Episode 12 - Candice Price

Kevin Knudson: Welcome to My Favorite Theorem. I am Kevin Knudson, professor of mathematics at the University of Florida, and I am joined by my cohost.

Evelyn Lamb: Hi. I’m Evelyn Lamb. I’m a math and science writer in Salt Lake City, Utah.

KK: How’s it going?

EL: Yeah, it’s going okay. It’s a bit smoky here from the fires in the rest of the west. A few in Utah, but I think we’re getting a lot from Montana and Oregon and Washington, too. You can’t see the mountains, which is a little sad. One of the nice things about living here.

KK: Yeah. Well, Hurricane Irma is bearing down on Florida. I haven’t been to the grocery store yet, but apparently we’re out of water in town. So I might have waited a couple days too late.

EL: Fill up those bathtubs, I guess.

KK: I guess. I don’t know. I’m dubious. You know, I lived in Mississippi when Katrina happened, and the eye came right over where we lived, and we never even lost Direct TV. I’m trying not to be cavalier, but we’ll see. Fingers crossed. It’s going to be bad news in south Florida, for sure. I really hope everybody’s OK.

EL: Yeah, definitely.

KK: Anyway.

EL: Fire, brimstone, and water recently.

KK: Anyway, we’re not here to talk about that. We’re here to talk about math. Today we’re thrilled to have Candice Price with us. Candice, want to say hi?

Candice Price: Hi everyone!

KK: Tell us a little bit about yourself.

CP: Sure. I’m currently an assistant professor of mathematics at the University of San Diego. I got my Ph.D. at the University of Iowa, and I study DNA topology, so knot theory applied to DNA, applied to biology.

EL: So that’s knot with a ‘k.’

CP: Yeah, knot.

KK: San Diego is a big switch from Iowa.

CP: Yeah, it is. In fact, I had a stopover in New York and a stopover in Texas before getting here. All over.

EL: You’ve really experienced a lot of different climates and types of people and places.

CP: Yeah. American culture, really.

KK: All right. You’ve told us. Evelyn and I know what your favorite theorem is, and I actually had to look this up, and I’m intrigued. So, Candice, what’s your favorite theorem?

CP: Sure. My favorite theorem is actually John H. Conway’s basic theorem on rational tangles. It’s a really cool theorem. What Conway states, or shows, is that there’s a one-to-one correspondence between the extended rational numbers, so rational numbers and infinity, and what are known as rational tangles. What a rational tangle basically is, is you can take a 3-ball, or a ball, an open ball, and if you put strings inside the ball and attach the strings to the boundary of the ball, so they’re loose in there but fixed, and you add these twists to the strings inside, if you take a count to how many twists you’ve added in these different directions, maybe the direction of west and the direction of south, and if you just write down how many twists you’ve done, first going west and then going south, and then going west, going south, all of those, all the different combinations you can do, you can actually calculate a rational number, and that rational number is attributed to that tangle, to that picture, that three-dimensional object.

It’s pretty cool because as you can guess, these tangles can get very complicated, but if I gave you a rational number, you could draw that tangle. And you can say that any tangle that has that same rational number, I should be able to just maneuver the strings inside the ball to look like the other tangles. So it’s actually pretty cool to say that something so complicated can just be denoted by fractions.

EL: Yeah. So how did you encounter this theorem? I encountered it from John Conway at this IAS program for women in math one year, and I don’t think that’s where we met. I don’t remember if you were there.

CP: I don’t think so.

EL: Yeah, I remember he did this demonstration. And of course he’s a very engaging, funny speaker. So yeah, how did you encounter it?

CP: It’s pretty cool, so he has this great video, the rational tangle dance. So it’s fun to show that. I started my graduate work as a master’s student at San Francisco State University, and I had learned a little bit about knot theory (with a ‘k’) as an undergrad. And so when I started my master’s I was introduced to Mariel Vazquez, who studies DNA topology. So she actually uses rational tangles in her research. That was the first time I had even heard that you could do math and biology together, which is a fascinating idea. She had introduced to me the idea of a rational tangle and showed me the theorem, and I read up on the proof, and it’s fascinating and amazing that those two are connected in that way, so that was the first time I saw it.

