I was an undergrad, I did not learn about SVD. So I think my undergrad class stopped just before that. And so I had to wait to learn about all of its wonders. So for people who are listening, maybe I could just say it's a fundamental result that says the following, simply put. Any matrix whatsoever can be written as a product of three matrices. And these three matrices have nice properties. Two of them, the ones on the left and the right, are unitary matrices, or orthogonal if your matrix is real. And then the middle matrix is a diagonal matrix. And the terminology is if you look at the columns of the two unitary matrices, these are called the singular vectors of your original matrix. And then the entries of the diagonal matrix, those are called the singular values of that matrix. So unlike something like an eigen decomposition, you don't have to make any assumptions about the matrix you started with. It doesn't have to have some special properties for this to work. It's just a blanket statement. Any matrix can be factored in this way.

EL: Yeah, and I, as we were saying, before we started recording, I also did not actually encounter this in any classes.

KK: Nor did I.

EL: And yeah, it’s something I've heard of, but not never really looked into because I didn't ever do linear algebra, you know, as part of my thesis or something like that. But yeah, okay, so it seems a little surprising that there aren't any extra restrictions on what kind of matrices can do this. So why is that? I don't know if that question is too far from left field.

TDB: Maybe that's one of the, you know, many amazing things about SVD is that you don't have to make any assumptions. So number one, in mathematics, we usually say multiplying things is pretty easy, but factorizing is hard. Like, it's hard to factor something. But here in linear algebra, it's like, oh, things are really nice. You just have this matrix, and you get a factorization. That's pretty amazing. I think, maybe to connect why is that—to connect this with maybe something that's more familiar, we could ask, what are those singular vectors? Where do they come from? Or, you know, what's the proof sketch of this?

EL: Yeah.

TDB: And essentially, what you do is you take your matrix, you multiply it by its transpose. And that thing is going to be this nice real symmetric matrix, and that has eigenvectors. And so the eigenvectors of that matrix are actually the singular vectors of your original one. Now, depending on like, if you multiply them the transpose of the matrix on the left or right, that will determine whether, you know, you get the left or right singular vectors. So, you might think that SVD is, like, second best: “Oh, not every matrix is square, so, we can't talk about eigenvectors, oh, I guess singular vectors will have to do.” But actually, it's like picking up on this nice spectral decomposition theorem that we like. And I think when one looks out into the mathematical/scientific/engineering landscape, and you see SVD sort of popping up all over the place, it's pretty ubiquitous. And so that sort of suggests it’s not a second-class citizen. It's really a first-class result.

EL: Yeah. Well, that's funny, because I did, when I was reading it, I was like, “Oh, I guess this is a nice consolation prize for not being an invertible square matrix, is that you can do this thing.” But you're telling me that that was—that’s not a good attitude to have about this?

TDB: Well, yeah, I think SVD, I wouldn't think of it as a consolation prize, I think it is quite something really fundamental. You know, if you were to invite linear algebra onto this podcast and ask linear algebra, what its favorite theorem is, just based on the ubiquity and prevalence of SVD in nature, I'd probably bet linear algebra would say singular value decomposition.

EL: Yeah, can can we get them next?

KK: Can we get linear algebra on? We’ll see. Okay, so I don't know if this question has—it must have an answer. So say your matrix is square in the first place. So you could talk about the eigenvalues, and you do this, I assume the singular values are different from the eigenvalues. So what would be the advantage of choosing the singular values over the eigenvalues, for example?

TDB: So I think if your matrix is square, and symmetric, or Hermitian, then the eigenvectors correspond to the singular vectors.

KK: Okay, that makes sense.

TDB: But, that's a good question, Kevin. And I don't have a good answer that I could confidently go on record with.

KK: That’s cool. Sorry. I threw a curveball.

TDB: That’s a great question.

KK: Because then singular values are important. The way I've always sort of heard it was that they sort of act like eigenvalues in the sense that you can line them up and that the biggest one matters the most.

TDB: Exactly, exactly. Right. And in fact, I mean, that sort of goes back to the proof that we were talking about. I was saying, oh, the singular vectors are the eigenvectors of this matrix multiplied by its transpose. And the singular vectors turn out to be the square roots of the eigenvalues of that square matrix that you got. So they're definitely related.

KK: Okay. All right. Very cool. So what drew you to this theorem? Why this theorem in particular?

TDB: Yeah, why this theorem? So this kind of goes back to what we were talking about earlier. I really like this theorem because it's very parallel to a construction in category theory.

KK: Yes.

TDB: Maybe people find that very surprising. We're talking about SVD. And all of a sudden, here's this category theory, curveball.

EL: Yeah, because I really do feel like linear algebra almost feels like some of the most tangible math., and category theory, to me, feels like some of the least tangible.

KK: So wait, wait, are you going to tell us this is the Yoneda lemma for linear algebra?

TDB: No. Although that was going to be my other favorite theorem. Okay, so I'm excited to share this with you. I think this is a really nice story. So I'm going to try my best because it can get heavy, but I'm going to try to keep it really light. But I might omit details, but you know, people can maybe look further into this.

So to make the connection, and to keep things relatively understandable, let's forget for a second that I even mentioned category theory. So let’s empty our brains of linear algebra and category theory. I just want to think about sets for a second. So let me just give a really simple, simple construction. Suppose we have two sets. Let's say they're finite, for simplicity. And I'll call them a set X and a set Y. And suppose I have a relation between these two sets, so a subset of the cartesian product. And just for simplicity, or fun, let’s think of the elements of the set X as objects. So maybe animals: cat, dog, fish, turtle, blah, blah. And let's also think of elements in the set Y as features or attributes, like, “has four legs,” “is furry,” “eats bugs,” blah, blah, blah. Okay. Now, given any relation—any subset of a Cartesian product of sets—you can always ask the following simple question. Suppose I have a subset of objects. You can ask, “Hey, what are all the features that are common to all of those objects in my subset?” So you can imagine in your subset, you have an object, that object corresponds to a set of features, only the ones possessed by that object. And now just take the intersection over all objects in your subset? That's a totally natural question you could ask. And you can also imagine going in the other direction, and asking you the same question. Suppose you have a subset of features. And you want to know, “Hey, what are all of the objects that share all of those features in that subset I started with?” A totally natural question you could ask anytime you have a relation.

Now, this leads to a really interesting construction. Namely, if someone were to give me any subset of objects and any subset of features, you could ask, “Does this pair satisfy the property that these two sets are the answers to those two questions that I asked?” Like, I had my set of objects and, Oh, is this set of features that you gave me only the ones corresponding to this set of objects and vice versa? Pairs of subsets for which the answer is yes, that satisfy that property, they have a special name. They're called formal concepts. So you can imagine like, oh, the concept of, you know, “house pet” is like the set of all {rabbits, cats, dogs}, and, like, the features that they share is “furry,” “sits in your lap,” blah, blah, blah. So this is not a definition I made up, you know, you can go on Wikipedia and look at formal concept analysis. This is part of that. Or you can usually find this in books on lattice theory and order theory. So formal concepts are these nice things you get from a relation between two sets.

Now, what in the world does this have to do with linear algebra or category theory, blah, blah, blah? So here's the connection. Probably you can see it already. Anytime you have a relation, that’s basically a matrix. It's a matrix whose entries are 0 and 1. You can imagine a matrix where the rows are indexed by objects and the columns are indexed by your features. And there's a 1 and the little x little y entry if that object has that feature and 0 otherwise.

KK: Sure.

TDB: And it turns out that these formal concepts that you get are very much like the eigenvectors of that 0-1 matrix multiplied by its transpose. AKA, they're like the singular vectors of your relation. So I'm saying it turns out—so I'm kind of asking you to believe me, and I'm not giving you any reason to see why that should be true—But it's sort of, when you put pen to paper paper and you work out all of the details, you can sort of see this. But I say it's like because if you just do the naive thing, and think of your, your 0-1 matrix as a linear map, like as a linear transformation, you could say, okay, you know, should I view this as a matrix over the reals? Or maybe I want to think of 0 and 1 as you know, the finite field with two elements. But if you try to work out the linear algebra and say, oh, formal concepts are eigenvectors, it doesn't work. And you can sort of see why that is. we started the conversation with sets, not vector spaces. So this formal concept story is not a story about linear algebra, i.e., the conversation is not occurring in the world of linear algebra. And so if you have mappings—you know, from sets of objects to sets of features—the kind of structure you want that to preserve is not linearity, because we started with sets. So we weren't talking about linear algebra.

So what is it? It turns out it's a different structure. Maybe for the sake of time, it's not really important what it is, or if you ask me, I'll be happy to tell you. But just knowing there's another kind of structure that you'd like this map to preserve, and under that right sort of context, when you're in the right context, you really do see, oh, wow, these formal concepts are really like eigenvectors or singular vectors in this new context.

Now, anytime you have a recipe, or a template, or a context, but you can just sort of substitute out the ingredients for something else, I mean, there's a bet that category theory is involved. And indeed, that's the case. So it turns out that this mapping, this sort of dual mapping from objects to features, and then going back features to objects, that, it turns out, is an example of adjunction in category theory. So there's a way to view sets as categories. And there's a way to view mappings between them as functors. And an adjunction in category theory is like a linear map and its adjoint, or like a matrix and its transpose. So in category theory, an adjunction is — let me say it this way, in linear algebra, an adjoint is defined by an equation involving an inner product. Linear adjoint, there's a special equation that your map and its adjoint must satisfy. And in category theory, it's very analogous. It's a functor that satisfies an “equation” that looks a lot like the adjoint equation in linear algebra. And so when you unravel all of this, it's almost like Mad Libs, you have, like, this Mad Lib template. And if you erase, you know, the word “matrix” and substitute in the whatever categorical version of that should be, you get the thing in category theory, but if you stick in “matrix,” oh, you get linear algebra. If you erase, you know, eigenvectors, you get formal concepts, or whatever the categorical version of that is, but if you if you have eigenvectors, then that's linear algebra. So it's almost like this mirror world between the linear algebra that we all know and love, and like, Evelyn, you were saying, it's totally concrete. But then if you just swap out some of the words, like you just substitute some of the ingredients in this recipe, then you recover a construction in category theory, and I am not sure if it's well known — I think among the experts in category theory it is — but it's something that I really enjoy thinking about. And so that's why I like SVD.

EL: So I think you may have had the unfortunate effect of me now thinking of category theory as the Mad Libs of math. Category theorists are just going and erasing whatever mathematical structure you had and replacing it with some other one.

KK: That’s what a category is supposed to do, right? I mean, it's this big structure that just captures some big idea that is lurking everywhere. That's really the beautiful thing, and the power, of the whole subject.

TDB: Yeah, and I really like this little Mad Lib exercise in particular, because it's kind of fun to think of singular vectors as analogous to concepts, which could sort of maybe explain why it's so ubiquitous throughout the scientific landscape. Because you have this matrix, and it’s sort of telling you what goes with what. I have these correlations, maybe I organize them into a matrix matrix, I have data and organize it into a matrix. And SVD sort of nicely collects the patterns, or correlations, or concepts in the data that's represented by our matrix. And, I think, Kevin, earlier you were saying how singular values sort of convey the importance of things based on how big they are. And those things, I think, are a little bit like the concepts, maybe. That’s sort of reaching far, but I think it's kind of a funny heuristic that I have in mind.

KK: I mean, the company you work for is very famous for exploiting singular values, right?

TDB: Exactly. Exactly.

KK: Yep. So another fun part of this podcast is we ask our guests to pair their favorite theorem with something. So what pairs well with SVD?

TDB: Okay, great question. I thought a lot about this. But I, like, had this idea and then scratched it off, then I had another idea and scratched it off. So here's what I came up with. Before I tell you what, I want to pair it pair this with, I should say, for background reasons, this, Mad Libs or ingredients-swapping recipe-type thing is a little bit mysterious to me. Because while the linear algebra is analogous to the category theory, the category theory doesn't really subsume the linear algebra. So usually, when you see the same phenomena occurring a bunch of places throughout mathematics, you think, “Oh, there must be some unifying thread. Clearly something is going on. We need some language to tell us why do I keep seeing the same construction reappearing?” And usually category theory lends a hand in that. But in this case, it doesn't. There's no—in other words, it's like I have two identical twins, and yet they don’t, I don’t know, come from the same parents or something.

KK: Separated at the birth or something?

TDB: Yeah. Something like that. Yeah, exactly. They’re, like, separated to birth, but you're like, “Oh, where are their parents? Where were they initially together?” But I don't know that, that hasn't been worked out yet. So it's a little bit mysterious to me. So here it is: I'm going to pair SVD with, okay. You know, those dum-dum lollipops?

KK: Yeah, at the bank.

TDB: Okay. Yeah, exactly. Exactly. Just for listeners, that’s d-u-m, not d-u-m-b. I feel a little bit—anyway. Okay, so the dum-dum lollipops, they have this mystery flavor.

KK: They do.

TDB: Right, which is like, I can't remember, but I think it's wrapped up with a white wrapper with question marks all over it.

EL: Yeah.

TDB: And you're letting it dissolve in your mouth. You're like, well, I don't really know what this is. I think it’s, like, blueberry and watermelon? Or I don't know. Who knows what this is? Okay. So this mystery that I'm struggling to explain is a little bit like my mathematical dum-dum lollipop mystery flavor. So, you know, I like to think of this as a really nice, tasty mathematical treat. But it's shrouded in this wrapper with question marks over it. And I'm not quite really sure what's going on, but boy, is it cool and fun to think about!

EL: I like that. Yeah, it's been a while since I went to the bank with my mom, which was my main source of dum-dum lollipops.

TDB: Same, exactly. That's funny, with my mom as well.

EL: Yeah. That that's just how children obtain dum-dums.

KK: Can you even buy them anywhere? I mean, that’s the only place that they actually exist.

EL: I mean, wherever, bank supply stores, you know, get a big safe, you can get those panic buttons for if there's a bank robber, and you can get dum-dum lollipops. This is what they sell.

TDB: That’s right.

KK: No, it must be possible to get them somewhere else, though. When I was a kid trick-or-treating back in the 70s, you know, there would always be that cheap family on the on the block that would either hand out bubblegum, or dum-dums. Or even worse, candy corn.

EL: I must admit I do enjoy candy corn. It's not unlike eating flavored crayons, but I’m into it. Barely flavored. Basically just “sweet” is the flavor.

KK: That’s right.

EL: Yeah, well, so actually, this raises a question. I have not had a dum-dum in a very long time. And so is the mystery flavor always the same? Or do they just wrap up some normal flavor?

KK: Oh, that’s a good question.

EL: Like, it falls off the assembly line and they wrap it in some other thing. I never paid enough attention. I also targeted the root beers, mostly. So I didn't eat a whole lot of mystery ones because root beer is the best dum-dum.

KK: You and me! I was always for the root beer. Absolutely.

EL: And butterscotch. Yeah.

TDB: Oh, yeah. The butterscotch are good. So Evelyn, I was asking that same question to myself just before we started recording. I did a quick google search. And I think what happens, at least in some cases, like maybe in the past—and also don't quote me on this because I don't work at a dum-dum factory—but I think it was like, oh, when we're making the, I don't know, cherry or butterscotch flavored ones, but then the next in line are going to be root beer or whatever, we’re not going to clean out all of the, you know, whatever. So if people get the transition flavor from one recipe into the other, we’ll just slap on the “mystery.” I don't know, someone should figure this out.

KK: Interesting.

EL: I don't want to find out the answer because I love that answer.

KK: I like that answer too.

EL: I don't want the possibility that it's wrong, I just want to believe in that. That is my Santa Claus.

KK: And of course, now I’m thinking of those standard problems in the differential equations course where you’re, like, you're doing those mixing problems, right? So you've got, you know, cherry or whatever, and then you start to infuse it with the next flavor. And so for a while, there's going to be this stretch of, you know, varying amounts of the two, and then finally, it becomes the next flavor.

TDB: Exactly.

EL: Well, can you quantify, like, what amount and which flavor dominates and some kind of eigenflavor? I'm really reaching here.

TDB: I love that idea.

EL: Yeah. Oh, man. I kind of want to eat dum-dums now. That’s not one of my normal candies that I go to.

TDB: I know, I haven't had them for years, I think.

KK: Yeah, well, we still have the leftover Halloween candy. So this is, we can tell our listeners—What is this today? It's November 19?

EL: 19th, yeah.

KK: Right. So yeah, we bought one bag of candy because we never get very many trick-or-treaters anyway. And this year, we had one small group. And so we bought a bag of mini chocolate bars or whatever. And it's fun. We have a two-story house. We have a balcony on the front of our house. So this group of kids came up and we lowered candy from our balcony down. When I say “we” I mean my wife. I was cooking dinner. But we still have this bag. We're not candy-eaters. But you're right. I'm jonesing for for a dum-dum now. I do need to go to the bank. But I feel a little cheap asking for one.

EL: Yeah. I feel like, you know, maybe 15, 16, is where you kind of start aging out of bank dum-dums.

KK: Yep, yeah. Sort of like trick-or-treating.

EL: Well, anyway, getting back to math. Have we allowed you to say what you wanted to say about the singular value decomposition?

TDB: Yeah. I mean, I could talk for hours about SVD and all the things, but I think for the sake of listeners’ brains, I don't want to cause anyone to implode. I think I shared a lot. Category theory can be tough. So I mean, it appears in lots and lots of places. I originally started thinking of this because it cropped up in my thesis work, my PhD work, which not only involved a mixture of category theory, but linear algebra for, essentially, things in quantum mechanics. And so you actually see these ideas appear in sort of, you know, “real-world” physical scenarios as well. Which is why, again, it was kind of drawing me to this mystery. Like, wow, why does it keep appearing in all of these cool places? What's going on? Maybe category theory has something to say about it. So just a treat for me to think about.

EL: Yeah. And if our listeners want to find out more about you and follow you online or anything, where can they look?

TDB: Yeah, so they can look in a few places. Primarily, my blog mathema. com. I'm also on Twitter, @mathema as well, Facebook and Instagram too.

EL: And what is your book? Please plug your book.

TDB: Thank you. Thank you so much. Right. So I recently co-authored a book. It’s a graduate-level book on point-set topology from the perspective of category theory. So the title of the book is Topology: A Categorical Approach. And so this is really—we had in mind, sorry about this with John Terilla, who was my PhD thesis advisor, and Tyler Bryson, who is also a student of John at CUNY. And we really wrote this for, you know, if you're in a first-semester topology course in your first year of graduate school. So basic topology, but we were kind of thinking, oh, what's a way to introduce category theory that’s sort of gentler than just: “Blah. Here’s a book. Read all about category theory!” We wanted to take something that people were probably already familiar with, like basic point-set. Maybe they learned that in undergrad or maybe from a real analysis course, and saying, “Hey, here's things you already know. Now, we're just going to reframe the thing you already know in sort of a different perspective. And oh, by the way, that perspective is called category theory. Look how great this is.” So giving folks new ways to think and contemplate things they already know, and sort of welcoming them or inviting them into the world of category theory in that way.

KK: Nice.

EL: Yeah. So definitely check that out if you're interested in—the way you said like “Blah, category theory” —he other day, for some reason, I was thinking about the Ice Bucket Challenge from, like, I don't know, five or six years ago, where people poured the ice on their head for ALS research. (You’re also supposed to give money because pouring ice on your head doesn't actually help ALS research.)

TDB: Right.

EL: But yeah, it's like this is an alternative to the Ice Bucket Challenge of category theory.

TDB: That’s right. That's a great way to put it. Exactly.

EL: Yeah. Well, thank you so much for joining us. It was fun.

KK: This was great fun. Yeah.

On this episode, we had the pleasure of talking with Tai-Danae Bradley, a postdoc at X, about the singular value decomposition. Here are some links you might find relevant:

Bradley's website, math3ma.com
Her Twitter, Facebook, and Instagram accounts

The book she co-wrote, Topology: A Categorical Approach

Episode 61 - Yoon Ha Lee

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, coming at you from the double hurricane part of 2020 today. I mean, I'm not near the Gulf Coast so it's it's not quite as relevant for my life, but that is the portion of the year we are in right now. I am one of your hosts, Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And here's your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. It's just hot here. But you know, there have been, like, fire tornadoes, right, in California? This is all very on-brand for 2020. This year can’t end soon enough.

EL: Yeah, we say that. I feel like I've said that at the end of many previous years, and then it's not great.

Yoon Ha Lee: As a science fiction writer, I have to say never assume it's the worst. It can always get worse.

EL: Yes.

KK: Right, right, right.

EL: Yes. And that is our guest, Yoon Ha Lee. So yeah, would you like to introduce yourself, tell us a little bit about yourself, and maybe talk about your writing a little bit, how you got to writing from the degrees that you have in math.

YHL: So my name is Yoon Ha Lee. I'm from Houston, and I'm a science fiction and fantasy writer. I actually went to Cornell to get a degree in history, and then I realized that history majors starve on the street. So I switched to math, so that I could have an income and ended up not becoming a mathematician. My best-known books are probably the Machineries of Empire trilogy, which is Ninefox Gambit, Raven Stratagem and Revenant Gun. It's space opera, lots of ships blowing up everywhere. And then a kid's book, Dragon Pearl, which is out from Disney Hyperion in the Rick Riordan Presents series. And that one is also a space opera, because ships blowing up is just fun.

EL: Yeah, well, and that's funny. I think I just put together—I had seen the Rick Riordan publishing imprint before, and I just started reading Percy Jackson the other day. And so it's like, oh, that's who that guy is.

KK: And I think I might be the only one among us who is old enough to have seen the biggest space opera, Star Wars, in the theater in its first release.

YHL: Yeah, my parents let me see it on the television when I was six years old, and I was terrified at the point where Luke gets his hand cut off.

KK: That’s Empire.

YHL: I think the second one? I forget which movie it was, but he gets his hand cut off and I had nightmares for weeks. And I'm like, Mom and Dad, Why? Why? Why did you think this was an appropriate movie for a six-year-old? And then I got all the storybooks and I wanted the lightsaber and everything, so I guess it worked out.

KK: Of course, yeah. Well, my movie story—we’re getting off track, but it's it's a good movie story. So when I was six years old, in 1975, my parents thought it would be a good idea to take me to the drive-in to see Jaws. And I had nightmares for months that there was a shark living under my bed, a huge shart that was going to get me.

EL: My husband was born I think right around the time one of them was released. I don't remember which one now. But we were talking with one of his colleagues one time and figured out that on the day he was born, that colleague was going to see that movie, like, the day it came out.

KK: I’m going to guess it was I'm gonna guess it was Jedi. I don't know exactly how old you guys are. But that's that's my guess.

EL: That sounds right. Yeah, I'm not a big star wars person. But yeah, I guess I've always not been sure, like, “space opera.” The term is something that I feel like I know it when I see it. But I don't really know, like, how to describe it. Is it just—do you feel like a categorization of space opera is, like, ships blowing up?

YHL: Ships blowing up, generally bigger, larger-than-life characters, larger-than-life stakes, big galactic civilization types of things. It's basically the Star Wars genre.

EL: Yeah

KK: It works.

EL: yeah. And the Machinery of Empire—the reason that I invited you on here is because I just read Ninefox Gambit a few weeks ago and just thought, you know, this person sure uses a lot of math terms for a novel! So mathematicians might be especially interested in reading this one, it has shenanigans with calendar systems that are based on math and arithmetic and stuff. So yeah, that was fun. So you, in addition to getting a bachelor's degree in math, you got a master's in math education, right?

YHL: Yes, at Stanford. And I ended up not using it for very long. I was a teacher for, like, half a year before I left the profession.

EL: Okay, and was it just that your writing was taking off and you wanted to do that more? Were there other reasons?

YHL: A kid came along. That was the big reason. Yeah.

EL: Oh. Yeah. That definitely can take a lot of time.

KK: Ah yeah, just a little bit.

EL: Well, that's great. So what is your favorite theorem?

YHL: My favorite theorem is Cantor's diagonalization proof. And I discovered it actually in high school as a footnote in Roger Penrose’s The Emperor's New Mind. It was really just sort of a sidelight to the extremely complicated and hard-to-follow argument that he was making in that book on the nature of consciousness and quantum physics, which, as a high schooler, you know, it basically went over my head. But I was sitting there staring at this footnote and going “I don't understand this at all.” He said in the footnote that Cantor had proven that the real numbers, the set of real numbers, has a cardinality greater than the set of natural numbers. And of course, I was a high schooler. I hadn't had a lot of math background. So my understanding of these concepts was very, very shaky. But he said if you make a list of, you know—pretend that you have a list of all the real numbers and you put them, you know, 1, 2, 3, 4, you put them in correspondence with the natural numbers, and then you go down diagonally, first digit of the first number, second digit of the second number, third digit of the third number, and so on. And then you shift it by one. So if the numeral in that place is two, it becomes three, if it's nine, it becomes zero, and so on. So you can construct a number that is not on the list, even though your premise is that you have everything on the list. And I think this was the first time that I really understood what a proof by contradiction was. My math teachers had attempted very hard to get this concept into my head. And it just did not go through until I read that proof and meditated upon it. And it's funny, because I spent most of my life as a kid thinking that I hated math. And yet there I was in the library reading books about math, so I guess I didn't hate it as much as I thought I did.

EL: Yeah, I was thinking a high schooler reading that Penrose book is definitely—yeah, you had some natural curiosity about math, it sounds like.

KK: Yeah, I'm sort of sort of surprised that your high school teachers were trying to teach you proofs by contradiction. That's kind of interesting. I don't remember seeing any of that until I got to university.

YHL: I don't know that they got into depth about it. But this was at Seoul Foreign School, which was a private, international school in South Korea. And they tried to make the curriculum more advanced, with mixed results.

KK: Sure. It’s worth a shot.

EL: Yeah, and this, this is really one of those Greatest Hits. Like if you're putting together the like, record that you're going to send out or something, like, Math’s Greatest Hits with would include this diagonalization argument. It's so appealing. And we've had another guest select that too, Adriana Salerno a few months ago and yeah, just people. I think a lot of people who eventually do become mathematicians, this is one of those first moments where they feel like they really understand some some pretty high-concept math kind of stuff. So did you see this this proof later in school?

YHL: No. Ironically, most of what I was interested in doing when I did my undergraduate degree was abstract algebra. So I didn't even take a set theory course at all. But I knew it was sort of out there in the water, and I don't know, one of the things I loved about math and that led me to switch my major to math was the idea that there were these beautiful ideas and these beautiful arguments, and just sort of the elegance of it, which was very different from history, where—I love history, and I love all the battles and things, like the defenestration of Prague and all the exciting things happening. But you can't really prove things in history. Like you can't go back and run the siege of Stalingrad again, and see what happens differently.

KK: Maybe we could though, right? We have the computing power now. Maybe we could do that. This sounds like your next novel, right? So simulation of Stalingrad, and this time, the Nazis win or something? I don't know.

YHL: Oh no. I mean, science fiction writers totally do that. There's this whole strand of alternate history, science fiction or fantasy. Harry Turtledove is one author who, he likes to have the story where aliens invade during World War II and then the Nazis and the allies have to have to team up against the aliens kind of stories there. There is a set there is a readership for these things. Sure.

EL: So you use a lot of math concepts in your writing, your fiction writing. So have you ever tried to work in diagonalization, or this kind of idea, into any of your stories?

YHL: This one? No. I mean, occasionally, I remember writing a story in college, actually, called Counting the Shapes. And it was just everything in the kitchen sink, because I was taking point-set topology, and so I used it as a metaphor for a kind of magic that worked that way, and other ideas, like, I don't know, I had recently read James Gleick’s Chaos. So I was really interested in chaos theory and fractals. And I don't know that I was super systematic about it, and I sort of suspect that a real mathematician would look at it and poke holes. You know, I'm using this as a magic system, not as rigorous math, more as a metaphor, I guess, or flavor.