KK: Since I hadn’t heard of this theorem before, I looked it up, and I found this really cool classroom activity to do with elementary school kids. You take four kids, and you hand them two ropes. You allow them to do twists, the students on one end of the ropes interchange, and there’s a rotation function.

CP: Yeah.

KK: And then when you’re done you get a rational number, and it leads students through these explorations of, well, what does a twist do to an integer? It adds one. The rotate is a -1/x kind of thing.

CP: Right.

KK: So I was immediately intrigued. This really would be fun. Not just for middle school kids, maybe my calculus students would like it. Maybe I could find a way to make it relevant to my undergrads. I thought, what great fun.

CP: Yeah. I think it’s even a cool way to show students that with just a basic mathematical entity, fractions or rational numbers, you can perform higher mathematics. It’s pretty cool.

KK: This sort of begs the question: are there non-rational tangles? There must be.

CP: Yes there are! It categorizes these rational tangles, but there is not yet a categorization for non-rational tangles. There are two types. One is called prime, and one is called locally knotted. So the idea of locally knotted is that one of the strands just has a knot on it. A knot is exactly what you think about where you have a knot in your shoestring. Then prime, which is great, is all of the tangles that are not rational and not locally knotted. So it’s this space where we’ve dumped the rest of the tangles.

KK: That’s sort of unfortunate.

CP: Yeah, especially the choice of words.

KK: You would think that the primes would be a subset, somehow, of the rational tangles.

CP: You would hope.

EL: So how do these rational tangles show up in DNA topology?

CP: That’s a great question. So your DNA, you can think of as long, thin strings. That’s how I think about it. And it can wrap around itself, and in fact your DNA is naturally coiled around itself. That’s where that twisting action comes, so you have these two strings, and each string, we know, is a double helix. But I don’t care about the helical twist. I just care about how the DNA wraps around itself. These two strings can wind around, just based on packing issues, or a protein can come about and add these twists to it, and naturally how it just twists around. Visually, it looks like what is happening with rational tangles. Visually, the example that Kevin was mentioning, that we have the students with the two ropes, and they’re sort of twisting the ropes around, that’s what your DNA is doing. It turns into a great model, visually and topologically, of your DNA.

KK: Very cool.

CP: I like it.

KK: Wait, where does infinity come from, which one is that? It’s the inverse of 0 somehow, so you rotate the 0 strand?

CP: Yes, perfect. Very good.

KK: So you change your point of view, like when I’m proving the mean value theorem in calculus, I just say, well, it’s Rolle’s theorem as Forrest Gump would look at it, how he tilts his head.

CP: Right. I’m teaching calculus. I might have to use that. That’s good. I mean, hopefully they’ll know who Forrest Gump is.

KK: Well, right. You’re sort of dating yourself.

CP: That’s also a fun conversation to have with them.

KK: Sure. So another fun conversation on this podcast is the pairing. We ask our guests to pair their theorem with something. What have you chosen to pair Conway’s theorem with?

CP: So I thought a lot about this. So being in California, right, what I paired this with is a Neapolitan shake from In n Out burger. And the reason for that is, you’ve sort of taken these three different flavors, equally delicious on their own, right, rational numbers, topology, and DNA, and you put them together in this really beautiful, delicious shake. So the Neapolitan shake from In n Out burger is probably my favorite dessert, so for me, it’s a good pairing with Conway’s rational tangle theorem.

KK: I’ve only eaten at In n Out once in my life, sadly, and I didn’t have that shake, but I’m trying to picture this. So they must not mix it up too hard.

CP: They don’t, not too hard. So there’s a possibility of just getting strawberry, just getting vanilla, just getting chocolate, but then you can at some point get all three flavors together, and it’s pretty amazing.

KK: So I can imagine if you mix it too much, it would just be, like, tan. It would just be this weird color.

CP: Maybe not as delicious looking as it is tasting.

KK: That’s an interesting idea.

CP: It’s pretty cool.

KK: So we also like to give our guests a chance to plug anything they’re working on. Talk about your blog, or anything going on.

CP: Sure. I am always doing a lot of things. I am hoping I can take this time to plug, in February we have a website—we is myself, Shelby Wilson, Raegan Higgins, and Erica Graham—a website called Mathematically Gifted and Black where we showcase or spotlight every day a contemporary black mathematician and their contributions to mathematics, and we’re working on that now. We’ll have an article in the AMS Notices in February coming up. It’s up now so you can see it. We launched in February 2017. It’s a great website. We’re really proud of it.

EL: Yeah. Last year it was a lot of fun to see who was going to be coming on the little calendar each time and read a little bit about their work. You guys did a really nice job with that.

CP: Thanks. We’re very proud, and I think the AMS will put a couple of posters around the website as well.

KK: Great. Well, Candice, thanks for joining us.

CP: Thank you.

KK: This has been good fun. I like learning new theorems. Thanks again.

CP: Yeah, of course. Thank you. I enjoyed it.

[outro]

Show notes (click here)

Episode 11 - Jeanne Nielsen Clelland

Kevin Knudson: Welcome to My Favorite Theorem. I’m Kevin Knudson, professor of mathematics at the University of Florida. I’m flying solo in this episode. I’m at the Geometry in Gerrymandering workshop at Tufts University, sponsored by the Metric Geometry, what is it called, Metric Geometry and Gerrymandering Group, MGGG. It’s been a fantastic week. I’m without my cohost Evelyn Lamb in this episode because I’m on location, and I’m currently sitting in the lobby of my bed and breakfast with my very old friend, not old as in age, just going way back, friend, Jeanne Nielsen Clelland.

Jeanne Clelland: Hi Kevin. Thanks for having me.

KK: So you’re at the University of Colorado, yes?

JC: University of Colorado at Boulder, yes.

KK: Tell everyone about yourself.

JC: Well, as you said, we’re old friends, going all the way back to grad school.

KK: Indeed. Let’s not say how long.

JC: Let’s not say how long. That’s a good idea. We went to graduate school together. My area is differential geometry and applications of geometry to differential equations. I’m a professor at the University of Colorado at Boulder, and I’m also really enjoying this gerrymandering conference, and I’m really happy to be here.

KK: Let’s see if we can solve that problem. Although, as we learned today, it appears to be NP-hard.

JC: Right.

KK: That shouldn’t be surprising in some sense. Anyway, hey, let’s put math to work for democracy. Whether we can solve the problem or not, maybe we can make it better. So I know your favorite theorem, but why don’t you tell our listeners. What’s your favorite theorem?

JC: My favorite theorem is the Gauss-Bonnet theorem.

KK: That’s awesome because if anybody’s gone to our Facebook page, My Favorite Theorem, or our Twitter feed, @myfavethm, the banner picture, the theorem stated there is the Gauss-Bonnet theorem. That’s accidental. I just thought the statement looked pretty.

JC: Yeah, and when I first looked at your page, I saw that. And I thought, well, I guess my favorite theorem is already taken since it’s your banner page, so I was really excited to hear that I could talk about it.

KK: No, no, no. In fact, I was doing one last week, and the person mentioned they might do Gauss-Bonnet, and I said no, no, no. I have an expert on Gauss-Bonnet who’s going to do it for us. So why don’t you tell us what Gauss-Bonnet is?

JC: OK. So Gauss-Bonnet is about a relationship between, so it’s in differential geometry. It comes from the geometry of surfaces, and you can start with surfaces in 3-dimensional space that are easy to visualize. And there are several notions of curvature for surfaces. One of these notions is called the Gauss curvature, and roughly it measures whether a surface is bowl-shaped or saddle-shaped. So if the Gauss curvature is positive, then you think the surface looks more like a bowl, like a sphere is the prototypical example of positive Gauss curvature. If the Gauss curvature is negative, then your surface is shaped more like a saddle, and if the Gauss curvature is zero, then you think your surface, well the prototypical example is a plane, a surface that’s flat, but in fact this is a notion that is metrically invariant, which means if you take a surface and bend it without stretching it, you won’t change the Gauss curvature.

KK: OK.

JC: So for instance I could take a flat piece of paper and wrap it up into a cylinder.

KK: Yes.

JC: And since that doesn’t change how I measure distance, at least small distances on that piece of paper, a cylinder also has Gauss curvature zero.

KK: So this is a global condition?

JC: No, it’s local.

KK: Right.