KK: Oh, but I mean, writers do that all the time, right? So I, I taught math and lit class with a friend of mine in languages a few years ago. And, you know, Borges, for example, you know, this sort of stuff is all over his work, these ideas of infinity and, and it's even embedded in Kafka and all this stuff, and it can be a wonderful way to to get your readers to think about something from a point of view they might not have thought of before.

YHL: Well, the interesting thing about Ninefox Gambit and the math terminology that I used for flavor is that 20 publishers turned the book down because they said it had too much math. And I my joke about this is that they saw the word diagonalization in the linear algebra matrix context, and they didn't know what that meant, and they ran away from it. Which was extremely discouraging when my agent at the time, Jennifer Jackson, and I were going out on submission with this book. And it's like, it's basically a space opera adventure where people blow each other up. You don't have to worry about the occasional math term. It's just there as flavor for the magic system. But a lot of people—I’m sure you have encountered the fact that a lot of people in the US have math phobia, and this really does affect the readership as well.

KK: Really?

EL: Yeah, that’s funny, because in some way, I mean, you definitely use the the math language to give a certain flavor to the system that this universe is in, but you could sub it out for, like, any Star Trek term,

YHL: Exactly.

EL: t’s just like, oh, yeah, you could put tricorders and dilithium crystals, or, you know, anything in to serve that that because you know, you're it's not a math textbook, no one's learning linear algebra from reading Ninefox Gambit.

YHL: No, exactly. I actually, when I was originally writing the book, like the rough draft, I had my abstract algebra textbooks out and ready to go. And I was going to construct sort of a game engine, a combat engine of how these battles were going to work in an abstract algebra sense. And my husband who, he's not afraid of math, he's actually a gravitational astrophysicist, and he's arguably better at math than I am. But he sat me down and said, “Yoon Ha, you can't do this. You're not going to have any readers because science fiction readers who want to read about big spaceships blowing each other up do not want to have to wade through a math textbook to get to the action.” And I mean, it turned out that he was absolutely correct. So I ended up not doing that and just using it as, you know, “the force,” except with math flavor.

KK: Linear algebra is the force. All right!

EL: That’s so interesting. I noticed on your website that you have a section for games. So do you also like to design games?

YHL: I do design games. And by design games, I mean tiny little interactive, interactive fiction text adventures or really small tabletop RPGs in the indy sense. You know, three page games for five people, no GM, that kind of thing. So I do enjoy doing that. And it is related to math, I think, but it's certainly not something that we learn to do in any of our math classes.

EL: Yeah, well, I mean, personally, I think it would be very cool. Have you have you written up this potential game, the abstract algebra game thing into an actual game? Or was that kind of abandoned on the editing floor while you were putting the book together?

YHL: It got abandoned on the editing floor. Also because it would have been a tremendous time suck. And, you know, it would have been a fun idea. But if I wasn't going to use it in a book, and it certainly wasn't going to be used in like a computer game or some something like that, there just didn't seem to be enough incentive to go ahead and do it.

EL: Yeah, probably the market of math mathematicians who read sci fi is, you know, not a tiny market but maybe not quite the demographic you're looking for. But I'm just imagining, like, hauling out the Sylow theorems to, like, explode someone’s battle cruiser or something. Just saying that, you know, if you were bored some time and wanted to sink a bunch of time into that.

YHL: if somebody else wrote it, I would definitely buy it and read it, I have to say.

KK: All right. The challenge is out there, everybody. Everybody should get on this.

EL: Yeah, very cool. Yep.

KK: So another thing we do on this podcast is we ask our guests to pair their theorem with something. So what pairs well, with Cantor's diagonalization argument?

YHL: Waffles.

KK: Waffles? Oh, well, yeah.

YHL: Because sort of that grid shape. I know, this is super visual. But the waffles I'm thinking of, my husband did his postdoc at Caltech, so we lived in Pasadena. And when we were there, there was this delightful Colombian hotdog place. And they also made the best waffles with berries and fruit and syrup and whipped cream. And those are the waffles I think of when I think of the diagonal slash proof.

KK: Right. And so the grid is actually fairly small. Is it one of those waffle makers?

YHL: Yeah.

KK: Yeah. Okay, so I have a Belgian waffle maker, and it's fine. It makes four at a time, but those holes are pretty big. Right? I'm thinking of, like, the small, Eggo style, right? You can put a lot of digits.

EL: You could also, like, I guess, maybe a berry is too big to fit in them, but I'm just thinking you can put different things in all of them, make sure no two waffles have the same arrangement of syrup and berries and cream.

KK: This is a good pairing. I'm into this one a lot.

YHL: I’m hungry now.

KK: Yeah.

EL: Yeah. I just had lunch, so for once I don't leave this ravenous. So would you like to let people know where they can find you online?

YHL: Online I’m at yoonhalee.com. I'm also on Twitter as @deuceofgears and also on Instagram as @deuceofgears.

KK: Deuce of gears. Is there a story there?

YHL: It’s the symbol of the crazy general in Ninefox Gambit. Okay. And also, because I'm Korean, there are five zillion other Yoon Ha Lees. So by the time I joined Twitter, all the obvious permutations of Yoon Ha Lee had already been taken, so I had to pick a different name.

EL: Yeah, and if I'm remembering correctly, there are sometimes cat pictures on your Twitter feed. Is that right?

YHL: Yes. So the thing that I post periodically to Twitter is that my Twitter feed is 90% cat pics by volume. There are people who, you know, they tweet about serious things, or politics, or so on, and these are very important, but I personally get stressed out really easily so I figure people could use an oasis of cheerful cat pictures.

EL: Yes, I just wanted to make sure our listeners have this vital information that if they are running low on cat pictures, this is a place they can go. It's definitely been an important part of my mental health to make sure to look at plenty of cat pictures during this—these stressful times as they say.

KK: Yeah, on Instagram, I follow a lot of bird watching accounts. So I just get a feed of birds all day. It's better for my mental health.

EL: Well maybe Yoon’s cat would like that,

KK: I suspect yes, that's right. That's right. Yep.

EL: Yeah, we were talking to a friend who said that they have some bird feeders outside, they just have indoor cats. And the cats will meow to get them to open the windows in the morning so they could watch the birds outside. It’s like, “Mom, turn on the TV.”

YHL: I tried putting on a YouTube video of birds, and my cat was just completely indifferent to the visuals. But she kept looking at the speaker where the bird sounds were coming from.

KK: Hmm.

EL: Interesting. I guess maybe hearing is like more of a dominant sense or something? Cats have pretty good vision, though, I think.

YHL: Yeah, I think she's just internalized that nothing interesting comes out of the moving pictures.

EL: Yeah. Well, thanks for joining us. I really enjoyed talking with you.

KK: This has been good.

YHL: It’s been an honor.

On this episode of My Favorite Theorem, we were happy to talk with Yoon Ha Lee, a sci-fi and fantasy writer with a math background, about his favorite theorem, Cantor's proof of the uncountability of the real numbers. Here are a few links to things we mentioned in the episode:
Yoon Ha Lee's website, Twitter account, and Instagram account

Our episode with Adriana Salerno, who also loves this theorem
Roger Penrose's book The Emperor's New Mind

James Gleick's book Chaos
Harry Turtledove

Episode 60 - Michael Barany

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast for your quarantine life. I'm Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer in beautiful Salt Lake City, Utah.

KK: Yeah.

EL: How are you, Kevin?

KK: I'm okay. I had my—speaking of quarantines, I had my COVID swab test this morning.

EL: How was it?

KK: Well, you know, about as pleasant as it sounds. But yeah, I'm sure you've been to the pool and gotten water up your nose. That's what it feels like.

EL: Yeah.

KK: And then it's over. And it's no big deal. I should have the results within 48 hours. It’s part of the university's move to get everybody back to campus, although I don't expect to go back to the office in any serious way before August. But this is late May now for our listeners, who will probably be hearing this in December or something, right?

EL: Yeah. Who even knows? Time has no meaning.

KK: Hopefully this will all be irrelevant by the time our listeners hear this. [Editor’s note: lolsob.] We'll we'll have a vaccine and everything. It will be a brave new world and everything be fine.

EL: It’ll be a memory of that weird time early in year.

KK: That’s right. The before times. So anyway, today, we are pleased to welcome Michael Barany. Michael, why don’t you introduce yourself and let us know who you are and what's up.

Michael Barany: Hi. So I'm a historian of mathematics. I'm super excited to be on this podcast. I feel like I've been listening long enough that the Gainesville percussionists must be in grad school by now.

KK: No. One of them is my son, and he just finished his third year of college.

MB: Okay, yeah. So older than he was anyway.

EL: Yeah.

MB: Yeah, so I’m a historian of mathematics. I'm based at the University of Edinburgh, where I'm in a kind of interdisciplinary social science of science and technology department. So I get to teach students from all over the university how to think about what science means when you step back and look at the people involved and how they relate to society, how ideas matter, how technology's changed the world, all that fun stuff that gets people to really rethink their place in the world and the kind of things they do with their science.

KK: That’s very cool.

EL: And I know some people who are historians of math will get a degree through a math department and some get it through a history department, I assume. And which are you? I always wonder what the benefits are of each approach.

MB: Yeah, that's great. History of mathematics is a really strange field. It’s actually, as a field, a lot older than history of science as a field, and even older than history as a profession.

EL: Huh.

MB: So history of mathematics started as a branch of mathematics in the early modern period. So we're talking like the 1500s, 1600s. There are always debates about what you classify as this or that. And it started as a way of trying to understand how mathematical theories came about, how they naturally fit together. The idea was that if you understood how mathematical theories emerged, you could come up with better mathematical theories, and you could understand the sort of natural order of numbers and the universe and everything else that you want to understand with mathematics. And then more toward the 19th and the 20th century, there are all these different variations of history of mathematics that branched out of fields like history and philosophy, and philosophy of science and history of science. So my undergrad training was in mathematics. My PhD is from a history department, but from a history of science program in that department. But it's possible to get a PhD in history of mathematics from a mathematics department, it's possible to sort of straddle between different departments. And it makes it a really rich and interesting field. Mathematics education departments or groups sometimes give PhDs in history of mathematics. And they really use the history for different purposes. So if your goal is to make mathematics better, you're taking the perspective of someone doing it from a mathematics department. If your goal is to become a better educator, then you can use history for that in a math education context. I tend to do history as a way of understanding how things fit together in the past and trying to make sense of social values and social structures and ideologies and ideas and how those fit together. And that's the approach that that you come at from a history or history of science perspective.

KK: Very cool. And How did you end up in Edinburgh of all places?

MB: Well, so the academic job market is bad enough in mathematics, right, but in history of mathematics, in a good year, there may be two to three openings in history of science jobs in general. So that's the cynical answer. The more idealistic answer is Edinburgh has this really important place in the sociology of science. In the 1970 s and 80s especially, there was this group of kind of radical sociologists in at the University of Edinburgh who sat down. It was called the Edinburgh School of the sociology of scientific knowledge, which is known for this sort of extreme relativism and constructivism view of how politics and ideology shape scientific knowledge. And I did a master's degree in that department many years later, in 2009-2010, sort of getting my feet wet and starting to learn that discipline. And that approach has been really formative for me and my scholarship. And so it was an incredible stroke of luck that they just happened to have an opening in my field while I was on the market. And I was even even more lucky to have the chance to go there.

KK: Wow, that's great. I’ve always wanted to go there. I've never been to Edinburgh,

MB: It’s the most beautiful city in the world.

KK: Yeah, it looks great. All right, well, being a historian of math, you must know a lot of theorems. So the question is, do you actually have a favorite one? And if so, what is it?

MB: So my favorite theorem is more of a definition. But I guess the theorem is that the definition works.

KK: Okay, great.

EL: That works.

MB: Which, actually—saying what it means for a definition to work is actually a really hard problem, both historically and mathematically. So it's interesting in that regard. Ao the definition is the definition of the derivative of a distribution.

KK: Okay.

MB: So distributions, as you’ll recall from, from analysis—I guess, grad analysis I is usually when you meet them.

EL: Yeah, I think it wasn't until grad school for me at least.

KK: I don't know if I've ever met them, really.

MB: So distributions were invented in 1945, more or less. And in the early years, actually, people were saying you could teach this as a replacement for your basic calculus. So the idea was, this would be something that even beginning college students or even high school students would be learning. So it's interesting to see how they have people pitched that the level of a theory or the the relevant audience, and that's part of the story, too. But in earlier stages of one's calculus education, you learn that there are functions that are integrable but not continuous; continuous but not differentiable; differentiable but not continuously differentiable, and so on. And so a big problem is how do you know something's differentiable when you're studying a differential equation or trying to prove some theorem that involves derivatives. And distributions were the kind of magic wand that was invented in the middle of the 20th century to say that's not actually a problem. Basically, if you pretend everything's differentiable, then all the math works out. And when it really is differentiable, you get the correct differentiable answer, and when it's not, then you get another answer that's still mathematically meaningful. But it's sort of your magic passphrase to be able to ignore all of those problems.

So a distribution is this replacement for a function. Where functions have these sort of different degrees of differentiability, distributions are always differentiable and they always have antiderivatives, just like functions do, but every distribution can be differentiated ad nauseam for whatever differential equation you want to do. And the way you do that is through this definition—my favorite definition/theorem—which is you use integration by parts. So that's a technique you use in calculus class, too, as a sort of trick for resolving complicated integrals. And distributions actually don't tend to look at the things that make the calculus problems challenging or interesting, depending on what kind of student you are, or what kind of teacher you are. So you set them up in a way where you don't have to worry about boundary conditions, you don't have to worry about what the antiderivative things are, because you're working with things where you already know what the antiderivative is. And the definition of distribution uses this fact from integration by parts that you essentially move the derivative from one function to another. So we don't have an exact way of saying functionally what the derivative of a distribution is. You can still say if you multiply it by a function that's super-smooth and over a bounded domain—so you don't have any boundary conditions to worry about, and so you always know how to differentiate that—if you multiply that by a distribution, and take the integral, then if you want to take the derivative of that distribution, integration by parts says you can instead throw in a minus sign and take the derivative of that smooth function instead. And so using that kind of trick, of moving the derivative onto something that is always differentiable, you can calculate the effect of differentiating a distribution without ever having to worry about, say, what the values of of that distribution are after you’ve taken the derivative, because distributions are often things that don't have sort of concrete values in the way that we expect functions to have.

EL: And I hope this question isn't very silly. But when you think about integration by parts—you know, if you took calculus at some point and learned this, there's the UV, and then there's the minus the integral of something else. And so for this, we just choose a function that would be zero on the boundary, and that would get rid of that UV term. Is that right?

MB: Exactly? Yeah. So the definition of distribution sets up this whole space of really nice smooth functions. All of them eventually go to zero, and because you're always integrating over the entire domain, and it's always zero when you go far enough out into the domain, those boundary terms with that UV in the beginning just completely disappear, and you're just left with the negative integral, and then with the derivative flopped over.

EL: All right, great. So if anyone was worried about where their UV went, that's where it went. It was zero. Don't worry. Everything's okay. Yeah. Okay. So what is good about this? Or what do you like about this?

MB: Yeah. So I think this is a really interesting definition from a lot of different perspectives. One thing that I've been trying to understand in my research about the history of mathematics is what it means for mathematics to become a global discipline in the 20th century, so to have people around the world working on the same mathematical theory and contributing to the same research program. And this definition is really helped me understand what that even means and how to understand and analyze that historically. So we think, well, you know, a mathematical theory or a mathematical idea is the same wherever you look at it, and whoever's doing it. As long as they can manipulate the definitions or prove the theorem, it shouldn't matter where they are. But if you look historically, at actual mathematicians doing actual mathematics, where they are makes a huge difference in terms of what methods they're comfortable with, how they understand concepts, how they explain things to each other, how they make sense of new techniques. I mean, learning a new mathematics technique is actually really hard in a lot of cases. And so the question is, how do you form enough of an understanding to be able to work with someone who you can't go and have a conversation with over tea the next day to sort of work out your problems? And the answer is, basically, you use things like this definition and take something you're really comfortable with—integration by parts—and give it a new meaning. And by taking old meanings and reconfiguring them and relating them to other meanings, you make it possible for everyone to have their own sorts of mathematical universes where they're building up theories, but to interact in a way where they can all sensibly talk to each other and develop new ideas and share new ideas. So that's one of the things that that's really exciting about that the definition to me.

One of the other things is sort of how do you know what the significance of the definition is? I mean, a lot of people early on said, isn't this just like a pun? Isn't this just wordplay? Quite early on, when Schwartz was sharing this definition, and some people were getting really excited about it. Some people said, well, you know, it's a cool idea. But isn't this just basically integration by parts? What's new? What's interesting about this? And the history really shows this debate, almost, between people with different kinds of values and philosophies and goals for mathematics, for mathematics education, for the relationship between pure and applied mathematics, where they take different ideas of what's really going on with this definition. Is it something that's complex and difficult and profound and important in that way, or is it something that is utterly trivial and simple, and therefore really useful to people who may be, say, electrical engineers who are trying to work with the Heaviside calculus, and need some sort of magic way to make that all add up? And what made distributions and this definition really powerful is it could be these multiple things to multiple people. So you can have mathematicians in Poland, or in Manchester, or in or in Argentina come to these very, almost diametrically opposed views of what it is that's significant or challenging or easy about distributions, and they can all agree to talk to each other and agree that it's worth sharing their theories and inviting them to conferences, and reading their publications, and they can somehow all make a community out of these different understandings.

KK: I’ve never thought about the sociological aspects in that way. That's really interesting. So the theorem that basically says that this definition is a good one. Is that a difficult theorem to prove?

MB: So there are a lot of different parts. It’s not—I guess it doesn't even boil down to one statement.

KK: Yeah, sure. Yeah, that makes sense. Yeah.

MB: So there's the aspect that when you're dealing with a function, but dealing with it using the distributions definition, that anything you do is not going to ruin what's good about it being a function. So anything you do with a distribution, if you could have done it as though it were a regular function, you get the same answer. So that's one aspect of the theorem that sort of establishes this definition. Another aspect is that distributions are, in some sense, the smallest class of objects that includes functions where everything that is a normal function can be indefinitely differentiated. So that's one way of arguing that distributions are sort of the best generalization of functions, and this competition—I mean, there are a lot of different competing notions, or competing ideas for how you can solve this problem of differentiating functions that were circulating in the 1930s and 1940s. And distributions won this competing scene, in part by the aspects of the theorems about the definition that show it’s sort of the most economical, the simplest, smallest, the best in that sense. And then you have all the usual theorems of functional analysis, like everything converges as you expect it to; if you start with something that's integrable, you're not going to lose interpretability, in some sense.

EL: So this might be a little bit of a tangent, and we can definitely decide not to go down this path. But to make this really concrete—so when I think of a distribution, the example I think of—it’s been a while since I've thought of distributions actually, is the Dirac delta function. I naturally just call it a function, but it is really a distribution. And so this is a thing that, I always think of it, it's something that you can't really define what its value is, but it has a convenient property that if you integrate it, you get 1. Like, its area is 1 even though it's supported on only one point, and it is infinitely tall. And so zero times infinity, we want it to be 1 right here.

MB: And magically it turns out to be 1.

EL: Yeah. And basically, if you decide that this function, this distribution, has this property, then things work out, and it's great. Was that before or after Schwartz? Did this definition—was this kind of grandfathered into being a distribution? Or was it the inspiration?

MB: I love how you put that. Yeah. So this, this phrase that you said at the beginning, we call it a function, but it's really a distribution. I mean, that's evidence of Schwartz’s success, right? The idea that what it really is, what it fundamentally is, is a distribution rather than a function, that's the result of this really sort of deliberate—I mean, it's not it's not an exaggeration to call it propaganda in the second half of the 1940s by people like Laurent Schwartz and Marston Morse and Marshall Stone and Harald Bohr and all of these far-traveling advocates for the theory—to say, you think you've been working with functions, you think you've been working with measures, you think you've been working with operator calculus if you're an electrical engineer, for instance. Or you think you've been working with bra and ket, with Dirac calculus for quantum mechanics, but what you've really been doing ultimately, deep down without even knowing it, is working with distributions. And their ability to make that argument was part of their way of justifying why distributions were important. So people who had no problem just doing the math they were doing with whatever kind of language they were doing, all of a sudden, these advocates for distribution theory were able to make it a problem that they were doing this without having the kind of conceptual apparatus that distributions provided them. And so they were both creating a problem for old methods and then simultaneously solving it by giving them this distribution framework.

So, they did this to the Heaviside calculus, which is about 50 years older than distributions. They did this to the Dirac calculus, where the Dirac function comes from, which comes out of the 1920s and 30s. They did this to principal value calculus, which is also an interwar concept in analysis. Even among Schwartz's contemporaries, there were things like de Rham currents, which were—had Schwartz not come along, we would all be saying the Dirac function is really a de Rham current rather than a Schwartz distribution. But then there were even things that came after distributions, or sort of simultaneously and after, that Schwartz was able to successfully claim. Like there was this whole school of functional analysis and operator theory coming out of Poland associated with Jan Mikusiński. Where Schwartz was—because he was able to get this international profile so much more quickly and effectively—he was able to say all of this really clever research and theorems that Mikusiński is coming up with, that's a nice example of distribution theory, even though Mikusiński would have never put that in those terms. So a huge part of this history is how they're able to use these different views of what a distribution really is to sort of claim territory and grandfather things in and also sort of grandchild things, or adopt things into the theory and make this thing seem much bigger than the actual body of research that people who considered themselves distribution theorists themselves were doing.

EL: Okay. And so I think we also wanted to talk a little bit about—you mentioned in your email to us, I hope I'm getting this I'm not getting this confused with anything—how this theory goes with the history of the Fields Medal.

MB: Oh, exactly. Yeah. So this was a really surprising discovery, actually, in my research. I didn't set out—the Fields Medal kind of became one thing, one little bit of evidence that Schwartz was a big deal. I never expected in my research to come across some evidence that really changed how I understood what the Fields Medal historically meant. And this was just a case of stumbling into these really shocking documents, and then having built up all of this historical context to see what their historical implications were. So Schwartz was part of the second ever class of Fields Medalists in 1950. The first class was in 1936, then there's World War II, and then they sort of restart the International Congresses of Mathematics after the war. And Schwartz is selected as part of that second class. The main reason he's part of that class is because the chair of that committee is Harald Bohr, who is the younger brother of Niels Bohr. Actually, in the early 1900s, Harald Bohr was the more famous Bohr because he was a star of the Danish Olympic soccer team.

KK: Oh!

EL: Wow!

MB: He was a striker. His PhD defense had many, many, many more soccer fans that mathematicians. He was this minor Danish celebrity. And he went on to be a quite respectable mathematician. He had his mathematics institute alongside his brother's physics institute in Copenhagen. And during the interwar period especially, he established himself as this safe haven for internationally-minded mathematics in this period of immensely divisive conflict among different national communities. And because he kind of had that role as this respected figure known for internationalism, he was selected by the Americans who organized the 1950 Congress at Harvard to chair the Fields Medal committee. And Bohr, shortly before being appointed to that committee, had encountered Schwartz in a conference that was sponsored by the Rockefeller Foundation and took place in Nancy in France, and he was just totally blown away by this charming, charismatic young Frenchman with this cool-sounding new theory that seemed like it could unite pure and applied mathematicians, that could be attractive to mathematicians all over the world. And so Bohr basically makes it his mission between 1947 and 1950 to tell the whole world about distributions. So he goes to the US and to Canada, and he writes letters all around the world, he shares it with all his friends. And when he gets selected to chair this committee, what you see him constantly doing in the committee correspondence is telling all of his colleagues on the committee what an exciting future of mathematics Schwartz was going to be.

So the problem is, then sort of the question is, what is the Fields Medal supposed to be for? And they didn't really have a very clear definition of what are the qualifications for the medal. There was a kind of vague guidance that Fields left before he died. The medal was created after John Charles Fields’ death. And there was a lot of ambiguity over how to interpret that. So the committee basically had to decide, is this an award for the top mathematicians? Is this award an award for an up-and-coming mathematician? How should age play a factor? Should we only do it for work that was done since the last medal was awarded? A long time to consider there, so that didn't really narrow it down very much in in their case. And they go through this whole debate over what kinds of values they should apply to making this selection. And ultimately, what I was able to see in these letters, which were not saved by the International Mathematical Union, which hadn't even been formed at the time, they were kind of accidentally set aside by a secretary in the Harvard mathematics department. So they weren't meant to be saved. They just were in this unmarked file. And what those letters show is that Bohr basically constructs this idea of what the metal is supposed to be for in a strategic way to allow Schwartz to win. So there's this question, there's this kind of obvious pool of candidates, of outstanding early- to mid-career mathematicians, including people like Oscar Zariski and André Weil, and Schwartz's eventual co-medalist, Atle Selberg. And they are debating the merits of all of these different candidates, and basically, Bohr selects an idea of what the Fields Medal is for, to be prestigious enough to justify giving it to this exciting young French mathematician, but not so prestigious that he would have to give it to André Weil instead, who everyone agreed was a much better mathematician than Schwartz, and much more accomplished and much more successful and very close in age. He was about five years older than Schwartz.

KK: He never won the Fields Medal.

MB: And he never won the Fields Medal, right. And so what you see in the letters from the early years of the Fields Medal is actually this deliberate decision, not just by the 1950 committee, but I was also able to uncover letters for the 1958 Committee, where they consider whether the award should be the very best young mathematicians, and they deliberately decide in both cases that it shouldn't be, that that would be a mistake, that that would be a misuse of the award. Instead, they should give it to a young mathematician, but not a young mathematician that was already so accomplished that they didn't need a leg up.

EL: Right.

MB: And that was my really surprising discovery in the archives, that it was never meant to crown someone who was already accomplished, and in fact, being accomplished could disqualify you. So Friedrich Hirzebruch in 1958, everyone agreed was the most exciting mathematician. He was in his early 30s, sort of a very close comparison to like someone like Peter Scholze today. So already a full professor at a very young age, with a widely-recognized major breakthrough. And they considered Hirzebruch, and they said, No, he’s too accomplished. He doesn't need this medal. We should give it to René Thom or someone like that.

EL: Yeah. And, of course, people like me, who only were aware of the Fields Medal once they started grad school in math—I wasn't particularly aware of anything before that—Think of it as the very best mathematicians under 40 because it has sort of morphed into that over the intervening decades.

MB: Yeah. And one of the cool side effects is now you can now put an asterisk next to—Jean-Pierre Serre is known to brag about being the youngest-ever fields medalist. But the asterisk is that he won in a period when it was still a disqualification to be too accomplished at a young age.

KK: Yeah, but he still won.

MB: He did still win. He’s still a very important mathematician.

KK: You sort of couldn’t deny Serre, right?

MB: Well, they denied Weil, right?

KK: They did. But I think Serre is probably still—Anyway, we can argue about— we should have a ranking of best mathematicians of the ‘50s, right?

EL: I mean, yeah, because ranking mathematicians is so possible to do because it’s a well-ordered set.

KK: That’s right.

EL: Obviously in any field of life, there's no way to well-order people. I shouldn't say any field. I guess you can know how fast people can run some number of meters under certain conditions or something. But in general, especially in creative fields, it's sort of impossible to do. And so how do you choose?

MB: That’s what I love about studying the sociology of science and technology, is that you get these tools for saying—you know, even in fields like running, we think of sprinting as this thing where everyone has a time and that's how fast they are. But look at all of the stuff the International Olympic Committee has to do for anti-doping and regulating what shoes you can wear, like there are all of these different things that affect how fast you are that have to be really debated and controlled. They're kind of ultimately arbitrary. So even in cases like that, you know, it seems sort of more rankable than mathematics or art or something, and you can tell a great sprinter from someone like me who can barely run 100 meters. But at the same time, there are all of these different social and technical decisions that are so interrelated that even things that seem super objective and contestable end up being much more socially determined.

EL: Yeah.

KK: Yeah. All right. So part two of this podcast is you have to pair your theorem with something, or your definition or whatever we're going to call it your distribution, whatever it is.

EL: Yeah. If you treat it as a distribution, it’ll work fine.

KK: That’s right.

MB: Exactly.

KK: So what have you chosen to pair with distributions?

MB: So what I thought I would pair distributions with is a knock-knock jokes.

KK: Okay.

MB: So I did a little bit of research before coming on here, and I basically found there are no good math knock-knock jokes. I mean, someone please prove me wrong, like tweet at me. And yeah, tell me tell me.

KK: Are there good knock knock jokes, period?

EL: Oh, definitely.

MB: Yeah. So I did come up with one that sort of at least picks up on some of the historical themes. So Knock, knock.

KK and EL: Who’s there?

MB: Harold.

KK and EL: Harold who?

MB: Harold is the concept of a function anyway?

That's the best I could do.

EL: Okay.

MB: So why knock-knock jokes? They involve puns. So you're talking about shifting the meaning of something to come up with something new. They're dialogical: there’s a sort of fundamental interactive element. They sort of make communities. So sharing a knock-knock joke, getting a knock-knock joke, finding it funny or groan-inducing, tells you who your friends are, and who shares your sense of humor. And yeah, they fundamentally use this aspect of wordplay to to make something new and to make something social. And that's exactly what the theory of distributions does and what that definition does, just sort of expand your thinking. And they're also sort of seen as kind of elementary, or basic. It's kind of like a kid's joke.

EL: Right.

MB: It’s this question of distributions as this fundamental theory, your basic underlying theory. So I think it sort of brings together all of those aspects that I like about the definition.

KK: You thought hard about this. This is a really thoughtful, excellent pairing. I like this.

EL: Yeah, I like it. I'm trying to figure out what is the analogy to my favorite knock-knock joke, which is the banana and orange one, right, which is classic.

MB: It’s the only one I use in real life.

KK: Sure.

EL: It’s a great one!

KK: Yeah.

EL: Fantastic. But, like, what distribution is this knock-knock joke?

KK: The Dirac function, right? Excuse me, the Dirac distribution.

MB: Yeah. Aren't you glad I didn't say the Dirac distribution? Yeah, no, it's the only one you actually use all the time. Yeah, the Dirac distribution, or there's that theorem that any partial differential equation can be resolved as the sum of derivatives of these elementary distributions. That's your go-to ubiquitous, uses a pun, but uses in a way that kind of makes sense and is kind of groan-inducing, but also you just love to go back and to use it over and over and over again.

KK: Right.

EL: Nice.

KK: I think back in the 70s—dating myself here—I had a book of knock-knock jokes, and it actually had the banana and orange one in it. I mean, it's like, this is how basic of a book this was. So I might be ragging on knock-knock jokes, but of course, I had a whole book of them. So anyway.

EL: Oh, they're great. And especially when a child tells you one.

KK: That’s right. That’s what they’re there for.

MB: The best is when you have a child who hasn't heard the knock-knock joke you’ve heard 10 million times, and you get to be the person to share the groan-inducing pun with the child. I mean, that's how I imagine Schwartz going to Montevideo and explaining distribution theory, like the experience of sharing this pun and having them go “Ohhh” and slapping their forehead. There's this cultural resonance, to introduce something that you immediately grasp. And yeah, that's a really special experience.

KK: Yeah.

EL: So at the end of the show, we like to invite our guests to plug things, and I'll actually plug a couple of your things because we've sort of mentioned them already. You had a really nice article in Nature. I don't remember, it was a couple years ago—

MB: 2018.

EL: —about this history of the Fields Medal, focusing on Olga Ladyzhenskaya, who was on the short list in ’58 and would have been the first woman to get the Fields Medal if she had gotten it, but it was really interesting because it touches on these things about how the Fields Medal became what it is thought of now and how they made that decision at that time. So go read that. And you also have an article about this distribution stuff that I am completely now blanking on the title of, but it has the word “wordplay” in it, and you probably know the title.

MB: There’s “Integration by Parts” as the title.

EL: Okay.

MB: And then there's a long subtitle. So this is the thing any historian does, is they have some kind of punny title and then this long subtitle. I think one of the reasons I empathize with the theory of distributions is, like, this is how I think as a historian. I come up with a pun, and then I work out how all of the things connect together afterwards. You see that in all of my titles, basically, and papers, That's not that's on my website, mbarany.com, and the show notes.

EL: Yeah, we'll put those in the show notes. We'll link to your website and Twitter in the show notes. And yeah, anything else you want to mention?

MB: Yeah, so if you want all of this math and sociology and politics and stuff about academia and the values of mathematics, then my main Twitter account at @mbarany is the one to follow. If you just want sort of parodies and irreverent observations about math history, then @mathhistfacts is my parody account that I started in August, but the key to that is that behind every thing that looks like it's just a silly joke is actually something quite subtle about historical interpretation. And I always leave that as an exercise to the reader. But I do try to—this was my response to, you know, St. Andrews has this MacTutor archive of biographies of mathematicians that has hundreds and hundreds of mathematicians, these sort of capsule biographies. And they have these little examples, or these little summaries, like so-and-so died on this day and contributed to this theory, and it’s just kind of morbid to celebrate them for when they died. But then even the one that makes the rounds every year on Galileo's birthday, so Galileo is actually one of the—not Galileo, Galois. Galois is one of the few people who actually has an interesting death date, whose death is historically significant, and there's a Twitter account that tweets based on on these little biographical snippets, and does it for his birthday rather than his death day and then says, like, “Galois made fundamental contributions to Galois theory.” So this was my response to that account, those tweeting from these biographical snippets saying there's there's more to history than just when people died and what theory named after them they contributed to, and tried to do something a bit more creative with that.

EL: Yeah, that is fun. I felt slightly personally attacked because I did just publish a math calendar that has a bunch of mathematician’s birthdays on it, but I did choose to only do like a page about a mathematician on their birthday rather than their death day because it just seemed a lot less morbid.

MB: Very sensible. There are some mathematicians with interesting death days. So Galois, Cardano. Cardano used mathematics to predict his death day, so it's speculated that he also used some poison to make sure he got his answer right.

EL: Yikes! That’s a bit rough.

MB: But yeah, there are a few mathematically interesting death days. But yeah, I mean, birthdays are okay, I guess. I'm not super into mathematical birthdays anyway, but better than death days.

EL: Yeah. I mean, when you make a calendar, you've got to put it on some day. And it's weird to put it on not-their-birthday. But yeah, that's a fun account. So yeah, this was great. Thanks for joining us, Michael.

MB: Thanks. This was super fun.

On this episode of My Favorite Theorem, we were happy to talk with University of Edinburgh math historian Michael Barany. He told us about his favorite definition in mathematics: distributions. Here are some links you might find interesting.

Barany’s website and Twitter account
His article “Integration by Parts: Wordplay, Abuses of Language, and Modern Mathematical Theory on the Move” about the notion of the distribution

His Nature article about the history of the Fields Medal
Distributions in mathematics

The Dirac delta function (er, distribution?)

The Danish national team profile page of mathematician and footballer Harald Bohr

Episode 59 - Daniel Litt

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast and so much more. I'm Kevin Knudson, professor of mathematics at the University of Florida. Here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a math and science writer in Salt Lake City, Utah. I have left the county two times since this all happened. We don't have a car, so when I leave my home, it is either on feet or bicycle, which is your feet moving in a different way. But I have biked out of our county now into two different other counties. So it's very exciting.

KK: Fantastic. Well, I do have a car. I bought gas yesterday for the first time since May 26, I think. And yesterday was June 30.

EL: Yes.

KK: And I've gotten two haircuts, but it looks like you've gotten none.

EL: Yes. That’s correct. I’m probably the shaggiest. I've been in a while. My I normally this time of year is buzzcut city, which I do at home anyway. But I don't know.

KK: I will say I’m letting it get a little longer actually. I know I said I got a haircut, but you know, Ellen likes it longer somehow. So here we go. This is where we are. My son's been home for three months, and we haven't killed each other. It's all right.

EL: Great. Yeah, everything's doing as well as can be expected, I suppose. If you're listening to this in the future, and somehow, everything is under control by the time we publish this, which seems unlikely, we are recording this during the 2020 COVID-19 pandemic, right, which—I guess it still stays COVID-19 even though it's 2020 now, to represent the way time has not moved forward.

KK: Right. Time has no meaning. And you know, Florida now is of course becoming a real hotspot, and cases are spiking. And I'm just staying home and, and I have four brands of gin, so I'm okay.

EL: Yeah. Anyway!

KK: Anyway, let's talk math. So we're pleased today to welcome Daniel Litt. Daniel, would you please introduce yourself?

Daniel Litt: Hey, thank you so much. It's really nice to be here. I'm Daniel Litt. I'm an assistant professor at the University of Georgia in Athens, Georgia, likewise, a COVID-19 hotspot. I also have not gotten gas, but I think I've beat your record, Kevin. I haven't gotten gas since the pandemic began.

KK: Wow. That’s pretty remarkable.

DL: I’ve driven, maybe the farthest away I've driven from home is about a 15-minute drive, but those are few and far between.

KK: Sure.

DL: So yeah, I'm really excited to be here and talk about math with both of you.

KK: Cool. All right. So I mean, this podcast is—actually, let’s talk about you first. So you just moved to Athens, correct?

DL: I started a year ago.

KK: A year ago, okay. But you just bought your house.

DL: That’s right. Yeah. So I actually live in northeast Atlanta, because my wife works at the CDC, which is a pretty cool place to work right now.

KK: Oh!

EL: Oh wow.

KK: All right. Is she an epidemiologist?

DL: She does evaluation science, so at least part of what she was doing was seeing how the CDC’s interventions and deployers, how effective they were being help them to understand that.

KK: Very cool. Well, now it would be an interesting time to work there. I'm sure it's always interesting, but especially now. Yeah. All right. Cool. All right. So this podcast is called my favorite theorem. And you've told us what it is, but we can't wait for you to tell our listeners. So what is your favorite theorem?

DL: Yeah, so my favorite theorem is Dirichlet’s theorem on primes in arithmetic progressions. So maybe let me explain what that says.

KK: Please do.

EL: Yes, that would be great.

DL: Yeah. So a prime number is a positive integer, like 1, 2, 3, 4, etc, which is only divisible by one and by itself. So 2 is a prime, 3 is a prime, 5 is a prime, 7, 11, etc. Twelve is not a prime because it's 3 times 4. And part of what Dirichlet’s theorem on primes in arithmetic progressions tries to answer, part of the question it answered, is how are primes distributed? So there is a general principle of mathematics that says that if you have a bunch of objects, they're usually distributed in as random a way as possible. And Dirichlet’s theorem is one way of capturing that for primes. So it says if you look at an arithmetic progressions—that’s, like 2, 5, 8, 11, 14, etc. So there I started at 2 and I increased by 3 every time. Another example would be 3, 6,9, 12, 15, etc—there I started at 3 and increased by 3 every time. So Dirichlet’s theorem says that if you have one of those arithmetic progressions, and it's possible for infinitely many primes to show up in it, then they do. So let me give you an example. So for 3, 6, 9, 12, etc, all of those numbers are divisible by 3. So it's only possible for one prime to show up there, namely 3.

EL: Right.

DL: But if you have an arithmetic progression, so a bunch of numbers which differ by all the same amount, and they're not all divisible by some single number, then Dirichlet’s theorem tells you that there are infinitely many primes in that sequence. So for example, in the sequence 2, 5, 8, 11, etc, there are infinitely many primes, 5 and 11 being the first two [editor’s note: the first primes after 2. But it’s just odd for an even number to be prime]. And it tells you something about the distribution of those primes, which maybe I won't get into, but just their bare existence is really an amazing theorem and incredible feat of mathematics.

EL: So this theorem, I guess, for some of our listeners, and for me, it probably sort of reminds them in some ways of like twin primes or something, these other questions about distributions of primes. Of course, twin primes, you don't need a whole arithmetic progression, you just need two of them. That would be primes that are separated by two, which other than 2 and 3 is the smallest gap that primes can have. And, of course, twin primes is not solved yet.

DL: Yeah, we don’t know that there are infinitely many.

EL: Yeah, people think there are but you know, who knows? We might have found the last one already. I guess that's unlikely. But Dirichlet was proved a long time ago. So can you give me a sense for why this is a lot easier than twin primes?

DL: Yeah, so part of the reason, I think, is that twin primes are much sparser than primes in any given arithmetic progression. So just to give you an example, if you have a bunch of numbers, one way of measuring how big they are is you could take the sum of 1 over those numbers. So for example, the sum of 1/n, where n ranges over all positive integers, diverges; that sum goes to infinity. And the same is actually true for the primes in any fixed arithmetic progression. So if you take all the primes in the sequence 2, 5, 8, 11, etc, and take the sum of one over them, that goes to infinity, since there's a lot of them. On the other hand, we know that if you do the same thing for twin primes, that sum converges to a finite number. And that number is pretty small, actually. We know, up to quite a lot of accuracy, what it looks like. And that already tells you that they're sort of hard to find. And if you have things that are hard to find, it's going to be harder to show that there are infinitely many of them. I mention this sum of reciprocals point of view because it's actually crucial to the way Dirichlet’s theorem is proven. So when you prove Dirichlet’s theorem, it's one of the these really amazing examples where you have a theorem that's about pure algebra. And you end up proving it using analysis. So in this case, the theory of Dirichlet L-functions. And understanding that sum of reciprocals is kind of key to understanding the analytic behavior of some of these L-functions, or at least it’s very closely related.

KK: So I didn't know that result about the reciprocals of the twin primes converging. So even though we don't know that there are infinitely many, somehow…

DL: Yeah, in fact, if there are finitely many then definitely that sum would converge, right?

KK: Yeah, right. That’s—and we even know an estimate of what the answer is? Okay. That’s fascinating.

DL: Yeah, and what you have to do to prove that is show that these primes are sufficiently sparse. And then and then you win. EL: So once again, I am super not a number theorist. So I'm just going to bumble my way in here. But to me, if I'm trying to show that something diverges, I show that it's sort of like 1/n, and if it converges, it's sort of like 1/n2 or, or worse, or better, or however, you want to morally rank these things. So I guess I could imagine it not being that hard to show that twin primes are sort of bounded by n2, or you're like bounded by 1/n2 squared, the reciprocals of that, would that be a way to do this? Or am I totally off?

DL: It’s something like that. You want to show they're very spread out. Yeah, with primes, I do want to mention, so you mentioned like you want to say something like between 1/n or 1/n2. So primes are much, much rarer than integers, right? So it's really somewhere between those two.

EL: Yeah.

DL: So for example, understanding the growth rate of those numbers—the growth rate of the primes and the growth rate of the primes in a given arithmetic progression—is pretty hard. Like that's the prime number theorem, it’s one of the biggest accomplishments of 19th-century mathematics.

KK: Right. Does that help you prove that, though? Maybe it does, right? Maybe not?

DL: Yeah, so proving that the sum of the reciprocals of the primes diverges is much, much easier than the prime number theorem. And as you can prove that in, like, a page or page and a half or something. But it's very closely related to the key input of the prime number theorem, which is that the Riemann zeta function, the subject of the Riemann hypothesis, has a pole at s=1.

KK: All right. Okay. So what's so compelling about this theorem for you?

DL: Yeah, so what I love about it is that it's maybe one of the earliest places, aside from the prime number theorem itself, where you see some really deep interactions between algebra and complex analysis. So the tools you bring in are these Dirichlet L-functions, which are kind of generalizations of the Riemann zeta function. And they're really mysterious and awesome objects. But for me, what I find really exciting about it is that it's like the classic oldie. And people have been kind of remaking it over and over again for the last, like, century. So there's now tons of different versions of the Dirichlet theorem on primes in arithmetic progressions in all kinds of different settings. So here's an example. In geometry, you have a Riemannian manifold, which is kind of a manifold with a notion of distance on it. There's a version of Dirichlet’s theorem for loops in a Riemannian manifold, the first cases of which are maybe do that Peter Sarnak in his thesis. There are versions for over function fields. So I'm not going to be precise about what that means, but if you have some kind of geometric object that's kind of like the integers, you can understand it well and understand the behavior of primes and that kind of object, and how they behave in something analogous to an arithmetic progression. There's something called the Chebotarev density theorem, which tells you if you have a polynomial, and you take the remainder of that polynomial when you divide by a prime, how does its factorization behave as you vary the prime? So there's all kinds of versions of it, and it's a really exciting and cool sort of theme in mathematics.

EL: So kind of getting back to the the more tangible number theory thing—which I guess it's kind of funny that we think of numbers as more tangible when they're sort of the first example of an incredibly abstract concept. But anyway, we'll pretend numbers are tangible. So how does this relate, I remember, and I don't even remember now, I must have been writing some article that related to this, but looking at your primes that are your 1 more than a multiple of 6 versus 1 less and looking at whether there are more or fewer of these. So these are two different arithmetic progressions. The one that's like, you know, 7, 13, let's see if I can add by 6, 19, this, that progression, versus the 5, 11, etc, progression. So is this related to looking at whether there are more of the ones that are one more one less or things like that?

DL: For sure.

EL: I feel like there are all these interesting results about these biases and the distributions.

DL: Yeah, so people call this prime number races.

EL: Yeah.

DL: So what you might do is you might take two different arithmetic progressions and ask are there more prime numbers, like, less than a billion, say, in one of those progressions as opposed to the other? And there are actually pretty surprising properties of those races that I think are not totally well understood. So like even even this recent work of Kannan Soundararajan and Robert Lemke Oliver on this kind of thing.

EL: Oh, yeah, that’s what I was writing about!

DL: Which, yeah, shows some sort of surprising biases. And so that's the reason people think those are cool, is exactly this principle I mentioned before, this general principle of math that things should be as random as they can be. And there are maybe some ways in which our random models of the primes are not always totally accurate. And so understanding the ways in which they're inaccurate and how to fix that inaccuracy, like how to come up with a better model of the primes, is a really big part of modern number theory.

EL: But I guess, the Dirichlet theorem is what you need before you start looking at any of these other things, is you need to know that you can even look at these sequences.

DL: Right. Exactly. Yeah. I mean, how do you study the statistics of a sequence you don't know is infinite? Yeah.

EL: Right.

DL: One thing I’ll mentioned, one cool thing about it is it lets you—it’s not just an abstract existence result. Like, sometimes you just need a prime which is, like, 7 mod 23 to do some mathematical computation. Okay, and if it's 7 mod 23, then it's pretty easy to find one. You can take 7. But if you need a prime, that's a mod b, its remainder upon division by b is a, it's sort of hard to make one in general. And the fact that Dirichlet’s theorem gives them to you is actually really useful. So at least for a mathematician who cares about primes, it's something that just comes up a lot in daily life.

KK: But it's not constructive, though.

DL: Yeah, that's, that's right. It does kind of guarantee that there will be one less than some explicit constant, so in some sense, it's constructive, but it doesn’t, like, hand one to you.

EL: But still, I guess a lot of the time, you probably don't actually need a particular one. You just kind of need to know that there is one.

DL: Yeah.

EL: And where did you first encounter this theorem?

DL: I guess it was, I was probably reading Apostol’s number theory book when I was in college. But I think for me, I didn't really grok it until some other more modern version of it, like one of these remakes showed up for me in my own work. So I wanted to make a certain construction of algebraic curves. So that's some kind of geometric objects defined by some polynomial equations, which have some special properties. And it turned out that for me, the easiest way to do that was to use some version of Dirichlet’s theorem in some kind of geometric context.

KK: Very cool.

DL: So that was really exciting.

KK: Yeah. Well, it's it's nice when, like you say, when the oldies come up on your jukebox. They're useful.

DL: Yeah, exactly.

KK: So another fun thing about this podcast is that we ask our guests to pair their theorem with something. And I mean, I think Evelyn and I are just dying to know what pairs well with Dirichlet’s theorem on primes in arithmetic progressions.

DL: So for me, it's the Arthur Conan Doyle stories about Sherlock Holmes.

KK: Okay.

DL: For a couple different reasons. So first of all, because he's all about making connections between these sort of seemingly unrelated things, just like Dirichlet’s theorem is about making connections between, somehow for the proof, it's about connecting these things in algebra, primes, to things in complex analysis, these L-functions, but then also because it's an oldie that's been remade over and over again. It's still constantly being remade, like with the new BBC Sherlock show.

KK: It’s the best. Yeah, I remember when that was coming out. My wife and I were just so excited every time a new season come out, you know, just “Sherlock! Yes!”

DL: Yeah, just like I'm so excited every time a new version of Dirichlet’s theorem on primes in arithmetic progression comes out.

EL: Yeah, I haven't watched any of the Sherlock TV or movies yet. But we're watching a little more TV these days, and that might be a good one for us to go look at.

KK: It is so good. I mean, the first episode…

EL: Is that the one with Benedict Cumberbatch?

KK: Yeah, but the first one, just, I mean, it just grabs you. You can't not watch it after that. It's really, really well done.

DL: Yeah, they're really fun. Although—oh, go on.

KK: I was going to say the last one, the very last episode, I thought was a bit much.

DL: I don't know that I watched the last season.

KK: Yeah, it was a little…yeah. But you know, still good.

DL: I was reading a couple of the old short stories in preparation for this podcast. Those are also, I highly recommend.

KK: Which ones did you read?

DL: My favorite one that I read recently was, I think it's called the Adventure of the Speckled Band.

KK: Mm hmm.

EL: Oh, yeah.

DL: It's one of the classics.

KK: Right. Yeah. And I think they based one of the episodes on that one, too.

DL: Yeah. that’s right. Yeah.

EL: Yeah, that's a good one. I haven't read all of the Sherlock Holmes it seems like they're practically infinitely many of them. But you know, I had this collection on my Nook and we were moving, so it was like light, and I could read it in the hotel room easily and stuff. And as we were moving to Utah, I think the very first Sherlock Holmes one is set in Utah, or like part of it is set in Utah.

DL: Yeah, maybe the Sign of Four?

EL: Yes, I think it’s the Sign of Four.

DL: Yeah, I think it's one of the first two novellas. So I’ve read every single Sherlock Holmes when I was when I was in high school or something.

EL: Okay. But I was just like, of all things. I didn't know, I hadn't ever read any Sherlock Holmes before. And, like, this British guy writing about this British detective, and it’s set in the state I’m about to move to. It just seemed incredibly improbable to me.

DL: Yeah, I guess he had some kind of fascination with the U.S. because there's that one, which is sort of set in Utah as it was being settled, I guess.

EL: Yeah.

DL: And then there's the case of the five orange pips or something, which actually in a timely way crucially involves the KKK. And so yeah, so there's a lot of sort of interesting interactions with American history.

EL: Yeah, I don't I don't remember if I've read the orange pips.

KK: That figures in the TV series too.

EL: Okay. Yeah, I kind of forgot about those. Those might be a fun thing to go back to, since unlike you, I have not read all of them, and there always seem to be more that I could kind of dive into. I think I kind of tried to read too many at one time, and I just got fed up with what a jerk he is. Self righteous, smug guy.

DL: Yeah, definitely.

EL: Which doesn't make it not entertaining.

DL: If you like this stuff, there's a nother thing I was thinking of pairing. pairing with the theorem. There's a novel by Michael Chabon about a sort of very elderly Sherlock Holmes. Which I don't quite remember the name but part of it is about, you know, what it's like to be Sherlock Holmes when you're 90 and all your friends have left you, and so maybe that might, might appeal to you if you find him sort of an annoying character.

EL: Yeah. Could that be the Yiddish Policeman's Union?

DL: I don't think so. It's a much shorter book.

EL: Okay. That’s the title I could remember.

DL: That one is also excellent. It just doesn't have Sherlock Holmes in it. [Editor’s note: the book is The Final Solution: A Story of Detection.]

EL: Okay. Well, when you were talking earlier about the theorem, you used the word, I think you used the word remake or sequel or something. So I was wondering if you were going to pick movies, or something like that for your pairing. But this kind of works, too, because each one, it’s not a not remakes exactly—I guess with the movies there are remakes, movies and TV shows. But the stories are all, like, some new sequel. Like, here's a slightly different adventure that Sherlock goes on. And slightly different clues that he finds.

DL: Yeah, exactly. That's one thing that I love about math in general is that so much of it is you look at something classic, and then you put a little spin on it. Like I do a little exercise with some of the grad students at UGA in one of our seminars where we take a classic theorem. I think most recently, we did Maschke’s theorem, which is something about representation theory. And then you highlight every word in the theorem that you could change, and then kind of come up with conjectures based on changing some of those words, or questions based on changing some of those words. That's a really fun exercise in, kind of, mathematical remakes.

EL: That does sound fun. And I mean, I think that's one of the things that you learn, especially in grad school, is just how to start looking at statements of theorems and stuff and seeing where might there be some wiggle room here? Or where could I sub out a different space or a different set of assumptions about a function or something and get something new.

DL: Right, exactly. Yeah, definitely. With Dirichlet’s theorem, that happens so many times.

EL: Yeah, well, that's very fun. Thanks for bringing that one up. Thinking about it, I’m a little surprised that we haven't had it already on the podcast.

DL: Yeah, it's classic.

EL: Yeah, it really is.

KK: So we also like to give our guests a chance to plug anything that they're working on. You're very on Twitter.

DL: Yeah, that's right. You can you can follow me @littmath.

KK: Okay.

DL: So what do I want to plug? I think aside from Sherlock Holmes, who maybe needs no plugging, first of all, I would like to plug the Ava DuVernay documentary 13th, which I really liked and I think everyone should should watch.

EL: Yeah, and I saw that's free on YouTube right now. I don't know if that's temporarily, but I’m not a Netflix subscriber.

DL: Yeah, it is on Netflix. And yeah, I don't know if it'll be available on YouTube but for free by the time this comes out, but probably a nominal cost. In terms of things I've done that I think people who listen to this podcast might like, I did a Numberphile video about a year ago on the on it one of Hilbert’s problems about cutting up polyhedra and rearranging them that someone might someone who likes this podcast might enjoy. So if you google “Numberphile the Dehn invariant,” that’ll come up.

EL: Oh, great.

KK: Cool. All right.

EL: We’ll put links to those in the show notes. Yeah.

KK: All right. Well, thanks for joining us.

DL: Thank you guys so much for having me. This was a lot of fun.

KK: I learned something. I learn something every time, but I'm always surprised at what I'm going to learn. So this is this has been great. All right. Thanks, Daniel.

DL: All right. Thank you so much.

On this episode of My Favorite Theorem, we were happy to get to talk to Daniel Litt of the University of Georgia about Dirichlet's theorem on primes in arithmetic progressions. Here are some links you might find useful as you listen:

Litt's website
Litt's Twitter profile
More about the Dirichlet theorem from Wikipedia
Tom Apostol's number theory book
The article Evelyn wrote about surprising biases in the distributions of last digits of prime numbers
Michael Chabon's novella The Final Solution: A Story of Detection
Litt's Numberphile video about the Dehn invariant
Ava DuVernay's documentary 13th

Episode 58 - Susan D'Agostino

Kevin Knudson: Welcome to My Favorite Theorem, a podcast about math and so much more. I'm one of your hosts, Kevin Knudson, professor of mathematics at University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a math and science writer in Salt Lake City, Utah. So how are you, Kevin?

KK: I’m fine. It's it's stay at home time. You know, my wife and son are here and we're sheltered against the coronavirus, and we've not really had any fights or anything. It's been okay.

EL: That’s great!

KK: Yeah, we're pretty good at ignoring each other. So that's pretty good. How about you guys?

EL: Yeah, an essential skill. Oh, things are good. I was just texting with a friend today about how to do an Easter egg hunt for a cat. So I think everyone is staying, you know, really mentally alert right now.

KK: Yeah.

EL: She’s thinking about putting bonito flakes in the little eggs and putting them out in the yard.

KK: That’s a brilliant idea. I mean, we were walking the dog earlier, and I was lamenting how I just sort of feel like I'm drifting and not doing anything. But then, you know, I've cooked a lot, and I'm still working. It's just sort of weird. You know, it's just very.