JC: It’s a function on the surface, so at every point you can talk about the Gauss curvature at a point. So of course the examples I’ve given you, the sphere, the plane, those are surfaces where the Gauss curvature is constant, but on most surfaces this is a function, it varies from point to point.

KK: Right, so a donut, a torus, on the inside it would be negative, right?

JC: Right.

KK: But on the outside,

JC: That’s exactly right, and that’s a great example. We’re going to come back to the example of the torus.

KK: Good.

JC: So at the other extreme for surface, particularly for compact surfaces, you have topology, which is your area. And there’s a fundamental invariant of surfaces called the Euler characteristic. And the way you can compute this is really fun. You draw a graph, and the mathematical notion of a graph is basically you have points, which are called vertices, you have edges joining your vertices, and then you have regions enclosed by these edges, which are called faces.

KK: Yes.

JC: And if you take a surface, you can draw a graph on it any way you like. You count the number of vertices V, the number of edges E, and the number of faces F. You compute the number V-E+F, and no matter how you drew your graph, that number will be the same for any graph on a given surface.

KK: Which is remarkable enough.

JC: That is remarkable enough, right, that’s hugely remarkable. That’s a very famous theorem that makes this number a topological invariant, so for instance the Euler characteristic is 2, the Euler characteristic of a donut is zero. If you were to take, say, a donut with multiple holes, my son really loves these things called two-tone knots, which are donuts. A two-tone has Euler characteristic of -2, and generally the more holes you add, the more negative the Euler characteristic.

KK: Right, so the formula is 2 minus two times the number of holes, or 2-2g.

JC: Yes, and that’s for a compact surface.

KK: Compact surfaces.

JC: And it gets more complicated for non-compact. So the Gauss-Bonnet theorem in its simplest form, and let me just state it for compact surfaces, so I’m not worried about boundary, it says if you take the Gauss curvature, which is this function, and you integrate that function over the surface, the number that you get is 2π times the Euler characteristic.

KK: This blew my mind the first time I saw it.

JC: This is an incredible relationship, a very surprising relationship between geometry and topology. So for instance, if you take your surface and you wiggle it, you bend it, you can change that Gauss curvature a lot.

KK: Sure.

JC: You can introduce all sorts of wiggles in it from point to point. What this theorem says is that however you do that, all those wiggles have to cancel out because the integral of that function does not change if you wiggle the surface. It’s this absolutely incredible fact.

KK: So for example take a sphere. So we would get 4π.

JC: 4π.

KK: A sphere has constant sectional curvature 1. I guess, can you change that? You can, right?

JC: Sure!

KK: But if you maybe stretch it into an ellipsoid, the curvature is still maybe going to be positive, it’s going to be really steep at the pointy ends but flatter in the middle. So the way I always visualized this was that yeah, you might bend and stretch, which topologists don’t care about, and this integral—and the way we think about integrals is that they’re just big sums, right?—so you increase some of the numbers and decrease some of the numbers, so they’re just canceling out.

JC: Not only that, these numbers are scale invariant. So if you take a big sphere versus a small sphere, the big sphere has more area, but the absolute value of the curvature function is smaller, and those things cancel out. So the integral remains 4π.

KK: Right, so the surface of the Earth, for example, we can’t really see the curvature.

JC: Right.

KK: But it is curved.

JC: It is curved, and the area is so big that the integral of that very small function over that very large area would still be 4π.

KK: Right. So on the donut, right, we’re getting this cancelation. On the inside it’s negative, and it’s going to be 0 in some points, and on the outside it’s positive.

JC: Right. That’s really the amazing thing about the donut. It’s this unique surface where you get zero. So you have this outer part of the donut where the Gauss curvature is positive, the inner part where it’s negative, and no matter what you do to your donut, how irregularly shaped you make it, just the fact that it’s donut shaped means that those regions of positive and negative curvature exactly cancel each other out.

KK: Wow. Yeah, it’s a remarkable theorem. Great connection between geometry and topology. Do you want to talk about the noncompact case?

JC: This also gets interesting for surfaces with boundary. It actually starts, when I teach this in a differential geometry class, where this starts is a very classical idea called the angle excess theorem. And this goes back to Euclidean geometry. So everybody knows in flat Euclidean geometry, if you draw a triangle, what’s the sum of the angles inside the triangle?

KK: Yeah, 180 degrees.