EL: Yeah, time has no meaning.

KK: Yeah, it's it's been March for weeks, at least. I saw something on Twitter, Somebody said, “How is tomorrow finally March 30,000th?”

EL: Yeah.

KK: That’s exactly what it feels like. Anyway, today, we are pleased to welcome Susan D'Agostino to our show. Susan, why don't you introduce yourself?

Susan D’Agostino: Hi. Thanks so much for having me. I really appreciate being here. I’m a great fan of your show. So yeah, I'm Susan D’Agostino. I'm a writer and a mathematician. I have a forthcoming book, How to Free Your Inner Mathematician, which is coming out from Oxford University Press. Actually, it was just released in the UK last week and the US release will be in late May. And otherwise, I write for publications like Quanta, Scientific American, Financial Times, and others. And I'm currently working on an MA in science writing at Johns Hopkins University.

KK: Yeah, that's pretty cool. In fact, I pre-ordered your book. During the Joint Meetings, I think you tweeted out a discount code. So I took advantage of that.

SD: Yes. And actually, that discount code is still in effect, and it's on my website, which I'll mention later.

EL: Great. So you said you're at Hopkins, but you actually live in New Hampshire?

SD: Exactly. Yes. I'm just pursuing the program part-time, and it's a low-residency program. So I’m a full-time writer, and then just one class a semester. It creates community, and it's a great way to meet other mathematicians and scientists who are interested in writing about the subject for the general public.

EL: Nice. I went to Maine for the first time when I was living in Providence last semester and drove through New Hampshire, which I don't think is actually my first time in New Hampshire, but might have been. We did stop at one of the liquor stores there off the highway, which seems like a big thing in New Hampshire because I guess they don't have sales tax.

SD: No sales tax, no income tax, “Live Free or Die.” Yeah, and you probably test right around where I live because I live in New Hampshire has a very short seacoast, about 18 miles, depending on how you measure it. We live right on the seacoast.

EL: Oh yeah, we did pass right there. Wonderful. Yeah, the coast is very beautiful out there.

SD: I love it. Absolutely love it. I'm feeling very lucky because there's lots of room to oo outside these days. So, yeah, just taking walks every day.

EL: Wonderful.

KK: So you used to be a math professor, correct?

SD: Yes.

KK: And you just decided that wasn't for you anymore?

SD: Yeah, well, you know, life is short. There's a lot to do. And I love teaching. I had tenure and everything. And I did it for a decade. And then I thought, “You know, if I don't write the books I have in mind soon, then maybe they won't get done.” I've got my first one out already, only two years into this career pivot to writing, and I’m working on my next one. And I always had in mind, in fact, I have a PhD, but I also have an MFA. So I have a terminal degrees both in math and writing. And I always had one foot in the math world and one foot in the writing world, and I realized I didn't want to only live in one. So this is my effort to live fully in both worlds.

KK: That’s awesome.

EL: Yeah. Nice. So the big question we have now of course, is what is your favorite theorem?

SD: Okay, great. My favorite theorem is the Jordan curve theorem.

KK: Nice.

SD: Yeah. It’s a statement about simple closed curves in a 2-d space. So before I talk about what the Jordan curve theorem is, let's just make sure we're abundantly clear about what a simple closed curve is.

EL: Yes.

SD: So, a curve—you can think about it as just a line you might draw on a piece of paper. It has a start point, it has an end point. It could be straight, it could be bent, it could be wiggly, it could intersect itself or not. The starting point and the end point may be different or not. And because this is audio, I thought maybe we could think about capital letters in a very simple font like Helvetica, or Arial. So for example, the capital letter O is a is a curve. When you draw it, it has a start point and an end point that are the same. The capital letter C is also a curve. That one has a different starting and end point, but that's okay. It satisfies our definition. Capital letter P also. That one intersects itself in the middle, but it's still it's a curve.

Okay, so a simple curve is a curve that doesn't intersect itself along the way. It may or may not have the same starting and end point, but it won't intersect itself along the way. So capital letter O and capital letter C are both simple. But for example, the capital letter B is not simple, because if you were to start at the bottom, go up in a vertical line, draw that first upper loop and then the second upper loop, between the first and second upper bubbles of the B, you will hit that initial vertical line that you drew. So it's not simple because it touches itself along the way.

And a closed curve is a curve that starts and ends at the same point. So the letter O is closed, but the letter C is not because that one starts in one place ends in another.

KK: Right.

SD: Moving forward as we talk about the Jordan curve theorem, let's just keep in mind two great examples of simple closed curves: the letter O, and even the capital letter D. It's fine that that D has some angles, in the bottom left and upper left. So corners are fine, but it needs to start and end in the same place and doesn't intersect itself other than where it starts and ends.

Okay, so the Jordan curves theorem tells us that every simple closed curve in the plane separates the plane into an inside and an outside. So a plane, you might just think of as a piece of paper, you know, an 8 1/2 by 11 piece of paper, let's draw the letter O on it. And when you draw that letter O, you are separating that piece of paper, the surface, into a region that you might call inside the letter O and another region that you might call outside the letter O. And the second part of the Jordan curve theorem tells you that the boundary between this inside and that outside formed by this letter O is actually the curve itself. So if you're standing inside the O, and you want to get to the outside of the O, you've got across that letter O, which is the curve.

Okay, so that doesn’t sound very profound.

KK: It’s obvious. It’s just completely obvious.

EL: Any of us who are big doodlers—like, when I was a kid, at church, I was always doodling inside the letters in the church bulletin. That’s the thing. I know that there's an inside and outside to the letter O.

SD: You do. Yes. And you could ask your kid brother, kid, sister, whoever. Anyone—you probably didn't need a big mathematical theorem to assure you of this somewhat obvious statement when it comes to the letter O. Okay, so, I do want to tell you why I think it's really interesting beyond this fact that it seems obvious. But before I do, I just want to make two quick notes. And one is that you really do need the simple part, and you really do need the closed part of the theorem because, for example, if you think about a non-closed curve, like the letter C, and you're standing on the piece of paper around that letter C, maybe even inside, like where the C is surrounding you, it actually doesn't separate the piece of paper into an inside and an outside. And then you also need the non-simple part because if you think about the letter P, which is not simple because it intersects itself, if you think about the segment of the P that's not the loop, so the vertical bottom part of that P, that is part of the curve, the letter P, and that piece of the curve doesn't separate—so even though that P seems to have a little bit of a bubble up there, in the in the loop of the P, the bottom part of the P is part of the curve, and it's not the boundary between the inside, what you might consider the inside of the P, and the outside of the P. So you really do need the simple part and the closed part.

KK: Right, right.

SD: Okay, so the reason I think it's interesting, in spite of the fact that it seems obvious, is because it actually isn't very obvious. And it's not obvious when you talk about what mathematicians love to call pathological curves.

KK: Yeah. Okay. No, I know, I know, the theorem I just wanted to shrug my shoulders and say, “Oh, look, it's just a special case of Alexander duality.” Right? And so surely it works. But yeah, okay.

SD: And there are other poorly-behaved curves, or misbehaved curves, like another curve you might think about is the Koch snowflake. So one way of thinking about the Koch snowflake is—again, I'm going to wave my hands a little bit here because we're in audio and I can't draw you a picture—but if you think about the outline of a snowflake, and there's a prescribed way to draw the Koch snowflake, but I'm going to simplify it a little bit. Imagine the outline of a snowflake, so not the inside or the outside of the snowflake, just the outline of it. And on a Koch snowflake, that snowflake is going to have jagged edges. It's going to zig and zag as it goes along the outline of the snowflake. The Koch snowflake actually has an infinitely jagged curve, line, to draw it. So it's not that it has 1000 zigs and zags or 1 million or even 1 billion. It has an infinite number of zigs and zags going back and forth. So you know, it's a little bit easier to imagine the— what could loosely be defined as the inside of the Koch snowflake, and the outside of the Koch snowflake when you imagine one being drawn on a piece of paper. You know, right in the heart of the very dead center of that Koch snowflake, you could probably feel pretty confident saying, “Hey, I'm inside the Koch snowflake.” And then far outside, you could be confident saying, “I'm outside of the snowflake.” But if you think about yourself right up against the edge of this Koch snowflake. And put yourself right there. Then as you think about this boundary of the Koch snowflake, the boundary is supposed to be what separates the inside from the outside, but if you're right up close to that boundary, and in the process of drawing an infinite number of constructions to get the ultimate Koch snowflake. You continue zigging and zagging, you add more zigs and zags every time. Then even in the steps that it takes you to get to your drawing of the Koch snowflake, at some point, it might seem like “Hey, I'm inside. Oh wait, now they zigged and zagged and I’m outside. Oh, wait, they zigged and zagged some more. Now I'm inside again.” So it seems like even in the finite steps that you need to take to draw that Koch snowflake, to imagine what the it is in its infinite world, it seems like that boundary is not really clear. So again, another place where it makes you stop and say, “Wait a minute, maybe the Jordan curve theorem is not as obvious as it first looked.”

KK: Right. Why do you love this theorem so much?

SD: Yeah, so I love it. It actually it kind of goes along with your question of what do you pair it well with? So maybe I'll just jump ahead to what's sugar. Yeah. So, um, because even in my book and in the chapter that in which I discuss the Jordan curve theorem, I actually paired it with a poem. And the poem is by a New Hampshire native, Robert Frost, who actually went to Dartmouth, which is where I got my doctorate. And one of my favorite poems by Frost is called “The Road Not Taken.” And in the beginning of the poem, he's standing in front of this fork in the road, essentially, and he's looking at both options, realizing, “Okay, I've got to go left or I've got to go right.” You know, he starts off:

Two roads diverged in a yellow wood, And sorry I could not travel both And be one traveler, long I stood And looked down one as far as I could To where it bent in the undergrowth;

So he's standing here and he's saying, “Well, which path should I take?” And he notices one that he calls you know, “it was grassy and wanted wear” and had no leaves—what was what was the line—“in leaves no step had trodden black.” And he ultimately comes to the conclusion that he's going to take the past path less traveled. You know, at the very end of the poem, he says, “Two roads diverged in a wood and I—/ I took the one less traveled by,/ And that has made all the difference.” And it strikes me that what Frost is telling us, and what the Jordan curve theorem is telling us, is take the paths that are more unusual, that aren't well trodden, that people don't always look at first, that aren't as obvious or as paved for us. Maybe it's a path that's going to make you question whether you're inside or outside. Or maybe it’s going to have what feels like this amorphous boundary that you can't quite put your finger on. I guess it reminds me that sometimes making a non-traditional choice in life, or looking at pathological objects in math, is actually something very engaging to do, and can can make a life a little bit more interesting.

You know, when I first heard about this theorem, I had the same reaction that most everybody else does: Okay, so I can just draw a curve—you know, you say a curve and you think, “Oh, I can just draw a curve.” I'm just going to do a squiggle on a piece of paper. And as long as I make it simple and closed, then it might be the letter O or it might be some blob that doesn't intersect, but at least starts and ends where it ends where it started. You know, I remember thinking, wait, why does this theorem get its own name? Why isn’t it just lemma 113.7?

EL: An observation.

KK: Clearly.

SD: Why did it get its own name? A I remember asking, and a lot of people, at first everybody was happy to recite the theorem and and say what it was and laugh at how obvious it was, but then later, I kept searching and searching, and then finally I ended up discovering that in fact, it wasn't as obvious, but in order to appreciate how it’s not that obvious, you needed to look at the paths not taken, the more unusual lines and curves.

EL: Yeah, so this is a theorem that, of course, I I feel like I've known for a long time, not just in the “it's obvious” sense, but in the sense that it's been stated in classes that I took—and feel entirely unconfident about knowing anything about it's proof, at least in the general case. I feel like the the difference between how much I have used it and relied on it and what I actually understand of how to prove it is very large.

SD: Yeah, honestly I can say the same thing. My background is in coding theory, definitely not topology. And honestly, I never saw topology as my strength. It was always something that I was in awe of, but also found extremely challenging or less intuitive to me. But I had looked at the proof long ago. I haven't looked at them deeply recently. There are a number of different approaches. But yeah, I feel the same, that even—the statement sounds simple and it's not, and to my understanding, the proofs are also non trivial.

KK: Yeah. I mean, I was sort of being glib earlier and saying it's just a special case of Alexander duality, like that's easy to prove.

EL: Yeah. Right.

KK: I mean, I was teaching topology this this semester, and I was proving Poincaré duality, which is a similar sort of thing, and it's highly non-trivial. I mean, you break it into a bunch of steps, and it sort of magically pops out of it. And I think that's kind of the case here. It's like, you break it into enough discrete steps where each thing seems okay. But in the end, it is a lot of heavy machinery. And like even for Poincaré duality, in the end you use Zorn’s lemma I mean, there's some kind of choice going on. I think when when Jordan—actually, did Jordan even state this theorem? Or is this one of those things where where Jordan gets the credit, but it wasn't really him?

SD: Actually, I don’t know, and now I need to know that answer.

EL: I think he did.

KK: Did he?

EL: Yeah, not to toot my own horn but I’m, gonna anyway, the calendar that I published this year, the page-a-day calendar, still available for purchase, I think Camille Jordan’s birthday is pretty early. It's sometime in January, so I've actually even read this not too long ago. And I think he did publish it and did have a proof of it. And there's an interesting article, I believe by Thomas Hales, about his about Jordan’s proof of the Jordan curve theorem, I guess maybe to some extent defending from the claim some people have that that he never had a rigorous proof of it. I did read that for doing the calendar, but it was over a year ago at this point and I don't quite remember. But yeah, you can find a reference to it on my calendar. I will also include that in the show notes.

KK: And also the same Jordan of Jordan canonical form.

SD: Right.

KK: Pretty serious contributions there from one person.

SD: Absolutely.

KK: Yeah. All right. I actually like this pairing a lot.

EL: Yeah.

KK: And and since you live in New Hampshire, it's perfect.

SD: Yes. I have a number of New Hampshire references in my book because I just feel like I wanted to humanize math to the extent that I could, while still tackling pretty substantial ideas. But any time I had an invitation to bring in something from left field that was actually meaningful to me, I just went for it.

EL: Yeah.

SD: I’m sure Evelyn, too, it sounds like you're up on all of the mathematicians’ birthdays at this point because of your calendar.

EL: I know a few of them now. More than I did two years ago.

SD: Right.

KK: So it was like to give our guests a chance to plug anything. You’ve already plugged your book. Any other places we can find you online?

SD: Yeah, well, lately, I've been writing for Quanta magazine, which has been very exciting. And in fact, I have a few math articles already out this year. And I have a very special one—I can't tell you the topic. I'm not supposed to—it should be coming out April 15. And I'm very excited about that article that I believe is going to be on April 15, assuming everything is fine with the publication schedule, given the pandemic. But yeah, listeners can find links to my articles on my website, which is just susandagostino.com. And you can find information about my books and my articles and what I'm up to there. v KK: Cool. Well, thanks so much for joining us, Susan. This was a good one.

EL: Yeah, lovely to chat.

SD: Great. Well, thank you so much. And you know, I love the show, and really, it was my honor to be here. Thank you.

KK: Thanks.

On this episode of My Favorite Theorem, we talked with mathematician and science writer Susan D'Agostino. Here are some links you might find interesting as you listen.

D'Agostino's website
How to Free Your Inner Mathematician, her new book (find a discount code on her website)
Evelyn's article about the Koch snowflake
Thomas Hales' article about Camille Jordan's proof of the Jordan curve theorem (pdf)

Evelyn's page-a-day math calendar

The article D'Agostino was excited about towards the end of the podcast was this interview with Donald Knuth

Episode 57 - Annalisa Crannell

Evelyn Lamb: Welcome to My Favorite Theorem, joining forces today with Talk Math With Your Friends. I'm Evelyn Lamb. I co-host this podcast. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida, where it is boiling hot today, and I’m very happy to be in this—how would they put this on on TV?—crossover event, right?

EL: Yeah.

KK: So like, I think last night on NBC, on Wednesday nights, there are all these shows that take place in Chicago: Chicago Med and Chicago PD and Chicago Fire, Chicago Uber, who knows what. Anyway, sometimes they'll just merge them all into one three-hour super show, right? So here we go. This is the math version of this, right?

EL: Yes. And I realized today that our very first episode of My Favorite Theorem, we published that in late July 2017. So this is our early third birthday! And we're so glad that people came to join us! And we are very happy today to have our guest Annalisa Crannell with us. Hi, Annalisa. Can you introduce yourself and tell us a little bit about yourself?

Annalisa Crannell: So hi, my name is Annalisa Crannell. I profess mathematics at Franklin and Marshall College, which is in south-central, southeastern Pennsylvania. It's a small liberal arts college. I got my PhD working in differential equations, partial differential equations, nonlinear differential equations, switched into discrete dynamical systems, topological dynamical systems, but for the past 10 or 15 years have been really thinking hard about projective geometry applied to perspective art.

KK: That’s quite the Odyssey.

AC: Yeah, I was really influenced by by Paul Halmos saying that one of the marks of a really good mathematician is that they can change fields. And so yeah, I feel like I'm trying to enjoy so many different aspects of what this profession allows us to do.

EL: And a fun story, at least it was fun for me, is that one time you were here in Utah giving a talk at BYU, which is down the street. And we went to an art gallery and you pulled out your chopsticks and showed me how you use your chopsticks to help you know where to stand to best appreciate art, and it was just so amazing to me that that was this thing that you could do. So that was that was a lot of fun. And I think it just, to me, sums up the Annalisa experience.

AC: Thank you. Yeah, summing, I guess, is a good thing for mathematicians. I think everybody should carry chopsticks with them. I mean, it's great. It's frugal. It helps you avoid to trash, but it also helps you do really cool mathematics. So what's what's not to love about them?

EL: Yeah. So what is your favorite theorem?

AC: So if you had asked me about five years ago, I would have said the intermediate value theorem. But today, I am going to say no, Desargues’ theorem. So Desargues’ theorem first came into human knowledge in the 1640s. And it's a theorem that sounds like it's sort of about planar geometry, but I really think of it as being about perspective. So is this when I'm supposed to tell you what the theorem says?

KK: Yes, please.

EL: Yeah. Okay, should we all get out our—so this is one, I feel like I always need like a piece of paper. (I’m trying to hold it up, but I’ve got a Zoom background.) But I got my piece of paper out so I can hopefully follow along at home.

AC: Yeah. If you had a piece of paper or a chalkboard right behind you, you could imagine that you would have a triangle, like, standing up on a glass pane. And then on one side of this glass pane would be maybe a magician or somebody holding a light. Maybe your granddaughter drew the magician. (Okay, for people in the podcast, I'm showing a picture that my granddaughter drew on the chalkboard.) If this light shines on the triangle, then it casts a shadow, and the shadow is also a triangle. And so we say those two triangles are perspective from a point, the point is the light source. And we say that because the individual corners, the corresponding corners, are colinear with the light source. So A and the shadow of A are collinear with a light. B and the shadow of B are colinear with a light. But it turns out that those shadows, the triangle and its shadow, are also perspective from a line. And what that means is that if you think not about the points on the triangles, but the three lines on the triangles, and you really think of them as lines, not line segments, so going on forever, then the corresponding lines will also intersect along a line. And you can think of that second line, which we call the axis, as the intersection between the plane of glass that's sitting up in the air and the ground. So the interesting thing to me about Desargues’ theorem is that it pretends like it's a theorem about planar geometry, because this theorem holds when the two triangles are both in the same plane, in R2 or something, but the best ways of proving it, the most standard ways of proving it, are using essentially perspective, going out into three dimensions and proving it for two completely different planes and then pushing them back down into the regular plane. And so to me, this is a really interesting example of sort of how art informs math rather than the other way around. Or maybe they both inform each other.

EL: So going back a little bit, to me when I've I've looked at Desargues’ theorem before, somehow there's this big conceptual leap to me between perspective from a point and perspective from a line. Perspective from a point seems really easy to think about, and perspective from a line, I just have trouble getting it into my brain.

AC: Yeah, I do think perspective from a point is so much more intuitive. And so, so the minorly intuitive, the somewhat intuitive way of thinking of this axis, is you can sort of pretend like it's a hinge. So if these two triangles will sort of fold on to each other from the hinge—the triangle on the glass and the triangle on the ground can fold along this hinge—then they’re perspective from a line. So if you think about something that's in the real world, a flat thing in the real world, and its mirror image, then those two, it's hard to say whether they're perspective from a point, but the lines in the real world thing and the lines in the mirror will intersect along the line where the mirror hits the ground. And so that's that's another way of thinking of this axis, sort of three dimensionally.

KK: So I want to think about this in projective space, which probably isn't correct. Or maybe it is. I don't know. I mean, so these lines are points in projective spaces. This is this, how one might go at this in some other way? I asked the wrong question. I'm sorry.

AC: So that's not exactly the way that I think of it because I think of the line as a line in projective space.

KK: Okay.

AC: And the point is a point in projective space. So the point comes from, you could say, from a one-dimensional line in R-whatever.

KK: Okay.

AC: And so here's one of the interesting things about this theorem and about me loving this theorem. In 2011, one of my coauthors and I wrote a book on the mathematics of perspective art, and we used Euclidean geometry all the way through. We were giving a MathFest mini-course on this and a young mathematician came up to us and said, “I just love how you use projective geometry in art because I learned projective geometry and felt like it had to have something to do with art. And you guys are the ones that explained to me how it does.” And Mark and I turned to each other. We're like, “What kind of geometry?” So neither of us had ever taken a projective geometry class. Neither of us had ever learned any projective geometry. We did not know that it existed. And so this young mathematician ended up changing our lives. We ended up working with her and really learning a bunch of projective geometry in order to come out with our most recent book, which came out last December. And so when you ask questions that get into really deep projective geometry, I'm like, “Ooh, I have to write that one down because that's something else I have to go learn.” So for those of you young mathematicians out there, I just want to say learning new stuff and not knowing stuff is is really so much fun! Don't be afraid of starting something new, even if you don't know it all.

EL: And how did you first encounter Desargues’ theorem?

AC: Oh, man, so I first encountered Desargues’ theorem when, Fumiko Futamura, this young mathematician, had convinced me I needed to learn it. So I bribed an undergraduate to go through Coxeter’s Projective Geometry with me because it seemed like that was the standard book. And Coxeter is, like, the famous guy in this realm, and he is completely non-intuitive. So I found Desargues’ theorem in there, and I'm like, “I have no idea what this means.” The notation is awful. The diagrams are awful. Everything about this is awful. And so I read through his book trying to say, “What does this have to do with art?” And that was a really fun way to read it. So we just decided Desargues’ theorem is about shadows.

EL: Well, I was wondering. So I remember you have also given a talk about squares that kind of blew my mind, where I guess the the thesis of the talk is that all configurations of four points are a square, if you look at it from the right way. Is Desargues’ theorem related to that theorem? I feel like when you said the word shadow that is what reminded me of that other talk.

AC: Yeah, thank you. So that's really cool. So most of us know what the fundamental theorem of calculus says. Most of us know what the fundamental theorem of algebra says. The fundamental theorem of projective geometry in one sense really ought to be Desargues’ theorem. So you can think about these triangles, these points, these lines as objects. For mathematicians, we care about verbs. So a verb is the function. So you can think of a perspective mapping as mapping one set of points and lines to another set of points and lines with this particular rule that says that corresponding points have to line up with the sun, which you call the center, and corresponding lines have to line up with the axis, this hinge. But there's other functions that take points to points and lines to lines. So we know in linear algebra, you can do this all the time, and in linear algebra sets of parallel lines go to other sets of parallel lines. But there's other kinds of functions that do this. They're called colineations. So the fundamental theorem of projective geometry says that if you have four points and their images, and you know that points go to points and lines go to lines, then the entire rest of the function is pre-determined, we know that.

So Desargues’ theorem says that one kind of colineation is perspective mappings, right? Just, like, a shadow or mapping from the floor, this tiled floor onto your canvas through a window. We know from linear algebra, there are these other affine transformations. And so one really cool theorem that I totally love is if you have something that's not a linear algebra one, that's not an affine transformation, then it's automatically a perspective transformation together with an isometry. So you took a photograph and you moved it. That's this notion that every single thing that you do with four points going to four other points that determines a whole function. So yeah, so anytime you have four points connected by four lines, even if they look like a bow tie, or they look like Captain Kirk’s Star Trek logo, it turns out that's actually a weird perspective image of a square moved around somewhere.

EL: And you just have to figure out where you should stand to see it as a square.

AC: Yes, exactly.

KK: Are you guaranteed to be able to—so if it's on the wall, say, could you have to, like, lift it up into a third dimension to be able to see it correctly?

AC: So one of the weird things that happens is if you have a bow tie, we sort of think of a bow tie is that the inside of the bow tie, you would imagine that has to go to the inside of the square. And that is not actually the way it happens. So let's let's think about something that's much more familiar to us. Can you map a circle through perspective into other weird shapes like an ellipse? Well, Sure you can. So imagine that you've got a lampshade, and you've got a circular lampshade, and the shadow that it projects onto the wall is actually a hyperbola. We know that. And the light from the inside of the shadow goes to the part of the hyperbola that goes off towards infinity. Well, if you have the bow tie, think about the area outside of the “x” as almost a hyperbola. So this is when it would be so wonderful if I could actually draw pictures, but it's a podcast. On the on the bow tie, there's two sides that are parallel to each other, and then there's this weird “x” in the middle. The parallel sides, extend them out towards infinity from the bow tie. Right? That turns out to be where the square goes, so if you had a square lampshade, it would cast a shadow that would look like this outside of the bow tie. So the same way that a circular lampshade casts a shadow that looks like the outside of the hyperbola, the U shape of the hyperbola.

KK: My desk lamp is a rectangle, so I’m trying to see if it’s casting the right shadow here.

EL: Yeah, some experiments you can do. I feel like it's this “expand your mind on what a square is” kind of idea.

KK: Got to get rid of those old ideas, man.

EL: Yeah. I know that we we traveled a little bit from Desargues’ theorem and I want to give you a chance to circle back and for me

KK: Or square back. Sorry.

EL: Or square back, or projectively bow tie back to Desargues’ theorem, and I guess what do you love so much about it?

AC: What do I love so much about Desargues’ theorem? One of the things that I love is that it really tightly connects mathematics informing art and art informing mathematics. So Desargues himself, we don't know if he actually wrote this up and published it. We don't have a copy of his original manuscript. We do have something that came out from one of his, sort of acolytes, one of his followers, a guy named Bosse. And if you look at Bosse, okay, to draw Desargues’ diagram, you need 10 points: the three on the first triangle, the three on the second triangle, the sun, that gives you seven, and then the three along the axis. You also need 10 lines: the three on the triangle, the three on the other triangle, the three light rays, that gives you nine, and then the axis. When Bosse first published his diagram, his diagram was incomprehensible. It had 20 lines and 14 points, and it was just really a mess. And it was hard to even figure out where the heck the triangles were.

KK: Yeah, I don’t see them.

AC: And he ended up proving this not using sort of standard geometry, using using numerical stuff called cross-ratios. But the the proofs that make the most sense, that are convincing our proofs that allow us to think about things in three dimensions and use art. So that's one of the cool things, is that actually drawing, if you go and you shade in Bosse’s diagram in a cool artistic way, all of a sudden it sort of pops into 3-d and you can see it, but his original diagram not so much. The same is true of a lot of different proofs. If you try to imagine them as three-dimensional, if you draw them as three-dimensional, the proof becomes more obvious.

But also Desargues’ theorem is actually useful for artists because if you want to draw the shadow of something, if you want to draw the shadow of a kite, if you want to draw a reflection, shadows and reflections, they are projections, so projective geometry, and how do you know how to draw this? You have to use the fact that the shadow or the reflection, or this this projective image, however, you've made it, is perspective from a point and perspective from a line. So you're constantly using Desargues’ theorem to draw these images of images within your image. It just becomes so incredibly useful.

KK: My wife's an artist, but I can't imagine that she would use this. I mean, if you walked up to a typical artist, are they going to say, “Oh yeah, I use Desargues’ theorem all the time?” Or is it just a sort of an intuitive thing where people who are very good at drawing in perspective, can just kind of naturally draw it that way?

AC: Oh, yeah. So the truth is that Desargues’ theorem has really only pretty much been used by mathematicians, and occasionally misused by mathematicians. There's a description in a book by a guy named Dan Pedoe of Desargues’ theorem to draw the image of a pentagon on the top of a square, and he just completely gets it wrong. And Mark and I think that's hilarious. This book has been reproduced zillions of times. Anyway.

So no, actually artists have this incredible skill. One time, we had asked mathematicians and artists at one of our workshops to try to divide the image of a flag into three equal pieces perspectively. So imagine you're drawing the Italian flag going back into the distance, right? How do you do this? In the real world? This there's the three bars are evenly spaced, but in perspective, they're not. And the artists stood up and said, “Well, you just eyeball it, and you just put them here.” And I was horrified. This is not approved. This is not correct. My colleague Mark said, “Okay, this is good. But for those of us who can't just eyeball it, let's see if we can come up with a construction.” And eventually somebody did. They came up with a really cool geometric construction. And Mark had them put this up over the artist’s solution and it was spot on. As a mathematician, I decided to go take an art class. And one of the things we were supposed to do was to draw cans. And so the top of a can is circular, and so the image was going to be an ellipse. And I could not get the proportions right. My ellipses were so awful. So I would say that disarms is an incredibly useful tool for drawing things that look very accurate for people who do not know art, but who are good at math.

KK: Right.

AC: That’s a really long answer to your question. Yeah, artists don't tend to use it, but it really is a useful thing for drawing things that look correct.

KK: Cool. All right.

EL: And you've incorporated this into a class that you teach to help people, I don't know if the purpose of the class is more math or more perspective drawing, but it seems like an interesting mix.

AC: Yeah, we have a course called Perspective and Projective Geometry. We actually have a book that's come out that has Desargues’ theorem right on the cover up there. And it's aimed at the intro to proofs level. So it really teaches students to make conjectures about what they're seeing in the world and then to try to prove those conjectures, but also to try to draw. Ao it's actually sort of an applied course. So they, this students, when we introduce them to Desargues’ theorem, they're actually drawing the shadow of the letter A, and then discovering Desargues’ theorem, and then proving it using many colors and, yeah, lots of cool lines.

It's so much fun! It's a course that really attracts a very unusual swath of students. They all are students who love math, and who are art-curious. Almost none of them are good at art. But I tend to get more women than men in the class. I have often had my class being highly diverse in terms of races and ethnicities. And so for me, it's a fun class. I didn't do it just for the sake of promoting diversity in the math major, but it's sort of unintentionally has done that. And that's a really good feeling.

KK: Very cool. So another thing we like to do on this podcast is ask our guests to pair their theorem with something. So what pairs well with Desargues’ theorem?

AC: Yeah, so I think I already hinted at this, so anything that you can eat with chopsticks goes really well with Desargues’ theorem because chopsticks allow you to have wonderful food and do math at the same time, and what could be better?

KK: So basically, anything you can eat, then, right, you can eat anything with chopsticks?

AC: Soup is a little bit tricky, but yes.

KK: But you drink the soup, right? They give you the chopsticks, you’ve had ramen, right? There's the chopsticks for the noodles.

AC: Yes. Exactly.

EL: Do you have a favorite food to eat with chopsticks?

AC: Oh my goodness. Pretty much everything. I was just realizing ice cream is not so easy with chopsticks.

EL: Yeah.

AC: I think yesterday was national ice cream day. Yeah, I don't I don't know. I take my chopsticks with me in my in my planner bag, and a spoon. And so when I go to restaurants if they try to give me plastic things, I use my chopsticks. So basically, yes, anything I can eat with chopsticks, I will eat with chopsticks. If I can't, I'll use my spoon.

EL: Nice.

KK: We’re getting Thai takeout tonight. Now I'm really excited.

AC: I’m coming to your house.

KK: Sure, come on down. Although you know with all the COVID, I don't think Florida is really a place you want to be coming these days.

EL: So I guess this would be a good time to open the floor to questions. So Brian, I was thinking that I would be able to keep an eye on it, and I totally couldn't. So I'm glad that you were keeping an eye on it. So do you have any questions that our listeners would like to ask Annalisa?

Brian Katz: I’ve noticed three so far. One is from Joshua Holden, would Desargues’ theorem be useful for computer graphics?

AC: That’s a really good question. If I knew anything about computer graphics, I would be able to answer that better. I do know that my students who have gone on into computer engineering tell me that the course that I offered on projective geometry was one of their most useful courses, that this idea of ray tracing was was super, super helpful. So I don't know if Desargues’ theorem itself is specifically useful, but the idea of projective geometry is certainly how we understand the world through videos.

BK: We got a request from Doug Birbrower asking for you to hold up the line drawing while I asked the next one. I was wondering, so when we're talking about triangles, we have these vertices that are special points. How does this idea translate when you're talking about, say, shadows of more complicated objects that might be smooth? You talked a little about circles, but is there a special that happens when you generalize beyond polygons?

AC: One of the things that makes triangles really awesome is the same reason why triangular stools are so useful, is they're always stable, right? Whereas a four legged stool can wobble. If you try to draw the perspective image of an object With four points like a kite, it's really easy to make it be perspective from the sun without being perspective from a line. And if you do something like that, it'll look like maybe the kite is planar, but the shadow is curved, which might make sense on the ground. So in some ways, it's saying triangles really determine planes. Yeah, the question of drawing other curves is really interesting because of how you do or don't define curves in projective geometry. So one way you could think of a curve is a collection of points. You could also think of it as a collection of tangent lines. And so I think a way to generalize Desargues’ theorem to those would be to be talking about those collections of points and those collections of tangent lines.

BK: And then the third one that got some answers in in the chat was: I have a sense that, like, parallel things that when they're prospective from a point that means the point’s at infinity when we're talking about projective geometry. Is their geometric intuition about what it means for the line, perspective from a line, for that line to be infinity? And TJ suggested that it was that the objects are translations of each other.

AC: So if the line is it that that is at infinity, then either you could think of this as being translations, or you can think of it as being a dilation. And so it's a translation if both the axis where the two triangles meet is infinity and the center, that is what how you shine from one to another, is also off at infinity. And they’re are a dilation if the axis is off at infinity, but the center is what we call an ordinary point.

KK: This is new for us, having a Q and A. It's usually just the three of us, you know, me and Evelyn and whoever we're interviewing, but this is fun. I like this interaction.

EL: Yeah, I like that. And people have good questions. Yeah. Great. Thanks. Are there any more questions from the chat that we want to get to? Okay, looks like I'm seeing no. So I think this will sort of wrap up the…oh, Brian. Yeah.

BK: This one just appeared: Do cylindrical polar coordinates throw any light on this?

AC: Oh, so I was just about to say to everybody, “Thank you so much for asking me questions that I actually know the answers to!” And this one, I have no idea. I don't know. I don't know anything about cylindrical polar coordinates. Sorry. Now I'm going to write that one down and go check it out.

EL: But we can all appreciate the “throwing light” phrase of the question. That was very well done. Thank you.

KK: Clever.

EL: So, to wrap up the podcast portion of this, or the the episode with Annalisa portion of this, we will have show notes that are available. Our podcast listeners probably know where to find that at Kevin's website. And on that will include a link to your website, a link to the books that you have. Do you want to say the titles of the books that you've written that people might be interested in?

AC: So the first one, the one from 2011, is called Viewpoints with a subtitle “mathematical perspective, and fractal geometry in art,” and that's suitable for, like, a first-year seminar in math and art. So students don't need to really know anything at all about mathematics. And then the other one is called Perspective and Projective Geometry, and it came out in 2019.

EL: Yeah, so thank you for joining us, Annalisa. And for your doing it in this different fun format.

AC: I’m really flattered that you invited me to do this. Yeah, it's been so much fun trying to think about how to do this without drying gazillions of pictures. I appreciate that.

EL: Yeah.

KK: Thanks so much.

We were delighted to have a crossover event with Talk Math With Your Friends, an online math seminar that runs on Thursdays at 4 pm Eastern time. You can watch a video of this episode, which includes a collection of "flash favorite" theorems from the audience, here. Our guest for this episode was Annalisa Crannell from Franklin and Marshall College, who talked about Desargues' theorem. Below are some links you might find handy after listening to the episode.
Crannell's academic website
Her collaborator Fumiko Futamura's website 
Desargues' theorem on Wikipedia, which includes some helpful diagrams
The Image of a Square, a paper about the theorem that every quadrangle is a square if you look at it the right way. (Also available from Futamura's website.)
Viewpoints: Mathematical Perspective and Fractal Geometry in Art by Crannell and Mark Frantz
Perspective and Projective Geometry by Crannell, Frantz, and Futamura

During the episode, Crannell shared Bosse's original diagram for proving Desargues' theorem. It is here. Below is a version of the diagram colored in, making the triangles a little easier to see.

Episode 56 - Belin Tsinnajinnie

Evelyn Lamb: Hello, My Favorite Theorem listeners. This is Evelyn. Before we get to the episode, I wanted to let you know about a very special live virtual My Favorite Theorem taping. If you are listening to this episode before July 16, 2020, you’re in luck because you can join us. We will be recording an episode of the podcast on July 16 at 4 pm Eastern time as part of the Talk Math With Your Friends virtual seminar. Join us and our guest Annalisa Crannell to gush over triangles and Desargues’s theorem. You can find information about how to join us on the My Favorite Theorem twitter timeline, on the show notes for this episode at kpknudson.com, or go straight to the source: sites.google.com/southalabama.edu/tmwyf. That is, of course, for “talk math with your friends.” We hope to see you there!

[intro music]

Hello and welcome to my favorite theorem, the podcasts that will not give you coronavirus…like every podcast because they are podcasts. Just don't listen to it within six feet of anybody, and you'll be safe. I'm one of your hosts, Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. So if our listeners haven't figured out by now, we are recording this during peak COVID-19…I don’t want to use hysteria, but concern.

EL: Yeah, well, we'll see if it’s peak concern or not. I feel like I could be more concerned.

KK: I’m not personally that concerned, but being chair of a large department where the provost has suddenly said, “Yeah, you should think about getting all of your courses online.” Like all 8000 students taking our courses could be online anytime now… It's been a busy day for me. So I'm happy to be able to talk math a little bit.

EL: Yeah, you know, normally my job where I work by myself in my basement all day would be perfect for this, but I do have some international travel plans. So we'll see what happens with that.

KK: Good luck.

EL: But luckily, it does not impact video conferencing.

KK: That’s right.

EL: So yeah, we are very happy today to be chatting with Belin Tsinnajinnie. Hi, will you introduce yourself?

Belin Tsinnajinnie: Yes, hi. Yá’át’ééh. Shí éí Belin Tsinnajinnie yinishyé. Filipino nishłį́. Táchii’nii báshishchíín. Filipino dashicheii. Tsi'naajínii dashinalí. Hi, everyone. Hi, Evelyn. Hi, Kevin. My name is Belin Tsinnajinnie. I'm a full time faculty professor of mathematics at Santa Fe Community College in Santa Fe, New Mexico. I’m really excited to join you for today's podcast.

EL: Yeah, I'm always excited to talk with someone else in the mountain time zone because it's like, one less time zone conversion I have to do. We're the smallest, I mean, I guess the least populated of the four major US time zones, and so it's a little rare.

BT: Rare for the best timezone.

EL: Yeah, most elevated timezone, probably. Yeah, Santa Fe is just beautiful. I'm sure it's wonderful this time of year. I've only been there in the fall.

BT: Yeah, we're transitioning from our cold weather to weather where we can start using our sweaters and shorts if we want to. We're very excited for the warmer weather we had. We're always monitoring the snowfall that we get, and we had an okay to decent snowfall, and it was cold enough that we're looking forward to warm months now.

EL: Yeah, Salt Lake is kind of the same. We had kind of a warm February, but we had a few big snow dumps earlier. So tell us a little bit about yourself. Like, where are you from? How did you get here?

BT: Yeah. I am Navajo and Filipino. I introduced myself with the traditional greeting. My mother is Filipino, my father is Navajo, and I grew up here in New Mexico, in Na’Neelzhiin, New Mexico, which is over the Jemez mountains here in Santa Fe. I went to high school, elementary school, college here in New Mexico. I went to high school here in Santa Fe. I got my undergraduate degree from the University of New Mexico, and I ventured all the way out over to the next state over, to University of Arizona, to get my graduate degree. While I was over there, I got married and started a family with my wife. We’re both from New Mexico, and one of our biggest goals and dreams was to come back to New Mexico and live here and raise our families where our families are from and where we're from. And when the opportunity presented itself to take a position at the Institute of American Indian Arts here in Santa Fe, it's a tribal college serving indigenous communities from all over the all over the nation and North America, I wanted to take that. I feel very blessed to have been able to work for eight years at a tribal college. And then an opportunity came to serve a broader Santa Fe, New Mexico community, where I also serve communities that are near and dear to my heart, where I've been here for over 30 years. And I'm really excited to have this opportunity to serve my community in a community college setting.

So, going into academia, and going into mathematics, it's not necessarily a typical track that a lot of people have opportunities to take on, but I feel very blessed to be doing math that I love serving communities that I love, and being able to raise my families around the communities that I love to. So I feel like you have a special kind of buy-in by engaging in a career that serves my communities and communities that are going to raise my families as well, too.

KK: That’s great.

EL: Nice. So I see over your shoulder a little bit of a Sierpinski triangle. Is that related to the kind of math you like to think about? Or is it just pretty?

BT: Yeah. One, it’s pretty. When I was at the Institute of American Indian arts, most of the students there, they're there for art. They come from Native communities, and they're not there to do mathematics, necessarily. So part of my excitement was to think about ways to broaden the ideas of mathematics and to build off of their creative strengths. And that piece is a piece that one of my students did. They did their own take on a Sierpinski triangle. I have a few of those items from my office where they integrated visual arts and integrated creative aspects of mathematics from cultural aspects as well, too.

KK: So I always think of Native American artists being kind of geometric in nature. It feels that way to me, I mean, at least the limited bit that I've seen. Is that sort of generally true?

BT: The thing about Native art is that Native cultures are diverse in and of themselves too. So there are over 500 federally-recognized tribes, and in Mexico are over 20 tribes alone, 20 nations alone, and each of them have their own notions of geometry and their own notions of their kinds of mathematics that they engage in with respect to the place that their cultures, their identities, and their languages are rooted in. So, yeah, a lot of it is visual, and geometric, because that's what we see. But there's also many I imagine that we don't see, that's embedded in the languages and the practices. Part of my curiosity is seeing how we can recognize what we do and what our traditions are, how we can recognize that as mathematical. And it might be mathematical in the sense that we, as professional mathematicians, might not be accustomed to seeing or experiencing. And, you know, I'm still trying to understand my own cultures, languages and traditions too. So I know mathematics more than a lot of how I experience my own culture. So on one hand, I'm seeing things from a traditional mathematician brought through academia, but I’m also trying to understand things through the lens of someone who's trying to better understand my cultures and histories.

EL: So what is your favorite theorem?

BT: The theorem I chose today was Arrow’s impossibility theorem.

KK: Nice.

EL: Great. And this will be a timely one, at least for the US, because it will be airing—I mean, I guess the past two years basically have been part of the 2020 presidential season—but really in the thick of it. So yeah, tell us a little bit about what this is.

BT: So I'll say more about why I'm kind of drawn to this theorem. So it's a theorem that basically says that there is no perfect ranked voting system, or no perfect way of choosing a winner and, by extension, for me, it kind of brings up conversations about how democracy itself isn’t perfect and that it's really hard to say that a democratic system can accurately represent the will of the people. And I was drawn to this theorem because as I started thinking about the cultural aspects of mathematics and mathematics education, I'm also interested in the power dynamics and the political dynamics and the sociopolitical aspects of mathematics and math education. And a lot of what's out there and written about math education talks about using quantitative reasoning and quantitative analysis and statistical analysis to really engage in critical dialogues and examining inequities and injustices in the world. And all of that is rich and engaging and needed and necessary ways that we can use mathematics to view the world. But the mathematician part of me still misses the definition-proof-lemma aspect of engaging in mathematics. So this theorem kind of represents a way of engaging in politics through some of the theorem-definition- lemma aspects of it. So the way that I understand Arrow’s theorem, and I mentioned this to you before, that I don't know the ins and outs of this theorem, I just really like the ramifications of it and the discussions that it generates. But it basically starts with the idea that we can describe functions where we're considering a way of choosing a winner of an election from a list of candidates. And we're taking each voter’s ranked preference of those candidates. So one thing that we're assuming is that each voter can rank a list of n candidates, A1 through An, and if everyone can rank their preferences, then a voting system would be a way to take all of those, those ranks, or those ballots, and choosing an overall ranking that is supposed to indicate an overall preference for the group of voters.

And what Arrow’s impossibility theorem talks about is that we want values, and want to describe good ways of what a good voting system is. So we want to describe list of criteria that shows that we have a good voting system. So the list of criteria that involves Arrow’s impossibility theorem talks about 1) and unrestricted domain; 2) social ordering; 3) weak Pareto or unanimity; 4) a non-dictatorship; and 5) independence of irrelevant alternatives. And I'll go through what each one means. So basically, an unrestricted domain means that we want a voting system or a way of choosing a winner to be able to take any set of ballots with any number of candidates and be able to give some overall ordering, that these functions are well-defined. So the unanimity condition talks about if everyone prefers one candidate over another, where every single voter has one candidate ranked over another candidate, then the overall function that turns the ballots into an overall social ordering should indicate that that candidate is preferred over the other candidate. And we also don't want a dictatorship, right? And the idea of that mathematically defined is that we don't want one voter deciding exclusively what the overall social ordering is of the candidates. And so we don't want a dictatorship. And we want an independence of irrelevant alternatives, and what that what a lot of people think about as an example of is a “spoiler” candidate or a third party candidate, where even if everyone prefers one candidate over another, that a change in order of a third or other candidate, without disrupting that other order, shouldn't change the overall outcome of an election. They relate that to how sometimes third party candidates can be a spoiler for an election even though overall, it looks like a plurality of voters might prefer one candidate over another. But certain voting systems can have that characteristic where third or other other set of candidates can disrupt the outcome of that election.

KK: I’ve never heard of that.

EL: Wouldn’t it be terrible if that ever happened? [Note: These statements were delivered somewhat sarcastically, presumably referring to the 2000 Presidential election in the US]

BT: Right, right, right. So what Arrow’s impossibility theorem says is that those all may be desired characteristics of a voting system or a social choice function, but that it's impossible to have all of those criteria in a voting system. So the general outline of the proof is that if we have a system that has the unanimity criterion, and an independence of irrelevant alternatives, that if we have those two criteria in a social choice function, then the voting system must be a dictatorship. So if we add those assumptions, then we can go through and show that there is a voter whose sole ordering determines the overall ordering of the voting group, of the voters.

KK: That’s how I always learned this theorem, is that you set down these minimal criteria, and the only thing that works as a dictatorship, right?

BT: Right.

KK: These criteria are completely reasonable, right?

EL: You can’t have it all.

BT: Right, right. They're not outlandish. They're what we might think of as things that we might value in a democracy. And, of course, these, these things don't perfectly replicate what's going on in the real world, but the outcome is still fascinating to me that mathematically, we can show that we can’t have all these sets of what we think are reasonable criteria in a voting system.

KK: Recently, maybe in the last two years, I’ve been getting interested in gerrymandering questions. And there's there's a similar sort of theorem that got proved in the last year or two, which essentially says that, you know, people don't like these sort of weird-shaped districts, they think that's bad somehow, because it's on unpleasing to the eye. But apparently — and there’s also this idea of the efficiency gap, where you sort of want to minimize wastage. So if you laid out some simple criteria, like you want compact districts, and you want to make the efficiency gap, minimized that, then the theorem is you have to have weird shape districts, right? So it’s sort of an impossibility theorem in that way too. So these these kinds of ideas propagate through all of these these kinds of systems,

EL: The real world is impossible.

BT: Right. And even by extension, you know, in many voting theory classes, there's a districting problem, which relates to a good metric for measuring compactness. But then the apportionment issue as well, that it's very hard, if not impossible, to find a fair way of apportioning a whole number of representatives that's proportionate to the state's population, relative to the overall population of the country.

KK: Yeah.

BT: And so yeah, this is one of my favorite theorems because it kind of opens the door to those conversations and gives me another way of thinking about when representatives, or people who talk about the outcomes of elections, say things like “the people have spoken,” “this is the will of the people,” “we have a mandate now,” that I think these outcomes really complicate those claims and should really give us a critical eye and a critical way of really discussing what the will of the people is, and how those discourses really perpetuate the idea that voting, and voting alone, can accurately indicate the will of the people and that that's to be accepted, and that we move forward with them.

EL: Yeah. So have you gotten to use these Arrow’s paradox or any of these other things in classes?

BT: When I was at the Institute of American Indian Arts, I tried to develop a voting theory class. And we got into that and talked about that. And it interested me too because the voting system on the Navajo Nation, we vote for our own council and our own presidents too, and I use this as a way to think about how we have a certain candidate in Navajo Nation who's always running and is seemingly unpopular. And the voting system for president in Navajo Nation is that we have that two-party runoff system where we vote for our top choices and that the top two vote getters participate in a general runoff election. And for a few consecutive elections, this one candidate that is seemingly unpopular just gets enough votes to get into the top two for the runoff election and then gets overwhelmingly outvoted in the general election. So I think for me it was a fascinating way to engage in these kind of mathematical ideas, or mathematical discourses, while talking about some of the real outcomes that are going on in our nations, in our communities, in our efforts towards our self-determination and sovereignty. So I wanted to tie in something that's mathematical, where we can talk about mathematical discussions, with issues that are contemporary and real to our, our peoples.

EL: It’s something I always wonder about is, you know, we've got a theorem that says voting is impossible — or it says that, you know, it's impossible to actually say, like, this is the will of the people. But do you know if much research has been done about, like, real sets of choices that people have and what voting systems might be — do they really experience this paradox, or in the real world, do they have these strange orders of preferences that that confound ranked choice voting rarely?

BT: I imagine that there is research out there and there are people who have engaged in it much more than I have. But something that makes me curious are some of the underlying assumptions that go into Arrow’s theorem and what has been mathematized as necessary criteria, and the values that those might be representative of for certain groups of people. For example, I guess you could call it an axiom of many these voting theory theorems in mathematics is that one voter is one vote, and you know, there are systems where that might not be true. But one of your criteria is one person, one vote. And that one person votes for their own interests and their own interest only, and there are extensions of these criteria where if we have other non-ranked voting systems, then it can help.

But let me backtrack: one of the outcomes of Arrow’s theorem is that when people know that it's impossible for the outcome to really represent the will of the people, then it could result in people voting for candidates other than their first option because they know that voting for someone other than their true option because we election in favor of something that's not of their desire. So we have people voting against their own actual first choices. And that happens with ranked-choice voting, and some of the extensions of these conversations have been about voting systems that don't require ranked choice. So perhaps giving each candidate a rating, and it helps alleviate some of those issues with ranked-choice voting, and it helps alleviate those issues of third-party candidates, where you can still give your candidate five stars out of five, like an Amazon review, but still really give perhaps a better indication of your true view of the candidates, rather than a linear ranking. So it kind of reveals that there are some issues with just linear ranking of candidates, when the way that we think about in value and understand our preference of candidates might be much more complex than a simple 1 through n ranking. But kind of going back to what I think this could mean for communities and other societal perspectives, is in many democracies, that one vote-one choice is kind of an assumption that that's what we want. But for many communities, perhaps we want to vote for something that does benefit an overall view of the people. What would that look like as a criteria if we allowed for something like that? What would we do if we allow criteria, or embedded in our definitions, some way of evaluating how if when we register a vote, that we're all not only taking into account our own individual interests, but the interests of our land, of our communities, of our nations. So those are cultural values that are not assumed in the current conversations, but for many communities in many Indigenous nations, those are some things that are real and necessary to think about. What would that look like if we expand those and then be critical of those assumptions that are underlying these current conversations on voting theory in mathematics.

EL: So one of the other things we do on this podcast is We ask our guests to pair their theorem with something. What have you chosen to pair with this theorem?

BT: I have a ranking of three pairings.

EL: Great. I’m so glad! Excellent.

BT: So I have 1-2-3. So I'll give my third choice first. The third out of three pairings: green chili cheeseburgers.

EL: Okay.

BT: And in New Mexico, everyone has their favorite place to get a green chili cheeseburger, and we take pride in our green chili, and every year any contest about the green chili cheeseburger and who has the best green chili cheeseburger causes some conversation, and it causes some controversy and rich discussions over who has the best green chili cheeseburger. So, I think about that as a food that has a lot of controversy as to who has the best green chili cheeseburgers in New Mexico. The second pairing is another food item, the Navajo taco.

EL: Oh yeah. Those are good.

KK: What’s in those?

BT: So, well, what we call a Navajo taco is a piece of frybread with toppings often involving meat and cheese, with lettuce and tomato and maybe some chili. And this is another controversial discussion in Native communities because we call it a Navajo taco, but it's not just Navajos who make this kind of dish, because many communities make their own versions of frybread. And so some places call it Indian tacos, and there's a lot of controversy over which community first introduced the Navajo taco and why some people call it the Navajo taco and others call it Indian tacos. And so in Native communities, there's a lot of controversy over what constitutes the best version of this dish. And the other reason I'm pairing that is the frybread itself comes from a time where it was created out of necessity for survival, where the flour that had been rationed out to our communities was rancid, and in order to actually make it edible, it was deep fried. And so on one hand, it represents a point in time where our communities were just fighting for survival, and it also represents their ingenuity, and became a part of our everyday practice. But at the same time, it's a reminder that that was something that was imposed on our communities, much like voting systems nowadays. It's an act of our survival and our sovereignty, the voting systems that we have in place. But I think there's also need to come back and have other conversations about what's good for our communities.

And the first-ranked pairing is mathematics itself with Arrow’s theorem. So we have a lot of conversations about how mathematics is universal, mathematics is for everyone, that everyone can do mathematics, and that everyone can participate in mathematics. But for many people from from equity, justice and diversity perspectives, we want to be critical about who has access to mathematics, whose ideas of mathematics are represented in our mainstream ways of thinking about mathematics. Just like we think about democracy as being the will of the people and being a representation of all the people, that Arrow’s is kind of a critique of that notion of democracy. And I think mathematics, we can take a lesson from this theorem and think about what we mean when we say mathematics is universal or mathematics is for everyone or mathematics is for all, when this term itself is kind of a democratic take on mathematics, that everyone can do mathematics, and everyone can be an equal participant in mathematics. But, you know, we think the same thing about democracy, and this theorem says that there are some issues with that. So I'm interested in seeing how we can take this lesson and how we can think about how we can be more critical about the ways we think about mathematics itself.

EL: Yeah, well, you know, Arrow’s paradox is not about this, but we have issues with people who can't vote for various reasons and should be able to vote, or places that shut down polling places in certain communities to make it so people have to stand in line for six hours. Which is, you know, not easy to do if you've got a job that you need to get to. So yeah, there's so much richness. I love that you paired a ranking of three things with this. And now I feel like we should also vote on these, but I just don't think it's fair for one of them to be math. I mean, you’ve got two mathematicians here, three mathematicians here in total. I think it's going to be a blowout.

KK: No, tacos win every time, don’t they?

EL: I should have known.

KK: This is a really good pairing. I like this a lot.

EL: Yeah.

KK: We also like to give our guests a chance if they want to plug anything. Where can we find you online for example, or can we?

BT: Probably the best way to find me is on Twitter. My Twitter handle is @lobowithacause.