JC: 180 degrees, π, depending on whether you want to work in degrees or radians. This is a consequence of the parallel postulate, and in the history of developing non-Euclidean geometry, what happened is people had developed alternate ideas of geometry with alternate versions of the parallel postulate. So in spherical geometry, imagine you draw a triangle on the sphere. Say you’ve got a globe. Take a triangle with points: one vertex is at the north pole, and two vertices are at the Equator. Say you’ve moved a quarter of the way around the circle, and the straight lines in this geometry are great circles.

KK: Yes.

JC: So draw a triangle between those three points with great circles. That’s a triangle with three right angles.

KK: 270 degrees.

JC: Right, 270 degrees. What the angle excess theorem says is that the difference, and we use radians, so that has 3π/2 angle, instead of π. So it says that the difference of those two numbers is the integral of the Gauss curvature over that triangle.

KK: Oh wow, OK. OK, I believe that.

JC: As we were saying for a sphere, the total Gauss curvature integral is 4π. This triangle I’ve just described takes up an eighth of the sphere, it’s an octant. So its area is π/2, so that’s the difference of its Gauss curvature. So that’s why the difference of sum of those angles and π is π/2. So that’s where this theorem starts, and ultimately the way you prove the angle excess theorem, basically it boils down to Green’s theorem, which I was very excited to hear Amie Wilkinson talk about in one of your previous episodes. It’s really just Green’s theorem to prove the angle excess theorem. So from there, the way you prove the global Gauss-Bonnet theorem is you triangulate your surface. You cut it up into geodesic triangles, you apply the angle excess theorem to each of those triangles, you add them all up, and you count very carefully based on the graph you have drawn of triangles how many vertices, how many edges, and how many faces. And when you count carefully, the Euler characteristic pops out on the other side.

KK: Right, OK.

JC: It’s this very neat combination of classical things, the angle excess theorem and combinatorics. It’s fun teaching an undergraduate course when you tell them counting is hard.

KK: It is hard.

JC: And they don’t believe you until you show them the ways it’s hard.

KK: There’s no way. I can’t count.

JC: So it’s a really fun theorem to do with students. It’s the culmination of the differential geometry class that I teach for undergraduates. I spend the whole semester saying, “Just wait until we get to Gauss-Bonnet! You’re going to think this is really cool!” And when we get there, it really does live up to the hype. They’re really excited by it.

KK: Yeah. So this leads to the question. We like to pair our theorems with something. What have you chosen to pair the Gauss-Bonnet theorem with?

JC: Well the obvious thing would be donuts.

KK: Sure.

JC: And in fact I do sometimes bring in donuts to class to celebrate the end of the class, but you know, this is such a culminating theorem, I really wanted to pair it with something celebratory, like a fireworks display or some sort of very celebratory piece of music.

KK: I can get on with that. It’s true, donuts seem awfully pedestrian.

JC: They do. Donuts are great because of the content of the theorem. They’re a little too pedestrian.

KK: So a fireworks display with what, 1812 Overture?

JC: Something like that.

KK: Really, this is the end. Bang!

JC: I think it deserves the 1812 Overture.

KK: That’s a really good one, OK. And maybe we’ll try to get that into the podcast.

JC: That would be great.

KK: A nice public domain thing if I can find it.

[1812 Overture plays]

JC: Sounds great.

KK: So we like to give our guests a chance to plug something. So you published a book recently?

JC: I did. I recently published a book. It’s called From Frenet to Cartan: The Method of Moving Frames. It’s published in the American Math Society’s graduate series, and it’s basically designed to be a second course in differential geometry, so for advanced undergraduates or beginning graduate students who have had a course in curves and surfaces. Hopefully it’s accessible at that level, and it was really fun. It largely grew out of working with students doing independent study, so I really wrote this book in a way that’s intended to be very student-friendly. It’s informal in style and written the way I would talk to a student in my office. I’m very happy with how it came out, so if this is a topic that’s interesting to any of your listeners, check it out.

KK: That’s great. I took curves and surfaces from your advisor, Robert Bryant, who’s the nicest guy you’ve ever met.

JC: Oh, he’s wonderful.

KK: Everybody loves Robert. That was the last differential geometry course I took, so maybe I should read your book.