EL: Yeah. You'll see him popping up everywhere. Is that the mascot for the University of New Mexico?

KK: It is, the lobos.

EL: And I believe a talk that you gave at the Joint Math Meetings, is there video of that available somewhere?

BT: I was told that there would be video. I haven't found it yet. There was a video recorded. And I'll follow up with that and see that it gets out. I'll make an announcement on Twitter.

KK: I’ve noticed those have been trickling out kind of slowly. It'll show up, I think.

EL: Yeah, we'll try to dig it up by the time we put the show notes together so people can watch that. Unfortunately, I was still making my way to Denver when that happened, so I didn't get to see it. So selfishly I very much want to see it. I heard really good things about it. So thank you so much for coming on here and giving us a lot to think about.

BT: Oh, it was an honor. And you know, I love your podcasts.

KK: Thanks so much.

BT: I love what you’re doing. I had fun in listening to your other podcasts in preparation for this and loved hearing Henry Fowler and shout out to Moon Duchin too. I heard that you, Kevin, went to that gerrymandering work in Boston a few years ago. I was there too. And I had a great week there.

EL: Oh, nice.

KK: That was a big workshop. There was no way to meet everybody. Yeah,

EL: Thanks for joining us, and have a good rest of your day.

BT: Thank you. Thank you. You too.

In this episode of the podcast, we were happy to talk with Belin Tsinnajinnie, a professor at Santa Fe Community College, about Arrow's impossibility theorem, which basically says that a perfect voting system is impossible. Below are some links you might enjoy as you listen to the episode.
Arrow's impossibility theorem
Cardinal voting, an alternative to voting systems that are based on ranking the options
Our episode with Henry Fowler, who was at the time on the faculty of Diné College and is now at Navajo Technical University
Our episode with Moon Duchin, who studies gerrymandering, among other things
Belin Tsinnajinnie on Twitter

Episode 55 - Rebecca Garcia

Evelyn Lamb: Hello and welcome to my favorite theorem. Math podcast. I'm one of your hosts Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. How's it going?

EL: All right, it is a bright sunny winter day today, so I really like—I mean, I'm from Texas originally, so I'm not big on winter in general, but if winter has to exist, sunny winter is better than cloudy winter.

KK: Sure, sunny winter is great. I mean, it's a sunny winter day in Florida, too, which today means it is currently, according to my watch, 81 degrees.

EL: Oh, great. Yeah.

KK: Sorry to rub it in.

EL: Fantastic. It is a bit cooler than that here.

KK: I’d imagine so.

EL: So yeah. Anything new with you?

KK: No, no. Well, actually so so my I might be going to visit my son in a couple of weeks because he's studying music composition, right? And the the orchestra at his at his university is going to play one of his pieces, and so kind of excited about that.

EL: Very exciting! Yeah, that's awesome.

KK: Yeah, but that's about it. Otherwise, you know, just dealing with downed to trees in the neighborhood. Not in our yard, luckily, but yeah, stuff like that. That's it.

EL: Yeah. Well, we are very happy today to have Rebecca Garcia as a guest. Hi, Rebecca. How are you?

Rebecca Garcia: Hi, Evelyn. Håfa ådai, I should say, håfa ådai, Evelyn, and håfa ådai, Kevin. Thanks for having me on the program.

EL: Okay, and what—håfa ådai, did you say?

RG: Yeah, that's right. That's how we, that's our greeting in Chamorro.

EL: Okay, so you are originally from Guam, and is Chamorro the name of a language or the name of a group of people, or I guess, both?

RG: It’s both actually. Yes. That's right. And so Chamorro is the native language in the island. But people there speak English mostly, and as far as I'm able to tell I think I'm the first Chamorro PhD in pure mathematics.

EL: Well, you’re definitely the first Chamorro guest on our show. I think the first Pacific Island guest also.

KK: I think that's correct. Yeah.

EL: So yeah, how did you—so you currently are not in Guam. You actually live in Texas, right?

RG: I do. I'm a professor at Sam Houston State University, which is in Huntsville, Texas, north of Houston. And I'm also one of five co-directors of the MSRI undergraduate program.

EL: Oh, nice. That seems like it is a great program. So how did you how how did you get from Guam to Huntsville?

RG: Oh my goodness. Wow. That is a that is a long, long journey.

KK: Literally.

RG: I started out as a as a undergraduate at Loyola Marymount University, and I had the thought of becoming a medical doctor. And so I thought we were supposed to do some, you know, life science or you know, chemistry or biology or something along those lines. And so I started out as one of those majors and had to take calculus and fell in love with calculus and the professors in the math department. And I was drawn to mathematics. And that's how I ended up on the mathematics side. And one of the things that I learned in my undergraduate career was these really crazy math facts about the rational numbers. And so that's one of the things that interested me in mathematics, was just the different types of infinities the concept of countable, uncountable, those sorts of things.

EL: Yeah, those those seem to be the kinds of facts that draw a lot of people into this rich world of creativity and math that you might not initially think of as related to math when you're going through school. So I think this brings us to your favorite theorem, or at least the favorite theorem you want to talk about today.

KK: Sounds like it.

EL: Yeah, so what’s that?

RG: Yeah. So it’s more, I would say, more of a fun fact of mathematics that the rationals first of all are countable, meaning they are in one-to-one correspondence with the natural numbers. And so you can kind of, you know, label them, there's a first one and a second one in some way, not necessarily in the obvious way. But then, at the same time, they are dense in the real numbers. So that to me, just blows my mind, that between any two real numbers, there's a rational number.

EL: And yeah, so you can't like take a little chunk of the real line and miss all the rational numbers.

RG: That’s right.

KK: Right.

RG: That to me just blows my mind. Because—and then you just sort of start, you know, your brain just starts messing with you, you know, between zero and one there are infinitely many rational numbers and yet they're still countable. And it just, it just starts to mess with your mind a little bit. Right?

EL: Yeah. Well, and we were we were talking about this a little bit before and it's this weird thing. Like, yeah, there's, like a countable is like a smallness thing. And yet dense is like, they're, you know, they fill up the whole interval this way. I mean, it is really weird. So where did you first encounter this?

RG: This was in a class in real analysis. And, yeah, so that's where I started to…I thought I was going to be a functional analyst. I thought I was that's what I wanted to do this. I love real analysis. That didn't happen either. But it was in that class where we were talking about just these strange facts, like the Cantor set: that set is a subset of the reals that is uncountable and yet it’s sparse.

KK: Totally disconnected, as the topologists say.

RG: Totally disconnected. There you go. Yeah. Right. And so then all these weird things are happening. And you're just in this world where you thought you understood the real line, and then they throw these things at you like, the reals are dense. I mean, the rationals are dense in the reals, you have these weird uncountable sets that are totally disconnected. What's going on? Yeah, so that's where I started to hear about all these weird things happening.

KK: Right. So one of two things happens when people learn these things, right? It either blows their minds so much they can't keep going. Or it intrigues them so much that you want to learn more. But not be an analyst. Right?

RG: [laughing] That’s right. At some point I fell in love with computational algebraic geometry and these Gröbner bases, and how you can really get your hands on some of these things and their applications to combinatorics. So I ended up, I had an algebraist’s heart, but I was exposed to some really good analysts early in my career. And so I was very confused. But I've always, I stay true to my algebraic heart and follow that mostly.

EL: And so is this a fact that you get to teach to your students now ever?

RG: So no, this is not, but I do like to talk about the the different infinities and things along those lines. And I like to, before class I come in early, and I'll have a little chat with them about just the fact that—you know, they they don't understand that math is not “done.” So, there's still so much to do. And they have no idea that, you know, there's what, what is research like? What does that mean? And so I talk about open questions. And I bring some of that in the beginning of class. And these concepts that had also drawn me in, about the different kinds of infinities and these weird concepts about the rationals being dense and, you know, just things like that. I do get to talk about it, but it's not in a class that I would teach the material on.

EL: Yeah, just going back to this idea that you've got the rationals that are dense, so it's this, like, measure zero small set, but it's like everywhere. And then you've got the Cantor set, which is uncountable and sparse. It's like, we've got these various ways of measuring these sets. And you think that they line up in some natural way. And yet they don't. It's just like, you know, the density is measuring a different type of property of the numbers than the measure is.

RG: That’s exactly.

EL: And actually, I guess countability is a different thing. Also, I mean, it's, yeah, it's so weird. And it's hard to keep all these things straight. My husband does a lot of analysis and like has, yeah, all of these, like, what kinds of sets are what.

RG: And what properties they have. And yeah, I don’t have that completely straight.

KK: This is why I’m a topologist.

EL: But I mean, topology is like,

KK: Oh, it's weird too.

EL: It’s secretly analysis.

KK: Well…

EL: Analysis wishes it was topology, maybe.

KK: So my old undergraduate advisor—who passed away last summer, and I was really sad about that—but he always he always referred to topology as analysis done right.

EL: Shots fired.

KK: Which is cheap, of course, right? Because you prove all this stuff in topology Oh, the image of a connected set is connected. Yeah, that's easy now go off to the real line and prove that the connected sets are the intervals. That's the hard part. Right? So yeah, he's being disingenuous, but it was. It's a good line. Right.

RG: Right.

EL: So you said that you ran into this, was this an undergraduate class where you first saw these notions of countability and everything?

RG: Right, it was an undergraduate class where I ran into those notions and I was a junior, well, I guess it was in my second semester as a junior, where we were talking about these strange sets. And that's when I had also thought about going on to graduate school and wanting to do mathematics for the rest of my life. I mean, I was a major by then, of course, but I just didn't know what I was going to do. But it wasn't until then, when I learned about, well, this is this could be a career for you. This may be something you like to do. And of course, this was many, many years ago. And nowadays, you can do so much more with mathematics, obviously. I mean, we know that we can do so much more, I should say. We've always been able to do so much more. We just haven't been able to share that with our students so much. We never really spent the time to let them know there's so many careers and mathematics that one can do. But anyway, at that, at that time I was I was drawn into really thinking about becoming a mathematician, and that was one of the experiences that that made me think that there's so much more to this than than I originally thought.

EL: Yeah, well, I talk to a lot of people, you know, in my job writing and doing podcasts and stuff about math, and there's so many people who don't realize that, like, math research is a career you can do.

RG: Right.

EL: And the more we can share these kinds of “aha” moments and insights, the better and, you know, just show like, well, you can use, you know, kind of the logic and the rules of the game to like, find out these really surprising aspects of numbers.

RG: Right. And I think also, one of the experiences that I've had as an undergrad that really just sort of sealed the deal—I’m going to go into mathematics—was doing an undergraduate research program as a student. Well wasn't really at the time an undergraduate research program, it was just another summer program. This is many years ago, almost before all of that. And I had the chance to spend a summer just thinking about mathematics at a higher level with a cohort of other students who were like-minded as well, you know. And it was really—it was it was like, “Oh, I can do this for the rest of my life? Like how amazing is that?” And so, I was part of a summer program as an undergrad. And then when I was a graduate student, my lifelong mentor, Herbert Medina, was running a program in Puerto Rico and asked me to be a TA while I was a grad student. And so these were some of the things that led me to do what I do now, working with undergraduates, doing research and mathematics.

EL: And so that ties in to the MSRI program that you are part of, right?

RG: Right.

EL: I guess it I've seen it written like MSRI-UP. So I guess that's undergraduate program?

RG: Yes. Undergraduate Program. That's right. Yeah. Well, that that's sort of like, a different stage that I'm at now. But yeah, before that, I started my own undergraduate research program together with colleagues in Hawaii, at the University of Hawaii at Hilo. And we ran an undergraduate research program called PURE Math, and that was Pacific Undergraduate Research Experience in Mathematics. And we ran that for five years. And then, and then I ended up moving into the co-director role at MSRI-UP.

EL: Nice.

KK: That’s a great program.

EL: Yeah. So the other thing we like to ask our guests to do, is to pair their theorem with something. You know, just like the right wine can enhance that meal, you know, what would you recommend enjoying the density of the rationals with?

RG: Well, I did think about this a bit. And one of the things that I think, you know, you think the rationals are dense but they really shouldn't be? So, I think of foods that are dense, but they really shouldn't be, and one of those foods that comes to mind, especially being here in Texas, but also being married to a mathematician who is from Mexico, is tamales. So tamales really should not be dense. They should be fluffy and sumptuous, but here in Texas, you find really dense the most, unfortunately. But it It was strange to also discover that growing up in Guam, we also have our own version of tamales, and a lot of the foods are related in some way to foods from Mexico. So I feel like there's this huge rich connection between myself being from Guam, my husband being Mexican and there's just this strange richness that we share this culture, that I don't know, it just blows my mind too. So the same way that the rational is being dense in the reals blows my mind.

EL: All right, well, I have to ask more about this tamale like creation from in traditional Guam cuisine. What, is that wrapped in, like, banana leaves or something like that?

RG: It ought to be, and maybe traditionally it was. I think that nowadays it's not that way. They usually serve it in aluminum foil, and it's made—it's a mixture like tamales. So tamales in Mexico are made with corn, right?

KK: I was about to ask this. What are they made of in Guam?

RG: Yeah, yeah. And so in Guam we actually use, like, a rice product.

EL: Okay.

RG: It's ground up just like corn. And so instead of corn, we're using rice, and it's flavored in different ways.

KK: Interesting.

EL: All right. I have kind of in my mind because I'm more familiar with this like almost, is it kind of like a mochi texture? Because, I mean, that's a rice product, but maybe it's not maybe that's like more gelatinous than this would be.

RG: Yeah, I guess mochi is really pounded and yeah, so yeah, that's more chewy. I think that the tamal, well, you wouldn't say it like that, but the tamales in Guam are very soft and, gosh, I don't know how to describe it. But it's a very soft textured food.

KK: I would imagine the rice could be softer, and I mean, corn can get very dense, especially when you start to put lard in it and things like that.

RG: Yes.

KK: I mean, it’s delicious.

RG: It is delicious. And oh my, I can’t get enough tamales. Oh, well.

KK: Yeah, maybe you can.

RG: Yeah, I should learn.

EL: Yeah, well, nice. I unfortunately, we do have a couple restaurants in Salt Lake that are Pacific Island restaurants, but we have more people from Samoa and Tonga here. I don't know if we have a lot of people from Guam here. Yeah, there's actually like a surprising number of like, Samoans who live in Salt Lake. Who knew?

RG: Right.

EL: But yeah, it's it's because of like the history of Mormon missionaries.

KK: That’s what I was gonna say.

EL: Yeah, the world is very interesting, but yeah I don't know if I've seen this kind of food there. I will just have to, you know, if I'm ever in Huntsville I’ve got to get you to make me some of this. I’m just inviting myself over for dinner now. Hope you don't mind.

RG: That would be great. It would be wonderful to have you here.

EL: Is there anything else you'd like to share? We'd like to give our guests a chance to like, share, you know if they've got a website or blog or book or anything, but also if you want to share information about MSRI-UP, application information, anything like that for students? Anything you'd like to share?

RG: Oh, wow. That's a lot of stuff.

EL: Yeah, I know. I just rattled off a ton of things.

RG: Well, yes, I do have, I guess I would like to say for the undergraduate listeners in the audience, please consider applying to our MSRI-UP program, and just in general apply to a research program in the summer. These are paid opportunities for you to expand your mind and do some mathematics in a great environment, and so I highly recommend considering applying for that. And so this is the time right now of course by the time the listeners hear this, I’m sure it will be over, but consider doing some undergraduate research or using your summer wisely.

KK: I parked cars in the summer in college. I did.

EL: Well, you never know the connections that might happen though because I was talking to someone one time who basically his big break to get to go to grad school came because, like, somehow he was involved in like parking enforcement somewhere, and some math professor called in to complain about, like, getting a ticket, and one thing led to another and then he ended up in grad school. So really, you never know. Maybe that's not the ideal route to take. There are more direct routes, but yeah, there are many paths.

RG: Yes, there are. And there's also another, I guess another thing to flag would be, well, contributed to a book that Dr. Pamela Harris and others have put together on undergraduate research. So that just I guess that was just released. I'm not entirely sure now. I think it was accepted, and I don't know if if one is able to purchase it, but if you if you consider working with your students on undergraduate research, this is a great resource to use to get you going, I guess.

KK: Great.

EL: Oh, awesome. So this is like a resource for like faculty who want to work with undergraduates? Oh, that's great.

RG: Yes.

EL: We will find a link to that and put that in the show notes for people.

RG: That sounds good.

EL: Okay, great. Thanks so much for joining us.

KK: It’s been great.

RG: Thank you so much.

On this episode of My Favorite Theorem, we were happy to talk with Rebecca Garcia, a mathematician at Sam Houston State University, about the density of the rational numbers in the reals. Here are some links you might find helpful.

Her website
A biography of Garcia for SACNAS
MSRI-UP 
A Project-Based Guide to Undergraduate Research in Mathematics, the book she mentioned contributing to

Episode 54 - Steve Strogatz

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast and so much more. I'm Kevin Knudson, professor of mathematics at the University of Florida, and here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer based in Salt Lake City, where it is snowy, but I understand not as snowy as it is for our guest.

KK: I know, and we've been trying to make this one happen for a long time. So I'm super excited that this is finally going to happen. So today we are pleased to welcome Professor Steve Strogatz. Steve, why don't you introduce yourself?

Steve Strogatz: Well, wow, thank you. Hi, Kevin. Hey, Evelyn. Thanks for having me on. Yeah, I've wanted to be on the show for a very long time. And I think it's true what Evelyn just said, we have a very big snowstorm here today in not-so-sunny Ithaca, New York, upstate. I just took my dog out for a walk, and the snow was over my boots and going into them and making my feet wet.

KK: See, I have a Florida dog. She wouldn't know what to do. Actually, we were in North Carolina a few years ago at Christmas, and it snowed, and she was just alarmed. She had no idea what to do. And she's small, too, she just couldn't take it.

SS: Yeah, well, it would be more like tunneling than running.

KK: Right.

EL: Yeah. So we actually met quite a few years ago at this point — actually, I know the exact date because it was, like, two days before my brother's wedding the first time we met because you were on the thesis committee for my sister in law, who is a physicist, many years ago, and so we have this weird, it was when I had just moved to New York to work at Scientific American for the first time. So it was at the very beginning of my life as a math writer. And I remember just being floored by how generous you were with being willing to meet with a nobody like me.

SS: Well that’s nice.

EL: At this time when I was first starting.

SS: But actually, I had a crystal ball, and I knew you were going to become the voice of mathematics for the country, practically. I mean, so I let me brag on Evelyn’s behalf a little bit. If you go on Twitter, you—I wonder if you know this, Kevin, do you know this little factoid I'm going to unreel?

KK: I bet I do.

SS: You know where I'm going. On Twitter, if you ask “What mathematician do other mathematicians follow?” I think Evelyn is the number one person the last time I checked.

KK: She is indeed number one. That's right.

SS: Yeah.

EL: I like to say I'm the queen of math, Twitter, although I don't actually like to say this because it feels really weird.

SS: Well that’s okay. You didn't say it. But yeah, I do remember our meeting that day in my office. And right, it was on this happy occasion of a family, of a wedding. Okay, sorry, I interrupted you, Kevin.

KK: Oh, I don't know. I was going to say with the Twitter thing. I think you're not far behind, right? Like, aren't you number two, probably?

SS: I think the last time I looked I was number two.

KK: Yeah.

SS: So look at that. Okay, so look at that, the two tweet monsters here.

KK: And now the funny thing is I'm not even on that list. So here we go.

SS: Okay. Yeah, well you could catch up. I'm sure you'll be coming right on our heels.

KK: Maybe. I have over 1000 followers now, but apparently not that many mathematicians. So this is how this goes. Anyway, what weird times we live in, right?

SS: It's very weird. I mean, I don't know what this can get us, a cup of coffee or what.

KK: Maybe, maybe. Okay. Let's talk theorems. So Steve, you must have a favorite theorem. What is it?

SS: Yeah, I have a very sentimental attachment to a theorem and complex analysis called Cauchy’s theorem, or sometimes called Cauchy’s integral theorem.

KK: Oh, I love that theorem.

SS: It’s a fantastic theorem. And so I don't know. I mean, I feel like I want to say what I like about it mathematically and what I like about it personally. Does that work?

EL: Yeah, that’s exactly what we want.

SS: Well, okay. So then, the scene is, it's my sophomore year of college. Maybe I'll start with the emotional.

KK: Okay.

SS: It’s my sophomore year of college. I've just gotten very demoralized in my freshman year, taking the the honors linear algebra course that a lot of universities offer as a kind of first introduction to what college math is really going to be like. You know, a lot of kids in high school have done perfectly well in their precalculus and calculus courses, and then they get to college and suddenly it's all about proofs and abstraction. And it can be—I mean, we sometimes call it a transition course, right? It's a transition into the rigorous world of pure math. And so it was a shock for me. I had a lot of trouble with that course. I couldn't read the book very well, it didn't have pictures. And I'm kind of visual. And so I was always at a loss to figure out what was going on. And being a freshman I didn't have any sense about, why don't I look at a different book, you know, or maybe, maybe I should switch sections. Or I could ask my teaching assistant, or I could go to office hours. I didn't know to do any of that stuff.

So anyway, this is not my favorite theorem. I was very demoralized after this experience in linear algebra. And then when I took a second semester, also an honors course, that was a rigorous calculus course with the Heine-Borel theorem, and, you know, like, all kinds of—again, no formulas, it was all about, I remember hearing this stuff about “every open cover has a finite subcover,” and I thought, “I want to take a derivative! I can't do anything here. I don't know what to do!” So anyway, after that first year, I thought, “I don't have the right stuff to be a mathematician. And so maybe I'll try physics,” which I also always loved. I say all that as preamble to this complex analysis course that I was taking in sophomore year, which, you know, I still wanted to take math, I heard complex variables might be useful for physics, I thought it would be an interesting course. I don't know. Turned out it was a really great course for me because it really looked a lot like calculus, except it was f(z) instead of f(x).

KK: Right.

SS: You know, but everything else was kind of what I wanted. And so I was really happy. I had a great teacher, a famous person actually named Elias Stein.

KK: Oh.

SS: So Stein is a well-known mathematician, but I didn't know that. To me, he was a guy who wore Hush Puppies and, you know, had always kind of a rumpled appearance, came in with his notes. And he seemed nice, and I really liked his lectures. But so one day, he starts proving this thing, Cauchy’s theorem, and he draws a big triangle on the board. And he's going to prove that the integral of an analytic function f around this triangle is zero no matter what f is. All he needs is that it's analytic, meaning that it has a derivative in the sense of a function of a complex variable. It's a little more stringent condition—actually a lot more stringent than to say a function of a real variable is differentiable, but I didn't appreciate that at the time. I mean, that's sort of the big reveal of the whole subject.

KK: Right.

SS: That this is an unbelievably stringent condition. You can’t imagine how much stuff follows from this innocuous-looking assumption that you could take a derivative, but okay, so I'm kind of naive. Anyway, he says he's going to prove this thing, only assuming that f is analytic on this triangle and inside it. And that's enough. And then, you know, I feel like you don't have enough information, there's nothing to do! So then he starts drawing a little triangle inside the big triangle, and then little triangles inside the little triangle. And it starts making a pattern that today I would call a fractal, though I didn't know it at the time, and he didn't say the word fractal. And actually, nobody ever says that when they're doing this proof. But it’s—right, they don’t—but it's triangles inside of triangles in a self-similar way that doesn't actually play any particular role in the proof, other than it's just this bizarre move, like, What is going on? Why is he drawing these triangles inside of triangles? And by the end, I mean, I won't go into the details of the proof, but he got the whole thing to work out, and it was so magnificent that I started clapping.

And at that point, every kid in the room whipped their head around to look at me, and the professor looked at me, like what is wrong with you? You know, and yet, I thought, “Wow, why are you guys looking at me?” This was the most amazing theorem and the most amazing proof.” You know, so anyway, to me, it was a very significant moment emotionally because it made me feel that math was, first of all, something I could do again, something I could appreciate and love, after having really been turned off for a year and having a kind of crisis of confidence. But also, you know, aside from any of that, it's just, I think people who know would regard this proof —this is actually by a mathematician named Goursat, a French mathematician who improved on Cauchy’s original proof. Goursat’s proof of Cauchy’s theorem is just one of the great— you know, it's from “The Book” in the words of Paul Erdős, right? If God had a proof of this theorem, it would be this proof. Do you guys have any thoughts about that? I mean, I'm assuming you know what I'm talking about with this theorem and this proof.

KK: Well, this is one of my favorite classes to teach because everything works out so well. Right? Every answer is zero because of Cauchy’s theorem, or it's 2πi because you have a pole in the middle, right?

SS: Yeah.

KK: And so I sort of joke with my students that this is true. But then the things you can do with this one theorem, which does—you’re right, it's very innocuous-looking, you know, you integrate an analytic function on a closed curve, and you get zero. And then you can do all these wonderful calculations and these contour integrals and, like, the real indefinite integrals and all this stuff. I just love blowing students’ minds with that, and just how clean everything is.

EL: Yeah, I kind of—I feel like I go back and forth a little bit. I mean, like, in my Twitter bio, it does have “complex analysis fangirl.” And I think that's accurate. But sometimes, like you said, it's so many of these, you know, you're you're like teaching it or reading it and you're like, “Oh, this is complex analysis is so powerful,” but in another way, it's like our definition of derivative in the complex plane is so restrictive that like, we're just plucking the very nicest, most well-behaved things to look at and then saying, “Oh, look what we can do when we only look at the very most well-behaved things!” So yeah, I kind of go back and forth, like is it really powerful or are we just, like, limiting ourselves so much in what we think about?

KK: And I guess the real dirty secret is that when you try to go to two complex variables, all hell breaks loose.

SS: Ah, see, I've never done that subject, so I don't appreciate that. Is that right?

KK: I don't, either. Yeah. But I mean, apparently, once you get into two variables, like none of this works.

SS: Ohhh. But that's a very interesting comment you make there, Evelyn, that—you know, in retrospect, it's true. We've assumed, when we make this assumption that a function is analytic, that we are living in the best of all possible worlds, we just didn't realize we were assuming that. It seems like we're not assuming much. And yet, it turns out, it's enormously restrictive, as you say. And so then it's a question of taste in math. Do you like your math really surprising and really beautiful and everything works out the way it should? Or do you like it thorny and full of rich counterexamples and struggles and paradoxes? And I feel like that's sort of the essential difference between real analysis and complex analysis.

EL: Yeah.

SS: In complex analysis, everything you had dreamed to be true is true, and the proofs are relatively easy. Whereas in real analysis, sort of the opposite. Everything you thought was true is actually false. There are some nasty counterexamples, and the proofs of the theorems are really hard.

EL: Yeah, you kind of have to MacGyver things together. “Yeah, I got this terrible epsilon and like, you know, it's got coefficients and exponents and stuff, but okay, here you go. I stuck it together.

KK: But but that's interesting, Steve, that this is your favorite theorem because, you know, you're very famous for studying kind of difficult, thorny mathematics, right? I mean, dynamics is not easy.

SS: Huh, I wouldn't have thought that, that's interesting that you think that. I don't think of myself as doing anything thorny.

KK: Okay.

SS: So that's interesting. I mean, yes, dynamical systems in the hands of some practitioners can be very subtle. I mean, those are people who have a taste for those those kinds of issues. I've never been very sophisticated and haven't really understood a lot of the subtleties. So I like my math very intuitive. I’m on the very applied end of the applied-pure spectrum, so that sometimes people will think I'm not really a mathematician at all. I look more like a physicist to them, or maybe even, God forbid, a biologist or something. So yeah, I don't really have much taste for the difficult and the subtle. I like my math very cooperative and surprising. I like—well, not surprising for mathematical reasons, but more surprising for its power to mirror things in the real world. I like math that is somehow tapping into the order in the world around us.

EL: Yeah, so this it's interesting to me, also that you picked this because, yeah, as you say, you are a very applied mathematician. And I think of complex analysis as a very pure—I actually, I'm trying to not say “pure” math, because I think it's this weird, like, purity test or something. But you know, that like a very theoretical thing. So does it play into your field of research at all?

SS: Well, uh, not particularly. Yeah. So that's a good question. I mean, I have to say I was a little intimidated by the title of the podcast. If you ask me what's my favorite theorem, the truth is for me, theorems are not my favorite things.

KK: Okay.

SS: My favorite things are examples or mathematical models. Like there’s a model in my field called the Kuramoto model after a Japanese physicist Yoshiki Kuramoto. And if you asked me what's my favorite mathematical object, I would say the Kuramoto model, which is a set of differential equations that mirrors how fireflies can get their flashes in sync, or how crickets can chirp in sync, or how other things in nature can self-organize into cooperative, collective oscillation. So that's my favorite object. I've been studying that thing for 30 years. And I suppose there are theorems attached to it, but it's the set of equations themselves and what they do that is my favorite of all. So I don't know, maybe that's my real answer.

KK: Well, that’s fine. So yeah, it's true. We've had people who've done that in the past, they didn't have a favorite theorem, but they had a favorite thing.

SS: But still, I mean, I am still a mathematician, part of me is, and I do have theorems that I love, and one of the things I love about Cauchy’s theorem is that in the proof, with this drawing of all the nested triangles inside the big triangle, you end up using a kind of internal cancellation. The triangles touch other triangles except on their common edge, sometimes you're going one way, and sometimes you're going in the opposite direction on that same edge. And so those contributions end up cancelling. And you end up, the only thing that doesn't cancel is what's going on around the boundary. And then that can be sort of pulled all the way into a tiny triangle in the interior, which is where you end up using the local property that is the derivative condition to get everything that you need to prove the result about the big triangle on the outside.

But the reason I'm going into all that is that this is a principle, this internal cancellation, that is at the heart of another theorem that's been featured on your show, the fundamental theorem of calculus, which uses a telescoping sum to convert what's happening on the boundary to what's happening when you integrate over the interior. This idea of telescoping I think, is really deep. I mean, it's what we use to prove Stokes’ theorem. It's what we would use to prove all the theorems about line integrals. It comes up in topology when you're doing chains and cochains. So this is a principle that goes beyond any one part of math, this idea of telescoping. And I've been thinking I want to write an article, someday (I haven't written it yet) called “Calculus Through the Telescope” or “A Telescopic View of Calculus” or something like that, that brings out this one principle and shows its ramifications for many parts of math and analysis and topology. I think some people get it, people who really understand differential forms and topology know what I'm talking about. But no one ever really told me this, and I feel like maybe it should be mentioned, even though it is well-known to the people who know it.

KK: Right, it's the air we breathe, right? So we don't we don't think about it.

SS: I guess, but like, I think there are probably high school teachers, or others who are teaching calculus—like for instance, when I learned about telescoping series in my first calculus course, that's just seems like a trick to find an exact sum of a certain infinite series of numbers. You know, they show you, “Okay, you could do this one because it's a telescoping series.” And it seems like it's an isolated trick, but it's not isolated. This one idea—you can see the two- dimensional version of it in Cauchy’s theorem, and you can see the three-dimensional version of it in the divergence theorem, and so on. Anyway, so I like that. I feel like this idea has tentacles spreading in all directions.

EL: Yeah. Well, this makes me want to go back and think about that idea more because, yeah, I wouldn't say that I would necessarily have thought to connect it to this many other things. I mean, you did preface your statement with “those who really understand differential forms,” and my dark secret is that the word “form” really scares me. It's a tough one. It's somehow, that was one of those really hard things, when I started doing more, like, hard real analysis. It's like, I feel like I always had to just kind of hold on to it and pray. And you get to the end of it. You're like, “Well, I guess I did it.” But I feel like I never really got that full deep understanding of forms.

SS: Huh. I don't I don't claim that I have either. I'm reminded of a time I was a teaching assistant for a freshman course for the the whiz kids that—you know, every university has this where you throw outrageous stuff at these freshmen, and then they rise to the occasion because they don't know what you're asking them to do is impossible. But so I remember being in a course, like I say, as a teaching assistant, where it was called A Course in Mathematics for Students of Physics, based on a book by Shlomo Sternberg, at Harvard, and Paul Bamberg, who's a physicist there too, and a very good teacher. And that book tried to teach Maxwell's equations and other parts of physics with the machinery of differential forms and homology and cohomology theory to freshmen. But what was amazing is it sort of worked, and the students could do it. And in the course of teaching it, I came to this appreciation of integrating forms, and how it really does amount to this telescoping sum trick. And, anyway, yeah, it's true, that maybe it's not super widely appreciated. I don't know. I don't know if it is, I don't want to insult people who already know what I'm talking about. But I I do feel like there's a story to tell here.

KK: Okay. Well, we'll be looking for that.

EL: Yeah.

SS: Someday.

KK: In the New York Times, right?

SS: Well.

KK: So another thing we do on this podcast is we ask our guests to pair their theorem with something. And we might have sprung this on you, but you seem to have thought of a solution here. So what have you chosen to paired with Cauchy’s theorem?

SS: Cubist painting.

KK: Oh, excellent. Okay. Explain.

EL: Yeah, tell us why.

SS: Well, I'm thinking of Cubism. I don’t—look, I don't know much about art. So it might be a dumb pairing. But what I'm thinking is there's a there's a painting. I think it's by Georges Braque of a guy, or maybe it's Picasso. Someone walking down stairs. And maybe it's called a Nude Descending a Staircase, or something like that. You're nodding, do you know what I mean?

EL: I'm a little nervous about saying, I think it is Picasso, but I'm looking it up on my phone surreptitiously.

SS: I could try too. For some reason, I'm thinking it's George Braque, but that may be wrong. But so I'll describe the painting I have in my head and it may be totally not—

EL: No, it’s Marcel Duchamp!

SS: Oh, it's Marcel Duchamp?

EL: Yeah.

SS: And what's the name of it?

EL: Nude descending a staircase, number two. I think.

SS: Yeah, that's the one. Would that be considered Cubism?

EL: Yeah.

SS: It says according to Wikipedia, it’s widely considered a modernist classic. Okay, I don't know if it's the best example of what I'm thinking. But it's, let me just blow it up and look at it here. So, what hits me about it is it's a lot of straight lines. It's very rectilinear. And you don't see anything that really looks curved like a human form. You know, people are made of curved surfaces, our faces, our cheeks are, you know. What I like is this idea that you can build up curved objects out of lots of things made of straight lines. You know what you can do? mesh refinement on it. For instance, there's an old proof of the area of a circle where you chop it up into lots of pizza-shaped slices, right, and then you add up the areas of all those. And they can be approximated by triangles, and if you make the triangles thin enough, then those slivers can fill out more and more of the area, the method of exhaustion proof for the area of a circle. So this idea that you can approximate curved things with triangles, reminds me of this idea in Cauchy’s theorem that you first you prove it for the triangle, and then later Professor Stein proved the result for any smooth curve by approximating it with triangles, you know, a polygonal approximation to the curve, and then he could chop up the interior into lots of triangles. So I sort of think it pairs with this vision of the human form and it's sinuous descent down. You know, this person is smooth and yet they're being built out of these strange Cubist facets, or other shapes. I mean, think of other Cubist paintings you you represent smooth things with gem-like faceted structures, it sort of reminds me of Cauchy’s theorem.

KK: Okay, good pairing. Yup.

EL: Yeah, glad we got to the bottom of that before we made false statements about art on this math podcast.

SS: Yeah, it may not be the best Cubist example. But what are you gonna do? You invited a mathematician.

KK: So we also like to let our guests make pitches for things that they're doing. So you have a lot going on. You have a new podcast.

EL: Yeah, tell us about it.

SS: Okay. Yeah, thank you for mentioning it. I have a podcast with the confusing name Joy of X. Confusing because I also wrote a book by that name. And before that I had written an article by that name.

KK: Yes.

SS: So I did not choose that name for the podcast. But my producer felt like it sort of works for this podcast because it's a show where I interview scientists and mathematicians—in spirit, very similar to what we're doing here. And I talk to them about their lives and their work. And it's sort of the inner life of a scientist, but it could be a neuroscientist, it could be a person who studies astrophysics, or a mathematician. It's anything that is covered by Quanta Magazine. So Quanta Magazine, some of your listeners will know, is an online magazine that covers fundamental parts of math and science and computer science. Really, it's quite terrific. If people haven't read it, they might want to look at it online. It's free. And anyway, so Quanta wanted to start a podcast. And they asked me to host it, which was really fun because I get to explore all these parts of science. I've always liked all of the different parts of science, as well as math. And so yeah, that's the show. It's called the Joy of X where here, X takes on this generalized meaning of the unknown, not just the unknown in algebra, but anything that's unknown, and the joy of doing science and the scientific question. We'll be sure to link to that.

EL: Yeah.

KK: Also, I think Infinite Powers came out last year, right? 2019?

SS: That’s true. Yes, I had a book, Infinite Powers, about calculus. And that was an attempt to try to explain to the general public what's so special about calculus, why is it such a famous part of math. I try to make the case that it really did change the world and that it underpins a lot of modern science and technology as well as being a gateway to modern math. I really do think of it as one of the greatest ideas that human beings have ever come up with. Of course, that raises the question, did we discover it or invent it? But that’s a good one.

EL: Put that on a philosophy podcast somewhere. We don’t need that on this math podcast.

SS: Yeah, I don't really know what to say about that. That's a good timeless question. But anyway, yes, Infinite Powers was a real challenge to write because I'm trying to tell some of the history, but I'm not a historian of math. I wanted to really teach some of the big ideas for people who either have math phobia or who took calculus but didn't see the point of it, or just thought it was a lot of, you know, doing one integral after another without really understanding why they're doing it. So it's my love song to calculus. It really is one of my favorite parts of math, and I wanted other people to see what's so lovable and important about it.

KK: Yeah.

SS: The book, as I say, was hard because I tried to combine history and applications and big ideas without really showing the math.

KK: Yeah, that's hard.

SS: And make it fun to read.

KK: Right. It is. It's a very good book, though. I did read it.

SS: Oh, thanks.

KK: And I enjoyed it quite a bit.

EL: Well, it is on my table here under a giant pile of books to read, because people need to just stop publishing.

SS: That’s right.

EL: There’s too much. We just need to have a year to catch up, and then we could start going again but what's what's

KK: What’s that Japanese word, sort of the joy of having unread books? [Editor’s note: Perhaps tsundoku, “aquiring reading materials but letting them pile up in one’s home without reading them.”] There's a Japanese concept of like these books that you’ll, well, maybe even never read. But that you should have stacks and stacks of books. Because, you know, maybe you'll read them. Maybe you won't. But the potential is there.

SS: Nice.

KK: So I have a nightstand, on the shelf of my nightstand there's probably 20 books there right now, and I haven't read them all. I've read half of them, maybe, but I'm going to read them. Maybe.

SS: Yeah, yeah.

KK: Actually, you know, when you were talking about your sort of emotional feelings about Cauchy’s theorem, it reminded me of your—I don't know if it was your first book, but The Calculus of Friendship, about your relationship with your high school teacher.

SS: Well, how nice of you to mention it.

KK: Yeah. That was interesting to it, because it reminded me a lot of me, in the sense of, I thought I knew everything too when I was 18. Like, I thought, “Calculus is easy.” And then I get to university and math wasn't necessarily so easy. You know. And so these same sort of challenges, you know?

SS: Well, I appreciate that, especially because that book is pretty obscure. As far as I know, not many people read it. And it's very meaningful to me because I love my old teacher, Mr. Joffrey, who is now, let’s see, he's 90 years old. And I stayed in touch with him for about 35 years after college, and we wrote math problems to each other, and solutions. And it was really a friendship based on calculus. But over the course of those 35 years, a lot happened to both of us in our lives. And yet, we didn't tend to talk about that. It was like math was a sanctuary for us, a refuge to get away from some of the ups and downs of real life. But of course, real life has a way of making itself, you know, insinuating itself whether you like it or not. And so it's it's that story. The subtitle of the book is “what a teacher and a student learned about life while corresponding about math.” And I sometimes think of it as, like, there's a Venn diagram where there's one circle is people who want to read math books with all the formulas, because I include all the formulas from our letters.

KK: Yeah.

SS: And then there's people who want to read books about emotional friendships between men. And if you intersect those two circles, there's a tiny sliver that apparently you're one of the people in it.

KK: And your book might be the unique book in that in that Venn diagram too.

SS: Maybe. I don't know. But yeah, so it was it was clear it would not be a big hit in any way. But I felt like I couldn't do any other work until I wrote that book. I really wanted to write it. It was the easiest book to write. It poured out of me, and I would sometimes cry while I was writing it. It was almost like a kind of psychoanalysis for myself, I think, because I did have a lot of guilty feelings about that relationship, which, you know, if you do read the book, anyone listening, you'll see what I felt guilty about, and I deserved to feel guilty. I needed to grow up, and you see some of that evolution in the course of the book.

KK: Yeah. All right. Anything else you want to pitch? I mean?

SS: Well, how about I pitch this show? I mean, I'm very delighted to be on here. Really, I think you guys are doing a great thing helping to get the word out about math, our wonderful subject. And so God bless you for doing that.

KK: Well, this has been a lot of fun, Steve, we really appreciate you taking time out of your snow day. And so now do you have to shovel your driveway?

SS: Oh, yeah, that may be the last act I ever commit.

KK: Don’t you still have a teenager at home? Isn't that what they're for?

SS: My kids, I do have—you know what, that's a good point. I have one daughter who is still in high school and has not left for college yet, so maybe I could deploy her. She's currently making oatmeal cookies with one of her friends.

KK: Well, that's a useful, I mean that that's helping out the family too, right? I mean,

SS: They’re both able bodied, strong young women. So I should get them out there and with me, and we could all shovel ourself out. Yeah.

KK: Good luck with that. Thank you. Thanks for joining us.

SS: My pleasure. Thanks for having me.

On this episode of My Favorite Theorem, we were happy to talk with Steve Strogatz, an applied mathematician at Cornell University, about the Cauchy integral theorem. Here are some links you might find helpful.

Strogatz’s website, which includes links to information about his books and article
The Joy of X, the podcast he hosts for Quanta Magazine
The Cauchy integral theorem on Wikipedia
The Kuramoto model
Nude Descending a Staircase no. 2 by Marcel Duchamp

Episode 53 - Ruthi Hortsch

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the podcast that was already quarantined. I’m one of your hosts, Evelyn Lamb. I am holed up in my house in Salt Lake City, Utah, where I'm a freelance writer. So, honestly, I have worked in my basement, you know, every day for the past five years, and that hasn't changed. This is your other host.

Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida, which is open for business…But you can't go to campus.

EL: Okay.

KK: Yeah, we moved all of our classes online two weeks ago, I'm just teaching a graduate course this term, so that's sort of easier for me. I feel bad for the people who have to actually lecture and figure out how to do this all at once. My faculty have actually been great. They really stepped up. And, remarkably, I've had very few complaints from students, and I'm the chai,r so you know, they would come to me. And it's just really not—I mean, everybody has really taken the whole thing in stride. A lot of anxiety out there, though, among our students. Really, this is a really challenging time for everybody. And I just encourage my faculty to, you know, be kind to their students and to themselves. So let’s shelter in place and get through this thing, right?

EL: Yup. Yeah, we had an earthquake a week and a half ago to just, like, shake things up, literally. So it's just like, oh, as if I pandemic sweeping through town was not enough. We'll just literally shake your house for a while.

KK: Yeah, well, you know, we can go outside. We have a Shelter in Place Order, but it's been 90 degrees every day for the last week. And so you know, I like to go bird watching, but my favorite bird watching spot is a city park, and it's closed. So I have to just kind of sit on my back porch and see what's up. Yeah. Oh, well,

EL: Well, yes, we're making it through it. And I hope—I mean by the time this is—we have a bit of a backlog in our past episodes, and so who even knows what's going to be happening when this is airing. [Editor’s note: We decided to publish this one out of order, so we actually recorded it pretty recently.] But whatever is happening, I know our guests will be very thrilled to be listening to Ruthi Hortsch! Hi, Ruthi. How are you today?

Ruthi Hortsch: Hey, I'm managing.

EL: Yeah.

RH: It’s a weird time.

EL: Definitely. So what do you do, and where are you?

RH: Yeah, so I'm in New York City right now, which is kind of right now the hotbed of lots of new infections. But I've been in my apartment for the last two and a half weeks and haven't really directly been experiencing that.

I work for an organization called Bridge to Enter Advanced Mathematics. So we're a education nonprofit. We work with low-income and historically marginalized youth. And we're trying to create a realistic pathway for them to become mathematicians, scientists, engineers, programmers.

We start working with students when they're in middle school and we try to figure out, like, what are the things you need to get you to a place where you'll have a successful STEM career? And so we do a lot of different things, but they all are to that purpose.

EL: Yeah, and I'm so glad that we have you on the show to talk about this. Because, yeah, I've been thinking like, we really need to get someone from BEAM on here because I think BEAM is just such a great program. My spouse, and I donate to it every year. I mean, obviously not every year, I don't even know how old it is. But you know, we've made that part of our yearly giving, and yeah, I just think it does great work. So, does that have programs in both New York and LA now?

RH: Yes. So we started in New York City in 2011. And a few years ago, we expanded to LA. So the LA programs are still pretty new. They're building up, kind of starting with students in the first year of contact, and then adding in programming for the older students as that first class gets older. So they now have eighth graders, and that's their oldest class, and they'll continue to add in the ninth grade and the 10th grade program, et cetera, as it goes on. The other kind of exciting thing is, last year, we got a grant from the Gates Foundation. And that grant was to partner with other local programs and other cities to help them build up programs that could do some of the same things we do. So it's not the same comprehensive, really intensive support that we give our students in New York City and LA. But assuming summer camps don't get canceled this summer because of corona, there are going to be day camps in Albuquerque and Memphis that are advised by us.

EL: Oh, that's so great. Yeah, because that's the one thing about it is that it is so localized and, of course, important places for it to be localized. But, you know, the more the, the wider, the better. So that's awesome. And what's your role there? What do you do?

RH: Yeah, I have a hard time answering this question. So I work in programs, which is like, I work on things that are directly affecting students. I run one of our summer camps in the summer. So I run a sleepaway camp at Union College, in which students learn proof-based mathematics for the first time. The students at the sleepaway camp are all rising eighth graders, and so they get to learn number theory and combinatorics and group theory. They also do some modeling and programming and stuff.

During the year I do some managing our other programs team, so supporting other staff. I also do all of our faculty hiring. So certainly we hire a lot of people just for the summer, and most of them are—so we hire college, university students, we hire grad students, we hire professors in various different roles. And I handle all of the, like, hiring people to teach math courses.

EL: Wow.

KK: That’s a lot. Are your programs sort of face to face, or are they online? Is it sort of a combination of stuff?

RH: Yeah, so our summer we run six in-person summer camps each summer. So there's two in upstate New York that are sleepaway, one in Southern California that's sleepaway, and then one day camp in LA and two day camps in New York City. And those are all in-person, face to face. And then during the school year, we also have Saturday classes, which is a mix of life skills and enrichment. And we also do in-person advising. So we have office hours where students can come ask us anything, and then also kind of more intensive. Like, how do you apply to college? How do you get into other summer programs or other STEM opportunities? So most of our programs are face to face. Right now, we've had to cancel a bunch of our year-round stuff. So we don't have Saturday classes right now. We are doing one class for the eighth graders virtually, because we really thought it was critical. And at the moment, we're hoping the summer programs will still run, but it's really hard to say what's going to be going on in two weeks.

KK: Yeah, well, fingers crossed.

EL: But as wonderful as it is to talk about BEAM, what we're dying to know is what is your favorite theorem?

RH: Yeah, so this was actually really fast for me to think of. My favorite theorem is Falting’s theorem. So Falting’s theorem is also actually known as the Mordell conjecture, because Mordell originally conjectured it in the same paper in which he proved Mordell’s theorem, I believe, or at least during the same process of research for him.

EL: Yeah, and so for longtime listeners, was it Mathilde Lalín who, that was her favorite theorem?

RH: Mm-hmm.

EL: Okay, that's right. So we're kind of dovetailing right in.

RH: Yeah. So Mordell’s theorem is about—so when you look at elliptic curves, they have a finitely-generated abelian group. And Mordell’s theorem is the theorem that proves that it actually is finitely-generated.

KK: Right.

RH: So when I say the finitely-generated part, it's actually only looking at the rational points on the curve. So we care about algebraic curves, kind of in general. And then we want to think about, like, how do different algebraic curves behave differently? And because I'm trained as a number theorist, I also specifically care about how many rational points are on that curve and how they behave. So this intersects also with algebraic geometry. And in some sense, this is a statement about how the arithmetic part of the curves—the rational points—interacts with the geometry of it.

So one thing that people care about a lot in geometry is the notion of a genus. This is one of the ways to classify things. And of course, when you're looking at visual shapes, one way of thinking about the genus is how many holes does it have? So if you're just looking at a shape that’s, like, a big sphere, there's no way of poking a hole through it without actually breaking it apart. And so that has genus zero because there are zero holes. But if you're looking at a doughnut, a torus, that has one hole because there's like one place where you can poke something through. And then you can generalize from there that having more holes is higher genus. And so that's kind of a wishy-washy way of looking at things, and a very visual way. There are ways to define that formally in the algebraic sense, but in the places where both definitions make sense, the definition is the same.

And so when you look at algebraic curves, we can ask ourselves, how do genus zero curves act differently than genus one curves, act differently than genus two curves, and does that tell us anything about the number of rational points? And so it turns out that with genus zero curves, genus zero curves are actually really just conic sections. So basically the nice lines that you study in like algebra in high school. And those have infinitely many rational points, right? So when I say rational point, you can kind of think of it as being like the points where the components have rational values.

And genus one curves are actually exactly elliptic curves. So in that case, that's when Mordell’s theorem kicks in and the rational points are this finally generated abelian group. And sometimes they have infinitely many rational points, and sometimes they don't, and it kind of depends on what this algebraic structure, this algebraic group structure, looks like. So that's the most complicated weird point. And for genus two or higher curves, it turns out to be true that there are only finitely many rational points on a genus two or higher curve. And that's the statement of Falting’s theorem.

EL: Okay, and so I, there's something that I, you know, you hear like genus two or higher. And I always wonder, is there a limit to how high the genus can be of these curves? Or, like, is there a maximum complexity that these curves can have?

RH: So no. And actually, there's a statement in algebraic geometry that makes it really easy-ish— you know, “ish”— to calculate the genus, which is called Riemann-Roch. And it gives you a relationship between the degree of the equation defining it and the genus. And essentially, the genus grows quadratically with the degree. There's an asterisk on everything I'm saying. It’s mostly true.

KK: It’s mostly true.

EL: So if I'm remembering correctly, Mordell’s—let’s see, Mordell’s conjecture, Falting’s theorem—was really important for proving Fermat’s last theorem. Is that correct?

RH: I don't think so, no. But all of these things are related to each other.

EL: Okay.

RH: A lot of the common definitions and theorems that play into all these things, they share a lot, but it's not directly, like, one thing implied the other.

EL: Okay, yeah.

RH: In particular, Fermat’s Last Theorem was reduced to a statement about elliptic curves, which is about genus one curves, while Falting’s theorem is really a statement about genus two or higher curves.

EL: Okay.

KK: So was this a love at first sight kind of theorem?

RH: I think no. I think part of the reason that I really started appreciating it was because I had a mentor in undergrad who was really excited about it. And I didn't really understand the full implications and the context, but I was like, “Okay, this mentor I have is really about it, so I'm going to be really about it.”

And we actually used Falting’s theorem as a black box for the REU project I was working on. So we assumed it was true and then used that to show other things. And then later on in grad school, I had a number of things that I was really interested in that Falting’s theorem was related to. One of the things that I think is really cool that's being researched right now is there’s a bunch of like, tropical geometry that is being studied. And this is, like, relating algebraic verbs to kind of more combinatorial objects. So you can actually translate these lcurves that have a more—I don't want to say analytic, but a smooth structure, and then turning them into a question about, like, counting more straight-edged structures instead.

One of the things about Falting’s proof of Falting’s theorem is that it's not, it doesn't actually give you a bound. So it tells you that there are only finitely many points, but it doesn't give you a constructive way of saying, like, what does it actually bounded by, the number of finite points? And using tropical geometry, people have been able to make statements about bounds in certain situations, which is really cool.

KK: Okay, I always like these tropical pictures, you know, because suddenly everything just looks almost like Voronoi diagrams in the plane, these piecewise linear things. So I guess the idea of genus probably still makes sense there in some way, once you define it properly. Right?

RH: Yeah. And there's a correspondence between, there’s a notion of a tropical curve, which still looks like one of those Voronoi diagrams. There’s an actual correspondence, this curve in classical algebraic geometry gives you this particular diagram.

EL: Nice. And so you say it was very easy to choose this theorem. So what's your, like, elevator sales pitch for this theorem? Keeping in mind that no one is going to be in an elevator with anyone else anytime soon. We're staying far apart, but you know.

RH: Yeah. So, I think it’s kind of amazing that geometry can tell you something about the arithmetic of a curve. I think this is what drew me to arithmetic algebraic geometry, that there is this kind of relationship. When you think, okay, arithmetic, geometry, those are totally different fields, people study them in totally different ways, but in fact, it turns out that the geometry of a curve can tell you information about the arithmetic. And that's just bizarre, and also very powerful in that you can make a statement about how many rational solutions there are to an equation using correspondence in geometry.

The REU project that I worked on actually is a statement that I think is really easy to understand. If you have a rational polynomial, that gives you a function from the rationals to the rationals, right?

And so you can ask yourself: how many-to-one is that function? How many points gets sent to the same point? And if you look at only rational points, our REU project showed that it can't be more than four-to-one off a finite number points.

So if you are willing to ignore some finite number of points, then no rational polynomial is ever more than four-to-one.

KK: Interesting.

RH: And that feels like a very powerful statement. And it's because we had this hammer of Falting’s theorem to just smash it in the middle.

KK: That’s really fascinating. So no matter how high the degree it's no more than four-to-one? I wouldn’t have guessed that.

RH: Off a finite number of points.

KK: Yeah, sure. Generically. Yeah. Right. Interesting.

RH: I think the real powerful thing there is that Falting’s theorem comes in.

KK: Yes.

RH: Oh, actually, higher degree means high complexity means high genus.

KK: Okay, cool. So another thing we like to do is ask our guests to pair their theorem with something. So what pairs well with Falting’s theorem?

RH: Yeah, so this is a maybe a little bit of a stretch, but I've been living in New York City for four years, and I love bagels. They’re definitely one of the best parts of living in New York City. I'm always two blocks away from a really good bagel. Traditionally, bagels are genus one, so it's actually not quite appropriate. You have to, I don't know, do the fancy cut to increase the genus—there’s a way to cut a bagel to get higher genus. But I still think since we're thinking about genuses, we're thinking about complexity of things.

EL: Yeah. Well, like, you cut the bagel in in half, you know, to get like the cream cheese surface, and then just stick them together and you've got a genus two. Put a little cream cheese on the side. You know?

RH: Yeah. I mean, if we're cutting holes we can cut as we want.

EL: That’s true. So, are you more—what do you put on the bagel? What kind of bagel, also, do you prefer?

RH: Ao I mostly like everything bagels.

EL: Of course. Yeah. Great bagel.

RH: There is a weird thing that goes on where some bagel shops put salt on their everything bagel and some don't. And I feel like the salt is important.

KK: Yeah. Agree.

EL: As long as it's not too much. Like just the right amount of salt is—

RH: Yeah. It’s definitely important.

KK: Well a salt bagel is a pretzel.

EL: Yes.

RH: And I don't actually eat cream cheese. So I do eat fish sometimes, but I generally don't eat dairy. And I so I usually get, like, tofu scallion spread. And the tofu spread that gets sold in the bagel shops here is actually really good.

KK: Well yeah, I'm not surprised. I can't get a decent bagel in Gainesville. I mean, there's a couple of bagel shops, but they're no good.

RH: Yeah. This is what you get for leaving New York City.

KK: Right, right.

EL: Yeah, it's funny, actually one of our quarantine projects we're thinking about is making bagels. I've made bagels one other time. But, yeah.

KK: It's kind of a nuisance. You know that. That boiling step is really—I mean, it's crucial, but it just takes so much time and space.

EL: Yeah, I mean, they were not nearly as good as a real bagel shop bagel, but fun to play with.

KK: Yeah. So what's everyone doing to keep themselves occupied? So far I've got a batch of sauerkraut fermenting. I just started a batch of limoncello that'll be ready in a month. I made scones. Maybe that’s it. Yeah. How about you guys?

RH: Well, I'm still trying to work 40 hours a week.

KK: Yeah, I'm doing that too.

RH: We're still trying to help our students respond to the crisis and helping support them both academically, but holistically also.

KK: Yeah, it's very stressful.

RH: And at the moment, we're still doing all of our prep work for the summer, which is a huge undertaking? But when I have free time, I've been cooking more. And I'm actually also working on writing a puzzle hunt.

EL Ooh, cool. Well if that happens, we'll include a link to that in the show notes—if it's the kind of thing that you can do out of a particular geographical place.

RH: Yeah, so the puzzle hunt I'm helping write is actually for Math Camp.

EL: Okay.

RH: So before I worked for BEAM I worked for Canada-USA Math Camp, and in theory, they're running a camp this summer, and one of the traditional events there is [the puzzle hunt]. I think the puzzle hunt often gets put up after the summer, but I’m not sure.

EL: Oh, cool. The last thing that I, or library book that I got out from the library—it was actually supposed to be due, like, the day after the library shut down here—was 660 Curries, which is an Indian cookbook that—we don’t really cook meat at home, but it's got, I don't know, maybe a hundred-page section of legume curries and a bunch of vegetable curries, so we've been kind of working through that. We made one last night that was great. It was a mixture of moong dal and masoor dal. Yeah, we’ve been eating a lot of curry, and it just makes my early-this-year plan of, like, “Oh, I want to make more dal, so I've got to go stock up on lentils and rice,” brilliant plan, really has made it a lot easier. So yeah.

RH: I love dal, and I don't feel like anybody around me ever likes dal as much as I do.

KK: This is a dal-lover convention right here. It's one of my favorite things to eat. Yeah.

EL: Oh, yeah. Well, I can recommend, if you get a chance to get 660 Curries, I don't remember if it's called mixed red and lentil dal with garlic and curry leaves, or something like that.

KK: Yeah, I'm actually making curry tonight, but chicken curry so we'll we'll see.

EL: Yeah, so other than that, just panicking most of the time. It’s been a big pastime for me.

RH: I’ve had to, like, ban myself from reading the news in the evening.

KK: Good call.

EL: That is very smart.

RH: I haven’t done a good job keeping to it.

EL: Yeah, I have not done a good job with my self-control with that. So, I’m really trying to do that. I'm hoping to do some sewing projects too, maybe making some masks that I can leave out for people in the neighborhood to take. Obviously not medical grade, but maybe make people feel a little better.

KK: So yeah, Ellen, my wife, started doing that yesterday. She made, you know, probably 15 of them yesterday real quick.

EL: Nice.

KK: I went to the store yesterday and you know—

EL: Hopefully it gives people a little peace of mind and maybe decreases droplet transmission.

KK: Let’s hope.

EL: I’ve refrained from armchair epidemiology, which I encourage everyone to do. So yeah, I hope everyone stays safe and tries to keep keep a good spirit and help the people in your lives. I hope our listeners can do that too. And I hope they find some enjoyment in thinking about math for a little while with us.

KK: So yeah, thanks for joining us, Ruthi. We really appreciate it.

EL: Yeah, everyone go find BEAM online if you want to learn more about that.

RH: Yeah. Follow us on social media.

EL: Yeah. So what are the handles for that?

RH: Yeah, I should have this memorized. You can find it on our website. They're all linked to on our website, beammath.org. If you're in New York or LA, we have trivia night, which is a puzzle-y, mathy trivia, usually in the fall, that you can buy tickets to. So I definitely recommend that. And otherwise, sign up for our newsletter, which you can also do on our website.

EL: And you're on Twitter also, right?

RH: Yes, I am. You do have to know how to spell my last name, though.

EL: Okay.

RH: Yeah, I'm @ruthihortsch.

EL: All right. And that's H-O-R-T-S-C-H?

RH: Good job!

EL: Yeah, it’s funny, I was actually in a Zoom spelling bee last night. So yeah, I got second place.

KK: Good for you.

EL: Got knocked out on diaphoresis.

KK: Diaphoresis. Wow. Yeah, that's pretty—okay, anyway. All right. Well, thanks for joining us and take care everyone.

RH: Right. Yeah, it was nice to meet you.

EL: Bye.

[outro]

On today's episode of My Favorite Theorem, we had the privilege to talk with Ruthi Hortsch, a program coordinator at Bridge to Enter Advanced Mathematics (BEAM), a math program for low-income and historically marginalized middle- and high-school students. Dr. Hortsch lives in New York City, which is currently being hit hard by covid-19. We love all our listeners and guests, and right now we are especially thinking about those in New York and other virus hot spots. You may be sick, you may be worried about loved ones, you may be suddenly parenting or caregiving in ways you hadn't expected. We wish you the best, and we hope you enjoy thinking about math for a little bit instead of the news cycle. Stay strong and healthy, friends!

As you listen to this episode, you may find these links helpful.
The Bridge to Enter Advanced Mathematics website, Twitter, Facebook, and Instagram pages.
Ruthi Hortsch on Twitter
Faltings’s theorem, Dr. Hortsch's favorite theorem
Our episode with Matilde Lalín, whose favorite theorem was the closely-related Mordell's theorem.
660 Curries

Canada/USA Mathcamp

Tropical Geometry wikipedia page

Episode 52 - Ben Orlin

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast. I'm Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer, usually based in Salt Lake City, but currently still in Providence. I'll be leaving from this semester at ICERM in about a week. So trying to eat the last oysters that remain in the state before I leave and then head back.

KK: Okay, so you actually like oysters.

EL: Oh, I love them. Yeah, they're fantastic.

KK: That is one of those, it’s a very binary food, right? You either love them—and I do not like them at all.

EL: Oh, I get that, I totally get it.

KK: Sure.

EL: They’re like, in some sense objectively gross, but I actually love them.

KK: Well, I'm glad you've gotten your fill in. Probably—I imagine they're a little more difficult to get in Salt Lake City.

EL: Yeah, you can but it’s not like you can get over here.

KK: Might be slightly iffy. You don't know how long they've been out of the water, right?

EL: Yeah. So there's one place that we eat oysters sometimes there, yeah, that's the only place.

KK: Yeah, right. Okay. Well, today we are pleased to welcome Ben Orlin. Ben, why don't you introduce yourself?

Ben Orlin: Yeah, well, thanks so much for having me, Kevin and Evelyn. Yes, I'm Ben Orlin. I’m a math teacher, and I write books about math. So my first book was called Math with Bad Drawings, and my second one is called Change Is the Only Constant.

EL: Yeah, and you have a great blog of the same name as your first book, Math with Bad Drawings.

BO: Yeah, thank you. And I think our blogs are, I think almost birthday, not exactly but we started them within months of each other, right? Roots of Unity and Math with Bad Drawings.

EL: Oh, yeah.

BO: Began in, like, spring of 2013 which was a fertile time for blogs to begin.

EL: Yeah. Well, in a few years ago, you had some poll of readers of like, what other things they read and, and stuff and my blog was like, considered the most similar to yours, by some metric.

BO: Yeah, I did a reader survey and asked people, right, what what other sources they read, and mostly I was looking for reading recommendations. So what else do they consider similar? Overwhelmingly it was XKCD. Not so much—just because XKCD, it’s like if you have a little light that you're holding, a little candle you're holding up, and you're like, what does this remind you of? And like a lot of people are going to say the sun because they look up, and that’s where they see visible light.

KK: Sure.

BO: But I think in terms of actually similar writing, I think Toots of Unity is not so different, I think.

EL: Yeah. So I thought that was interesting because I have very few drawings on on mine. Although the ones that I do personally create are definitely bad. So I guess there’s that similarity.

BO: That’s the key thing, committing to the low quality.

KK: Yeah, but that's just it. I would argue they're actually not bad. So if I tried to draw like you draw, it would be worse. So I guess my book should just be Math with Worse Drawings.

BO: Right.

KK: You actually get a lot of emotion out of your characters, even though they're they're simple stick figures, right? There’s some skill there.

BO: Yeah, yeah. So I tried. I tried to draw them with a very expressive faces. Yeah, they're definitely still bad drawings is my feeling. Sometimes people say like, “Oh, but they've gotten so much better since you started the blog,” which is true, but it's one of these things where they could they could get a lot better every five-year interval for the next 50 years and still, I think not look like professional drawings by the end of it.

EL: Right. You're not approaching Rembrandt or anything.

KK: All right, so we asked you on here, because you do have bad drawings, but you also have thoughts about mathematics and you communicate them very well through your drawings. So you must have a favorite theorem. What is it?

BO: Yeah. So this one is drawn from my second book, actually, the second book is about calculus. And I have to confess I already kind of strayed from the assignment because it's not so much a favorite theorem as a favorite construction.

KK: Oh, that’s cool.

EL: You know, we get rule breakers on here. So yeah, it happens.

BO: Yeah, I guess that's the nature of mathematicians, they like to bend the rules and imagine new premises. So pretending that this were titled My Favorite cCnstruction, I would pick Weierstrass’s function. So that you know, first introduced in 1872. And the idea is it's this function which is continuous everywhere and differentiable nowhere.

EL: Yeah. Do you want to describe maybe what this looks like for anyone who might not have seen it yet?

BO: Yeah, sure. So when you're picturing a graph, right, you're probably picturing—it varies. I teach secondary school. So students are usually picturing a fairly small set of possibilities, right? Like you're picturing a line, maybe you're thinking of a parabola, maybe something with a few more squiggles, maybe as many squiggles as a sine wave going up and down. But they all have a few things in common one is that almost anything that students are going to picture is continuous everywhere. So basically, it's made of one unbroken line. You can imagine drawing it with your pencil without picking the pencil up. And then the other feature that they have is that they—this one's a little subtler, but there will be almost no points that are jagged, or sort of crooked, or, you know, if I picture an absolute value graph, right, it sort of is a straight line going down to the origin from the left, and then there's a sharp corner at the origin, and then it rises away from that sharp corner. And so those kind of sharp corners, you may have one or two in a graph a student would draw, but that's sort of it. You know, like sharp corners are weird. You don't can't draw all sharp corners. It feels like between any two sharp corners on your graph, there's going to have to be some some kind of non-sharp stuff connecting it, some kind of smooth bits going between them.

KK: Right.

BO: And so what sort of wild about about Weierstrass’s function is that you look at it, and it just looks very jagged. It’s got a lot of sharp corners. And you start zooming in, and you see that even between the sharp corners, there are more sharp corners. And you keep zooming in and there's just sharp corners all the way down. It's what we today call it fractal. Although back then that word wasn't around. And it's just it's the entire thing. Every single point along this curve is in some sense, a sharp corner.

EL: Yeah, it kind of looks like an absolute value everywhere.

BO: Yeah, exactly. It has that cusp at every single point you could look at.

KK: Right? So very pathological in nature. And, you know, I'm sure I've seen the construction of this. Is it easy to say what the construction is? Or is this going to be too technical for an audio format?

BO: It’s actually not hard to construct. There are there whole families of functions that have the same property. But Weierstrass’s is pretty simple. He starts with basically just a cosine curve. So you sort of have cosine of πx. So picture, you know, a cosine wave that has a period of two. And then you do another one that has a much shorter period. So you can sort of pick different numbers. But let's say the next one that you add on has a period that's 21 times faster. So it's sort of going up and down much quicker. And it's shorter, though, we've shrunk the amplitude also. So it's only about a third, let's say, as tall. And so you add that onto your first function. So now we've got—we started with just a nice, gentle wave. And now we've got a wave that has lots of little waves kind of coming off of it. And then you keep repeating that process. So the next, the second one in the iteration has a period of 21 cycles for two units. The next one has 212 cycles. And it's 1/9 the height of the original.

KK: Okay.

BO: And then after that, you're going to do you know, 213 cycles in the same span, 214 cycles. And so it goes—I don't know if you can hear my daughter is crying in the background, because I think she she finds it sort of upsetting to imagine the function that's has this kind of weird property.

EL: Fair.

BO: Especially because it's such a simple construction. Right? It's just, like, little building blocks for her that we're putting together. And one of the things I like about the construction, is it at no step, do you have any non-differentiable points, actually. It's a wave with a little wave on top of it and lots of little waves on top of that, and then tons and tons of little waves on top of that, but these are all smooth, nice, curving waves. And then it's only in the limit, sort of at the at the end of that infinite bridge, that suddenly it goes from all these little waves to its differentiable nowhere.

KK: I mean, I could see why that would be true, right?

BO: Yeah, right. Right. It feels like it's getting worse. And you can do—Weierstrass’s function is really a whole family of functions. He came up with some conditions that you need, basically that’s the basic idea. You need to pick an odd number for the number of cycles and then a geometric series for for the amplitude.

KK: So what's so appealing about this to you? It's just you can't draw it well, like you have to draw it badly?

KK: Yeah, that's one thing, right. Exactly. I try to push people into my corner, force them to have to drop badly. I do like that this is something—right, graphs of functions are so concrete. And yet this one you really can't draw. I've got it in my book, I have a picture of the first few iterations. And already, you can't tell the difference between the third step and the fourth step. So I had to, I had to, you know, do a little box and an inset picture and say, actually, in this fourth step, what looks like one little wave is really made up of 21 smaller waves. So I do sort of like that, how quickly we get into something kind of unimaginable and strange. And also, you know, I'm not a historian of mathematics. And so I always wind up feeling like I'm peddling sort of fairy tales about about mathematical history more than the complicated truth that is history. But the role that this function played in going from a world where it felt like functions were kind of nice and were something we had a handle on, into opening up this world where, like, oh no, there are all these pathological things going on out there. And there are just these monsters that lurk in the world of possibility.

KK: Yeah.

EL: Right. And was this it—Do you know, was this maybe one of the first, or the first step towards realizing that in some measure sense, like, all functions are completely pathological? Do you know kind of where it fell there, or, like, what the purpose was of creating it in the first place?

BO: Yeah, I think that's exactly right. I don't know the ins and outs of that story. I do know that, right, if you look in spaces of functions, that they sort of all have this property, right, among continuous functions, I think it's only a set of measure zero that doesn't have this property. So the sort of basic narrative as I understand it, leading from kind of the start of the 19th century to the end of the 19th century, is basically thinking that we can mostly assume things are good, to realizing that sometimes things are bad (like this function), culminating in the realization that actually basically everything is bad. And the good stuff is just these rare diamonds.

EL: Yeah, I guess maybe this slight, I don't know, silver lining, is that often we can approximate with good things instead. I don't know if that's like the next step on the evolution or something.

BO: Right. Yeah, I guess that's right. Certainly, that's a nice way to salvage some a silver lining, salvage a happy message. Because it's true, right? Even though, a simpler example, the rationals are only a set of measure zero and the reals, you know, they're everywhere, they're dense. So at least, you know, if you have some weird number, you can at least approximate it with a rational.

EL: Yeah, I was just thinking when you were saying this, how it has a really nice analogy to the rationals. And, and even algebraic numbers and stuff like, “Okay, start naming numbers,” you'll probably name whole numbers, which are, you know, this sparse set of measure zero. It’s like, o”h, be more creative,” like, “Okay, well, I'll name some fractions and some square roots and stuff.” But you're still just naming sets of measure zero, you’re never naming some weird transcendental function that I can't figure out a way to compute it.

BO: Yeah, it is funny, right? Because in some sense, right? We've imagined these things called numbers and these things called functions. And then you ask us to pick examples. And we pick the most unlikely, nicest hand-picked, cherry-picked examples. And so the actual stuff—we’ve imagined this category called functions, and most of what's in that category that we developed, we came up with that definition, most of what's in there is stuff that's much too weird for us to begin to picture.

EL: Yeah.

BO: Which says something about, I guess, our reach exceeding our grasp or something. I don't really know, but they are our definitions can really outrun our intuition.

EL: Yeah. So where did you first encounter this function?

BO: That’s a good question. I feel like probably as a kind of folklore bit in maybe 12th grade math. I feel like when I was probably first learning calculus, it was sort of whispered about. You know, my teacher sort of mentioned it offhand. And that was very enticing, and in some sense, that's actually where my whole second book comes from, is all these little bits of folklore, not exactly the thing you teach in class, but the little, I don't know, the thing that gets mentioned offhand. And you go “Wait, what, what was that?” “Oh, well, don't worry. You'll learn about that in your real analysis class in four years.” I don't want to learn about that in four years. Tell me about that now. I want to know about that weird function. And then I think the first proper reading I did was probably in a William Dunham’s book The Calculus Gallery, which is a nice book going through different bits of historical mathematics, beginning with the beginnings of calculus through through like the late 19th century. And he has the here's a nice discussion of the function and its construction.

KK: So when we were preparing for this, you also mentioned there are connections to Brownian motion here. Do you want to mention those for our audience?

BO: Yeah, I love that this turns out—so I have some quotes here from right when this function was sort of debuted, right when it was introduced to the world. You have Émile Picard, his line was, “If Newton and Leibniz had thought that continuous functions do not necessarily have a derivative, the differential calculus would never have been invented.” Which I like. If Newton and Leibniz knew what you were going to do to their legacy, they would never have done this! They would have rejected the whole premise. And then Charles Hermite? [Pronounced “her might, wonders if the pronunciation is correct]

KK: Hermite. [Pronounced “her meet”]

BO: That sounds better. Sounds good. Sure. Right. His line was, and I don't know what the context was, but, “I turn away with fright and horror from this lamentable evil of functions that do not have derivatives.” Which is really layering on I like the way people spoke in the 19th century. There was more, a lot more flavor to their their language.

EL: Yeah.

BO: And Poincaré also, he was saying 100 years ago prior to Weierstrass developing it, such a function would have been regarded as an outrage to common sense. Anyway, so I mention all those. You mentioned Brownian motion, right? The instinct when you see this function is that this is utterly pathological. This is math just completely losing touch with physical reality and giving us these weird intellectual puzzles and strange constructions that can't possibly mean anything to real human beings. And then it turns out that that's not true at all, that Brownian motion—so you look at pollen dancing around on the surface of some water, and it's jumping around in these really crazy aggressive ways. And it turns out our best models of that process, you know, of any kind of Brownian motions—you know, coal dust in the air or pollen on water—our best model to a pretty good approximation has the same property. The path is so jagged and surprising and full of jumps from moment to moment that it's nowhere differentiable, even though the particle obviously sort of has to be continuous. It can’t be discontinuous, I mean, it's jumping, like literally transporting from one place to another. So that's not really the right model. But it is non-differentiable everywhere, which means, weirdly, that it doesn't have a speed, right? Like, a derivative is a is a velocity.

EL: So that means maybe an average speed but not a speed at any time.

BO: Yeah, well, actually, even—I think it depends how you measure. I’d have to looked back at this, because what it means sort of between any two moments according to the model, between any two points in time, is traversing an infinite distance. So I guess it could have an average velocity, but the average speed I think winds up being infinite rates. Over a given time interval, you can just take how far it travels that time interval and divide by time, but I think the speed, if you take the absolute value of the magnitude? I think you sort of wind up with infinite speed, maybe? But really, it's just that you can’t—speed is no longer a meaningful notion. It's moving in such an erratic way. that n you can't even talk about speed.

KK: Well, because that tends to imply a direction. I mean, you know, it’s really velocity. That always struck me as that's the real problem, is that you can't figure out what direction it's going, because it's effectively moving randomly, right?

BO: Yeah, I think that's fair. Yeah. The only way I can build any intuition about it is to picture a single—imagine a baseball having a single non-differentiable moment. So like, you toss it up in the air. And usually what would happen is that it goes up in the air, it kind of slows down and slows down and slows down. There's that one moment when it's kind of not moving at all. And then it begins to fall. And so the non-differentiable version would be, like, you throw it up in the air, it's traveling up at 10 meters per second, and then a trillionth of a second later, it's traveling down at 10 meters per second. And what's happening at that moment? Well, it's just unimaginable. And now for Brownian motion, you've got to picture that that moment is every moment.

KK: Right. Yeah. Weird, weird world.

BO: Yeah.

KK: So another thing we like to do on this podcast is ask our guests to pair their, well in your case construction, with something. What does the Weierstrass function pair with?

BO: Yeah. So I think, I have two things in mind, both of them constructions of new things that kind of opened up new new possibilities that people could not have imagined before. So the first one, maybe I should have picked a specific dish, but I'm picturing basically just molecular gastronomy, this movement in in cooking where you take—one example I just saw recently in a book was, I think it was WD-50, a sort of famous molecular gastronomy restaurant in New York, where they had taken, the comes to you and it looks like a small, poppyseed bagel with lox. And then as it gets closer, you realize it's not a poppyseed bagel with lox, it's ice cream that looks almost identical to a poppyseed bagel with lox. So that's sort of weird enough already. And then you take a taste and you realize that actually, it tastes exactly like a poppyseed bagel with lox, because they've somehow worked in all the flavors into the ice cream.

KK: Hmm.

BO: Anyway, so molecular gastronomy basically is about imagining very, very weird possibilities of food that are outside our usual traditions, much in the way that Weierstrass’s function kind of steps outside the traditional structures of math.

EL: Yeah, I like this a lot. It's a good one. Partly because I'm a little bit of a foodie. And like, when I lived in Chicago, we went to this restaurant that had this amazing, like, molecular gastronomy thing. I’m trying to remember one of the things we had was this frozen sphere of blue cheese. And it was so weird and good. Yeah, you’d get you get like puffs of air that are something, and there’s, like, a ham sandwich, but it was like the bread was only the crust somehow there's like nothing inside. Yeah, it was all these weird things. Liquefied olive that was like in inside some little gelatin thing, and so it was just like concentrated olive taste that bursts in your mouth. So good.

BO: That sounds awesome to me the the molecular gastronomy food. I have very little experience of it firsthand.

KK: So you mentioned a second possible pairing. What would that be?

BO: Yeah, so the other one I had in mind is music. It's a Beatles album, Revolver.

KK: Great album.

BO: One of my favorite albums, and much like molecular gastronomy shows that the foods that we're eating are actually just a tiny subset of the possible foods that are out there, similarly what revolver did for for pop music and in ’65 whenever it came out.

KK: ’66.

BO: Okay. 66 Alright, thank you for that.

EL: I am not well-versed in albums of The Beatles. You know, I am familiar with the music of the Beatles, don’t worry. But I don't know what's on what album. So what is this album?

BO: So Kevin and I can probably go to track by track for you.

KK: I’d have to think about it, but it's got Norwegian Wood on it, for example.

BO: Oh, that's rubber sole, actually.

KK: Oh, that’s Rubber Soul. You're right. Yeah, I lost my Beatles cred. That's right. My bad. I mean, some would argue that—so Revolver was, some people argue, was the first album. Before that, albums had just been collections of singles, even in the case of the Beatles, but Revolver holds together as a piece.

BO: Yeah, that’s one thing. Which again, there's probably some an analogy to Weierstrass’s function there. Also, it begins with this kind of weird countdown where, I don’t remember if it's John or George, but they’re saying 1234 in the intro into Taxman.

KK: Yeah. Into Taxman, which is probably, it's not my favorite Beatles song, but it's certainly among the top four. Right.

BO: Yeah. So that one, already right there it’s a pop song about taxes, which is already, so lyrically, we're exploring different parts of the possibility space than musicians were before. Track two is Eleanor Rigby, which is, the only instrumentation is strings. Which again is something that you didn't really hear in pop. You know, Yesterday had brought in some strings, that was sort of innovative. Other bands have done similar things but, but the idea of a song that’s all strings, and then I’m Only Sleeping as the third track, which has this backwards guitar. They recorded the guitar and just played it backwards. And then Yellow Submarine, which is, like, this weird Raffi song that somehow snuck onto a Beatles album. Yeah, and then For No One has this beautiful French horn solo. Yes, every track is drawn from sort of a distant corner of this space of possible popular music, these kind of corners that had not been explored previously. Anyway, so my recommendation is, is think about the Weierstrass function while eating, you know, a giant sphere of blue cheese and listening to Taxman.

EL: Great. Yeah. I strongly urge all of our listeners to go do that right now.

BO: Yeah, if anyone does it, it'll probably be the first time that that set of activities has been done in conjunction.

EL: Yeah. But hopefully not the last.

BO: Hopefully not the last. That's right. Yeah. And most experiences are like that, in fact.

KK: So we also like to let our guests plug things. You clearly have things to plug.

BO: I do. Yeah. I'm a peddler of wares. Yes, so the prominent thing is my blog is Math with Bad Drawings, and you're welcome to come read that. I try to post funny, silly things there. And then my two books are Math with Bad Drawings, which kind of explores how math pops up in lots of different walks of life, like, you know, in thinking about lottery tickets or thinking about the Death Star is another chapter, and then Change Is the Only Constant is my second book, and it's all about calculus, and it’s sort of calculus through stories. Yeah, that one just came out earlier this year, and I'm quite proud of that one. So you should check it out.

KK: Yeah, so I own both of them. I've only read Math with Bad Drawings. I've been too busy so far to get to Change Is the Only Constant.

EL: And there were there been a slew of good pop—or I assume good because I haven't read most of them yet—pop math books that have come out recently, so yeah I feel like my stack is growing. It’s a fall of calculus or something.

BO: It’s been a banner year. And exactly, calculus has been really at the forefront. Steve Strogatz’s Infinite Powers was a New York Times bestseller, and then David Bressoud [Calculus Reordered] and others who I'm blanking on right now have had one. There was another graphic, like, cartoon calculus that came out earlier this year. So yeah, apparently calculus is kind of having a moment.

EL: Well, and I just saw one about curves.

KK: Curves for the Mathematically Curious. It's sitting on my desk. Many of these books that you've mentioned are sitting on my desk.

EL: So yeah, great year for reading about calculus, but I think Ben would prefer that you start that reading with Change Is the Only Constant.

BO: It's very frothy, it's very quick and light-hearted and should be—you can use it as your appetizer to get into the the, the cheesier balls of the later books.

KK: But it's highly non-trivial. I mean, you talk about really interesting stuff in these books. It's not some frothy thing. I mean it's lighthearted, but it's not simple.

BO: I appreciate that. Yeah, the early draft of the book I was doing pretty much a pretty faithful march through the AP Calculus curriculum. And then that draft wasn't really working. And I realized that part of what I wasn't doing that should be doing was since I'm not teaching, you know, you had to execute calculus maneuvers. I'm not teaching how to take derivatives. I can talk about anything as long as I can explain the ideas. So we've got Weierstrass’s function in there. And there's a little bit even on Lebesgue integration, and other sort of, some stuff on differential equations crops up. So since I'm not actually teaching a calculus course and I don't need to give tests on it, I just got to tell stories.

EL: Well, yeah, I hope people will check that out. And thanks for joining us today.

BO: Yeah, thanks so much for having me.

KK: Yeah. Thanks, Ben.

[outro]

Our guest on this episode, Ben Orlin, is a high school math teacher best-known for his blog and popular math books. He told us about Weierstrass’s construction of a function that is continuous everywhere but differentiable nowhere. Here is a short collection of links that might be interesting.

Ben’s Blog, Math with Bad Drawings

Math with Bad Drawings, the book

Change is the Only Constant