JC: Let me give him credit, too. Where this originally came from, when I was a new Ph.D., well relatively new, three years post-Ph.D., Robert invited me to give a series of graduate lectures with him at MSRI, and this book grew out of notes I wrote for that workshop many, many years ago. And Robert, when I very naively said to him, “You know, I have all these lecture notes I should turn into a book,” Robert, having written a book, should have laughed at me, but instead he said, “Yeah, you should!” And it became a back burner project for a long time.

KK: More than a decade, probably.

JC: Yeah, but eventually, I’ve had so much fun working with students on this project.

KK: I’ve written two books, and it’s really, it’s so much work.

JC: You don’t do it for the money.

KK: You really don’t do it for the money, that’s for sure. And of course it’s great you had such a model in Robert, as a teacher and an expositor.

JC: I count myself extremely fortunate to have had him as my advisor.

KK: Well, Jeanne, this has been fun. Thanks for joining us.

JC: Thanks for having me.

[outro]

Show notes (click here)

Episode 10 - Mohamed Omar

Kevin Knudson: Welcome to My Favorite Theorem. I’m your host Kevin Knudson, professor of mathematics at the University of Florida. And I’m joined by my cohost.

EL: I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah.

KK: Welcome home.

EL: Yeah, thanks. I just got back from Paris a week ago, and I’m almost back on Utah time. So right now I’m waking up very early, but not 3 in the morning, more like 5 or 6.

KK: Wait until you’re my age, and then you’ll just wake up early in the morning because you’re my age.

EL: Yeah. I was talking to my grandma the other day, and I was saying I was waking up early, and she said, Oh, I woke up at 4 this morning.

KK: Yeah, that’s when I woke up. It’s not cool. I don’t think I’m as old as your grandmother.

EL: I doubt it.

KK: But I’m just here to tell you, winter is coming, let me put it that way. We’re pleased today to welcome Mohamed Omar. Mohamed, why don’t you tell everyone a little bit.

MO: Great to be on the podcast. My name is Dr. Mohamed Omar. I’m a professor at Harvey Mudd College. My area of specialty is algebra and combinatorics, and I like pure and applied flavors of that, so theoretical work and also seeing it come to light in a lot of different sciences and computer science. I especially like working with students, so they’re really involved in the work that I do. And I just generally like to be playful with math, you know, have a fun time, and things like this podcast.

KK: Cool, that’s what we aim for.

EL: And I guess combinatorics probably lends itself to a lot of fun games to play, or it always seems like it.

MO: Yeah. The thing I really like about it is that you can see it come to life in a lot of games, and a lot of hobbies can motivate the work that comes up in it. But at the same time, you can see it as a lens for learning a lot of more advanced math, such as, like, abstract algebra, sort of as a gateway to subjects like that. So I love this diversity in that respect.

KK: I always thought combinatorics was hard. I thought I knew how to count until I tried to learn combinatorics. It’s like, wait a minute, I can’t count anything.

MO: It’s difficult when you have to deal with distinguishability and indistinguishability and mixing them, and you sort of get them confused. Yeah, definitely.

KK: Yeah, what’s it like to work at Harvey Mudd? That always seemed like a really interesting place to be.

MO: Harvey Mudd is great. I think the aspects I like of it a lot are that the students are just intrinsically interested and motivated in math and science, and they’re really excited about it. And so it really feels like you’re at a place where people are having a lot of fun with a lot of the tools they learn. So when you’re teaching there, it’s a really interactive, fun experience with the students. There’s a lot of active learning that goes on because the students are so interested in these things. It’s a lot of fun.

KK: Very cool. So, Mohamed, what’s your favorite theorem?

MO: First of all, my favorite theorem is a lemma. Actually a theorem, but usually referred to as a lemma.

KK: Lemmas are where all the work is, right?

MO: Exactly. It’s funny you mention combinatorics because this is actually in combinatorics. It’s called Burnside's Lemma. Yeah, so I love Burnside's Lemma a lot, so maybe I’ll give a little idea of what it is and give an idea in light of what you mentioned, which is that combinatorics can be quite hard. So I’ll start with a problem that’s hard, a combinatorial one that’s hard. So imagine you have a cube. A cube has six faces, right? And say you ask the naive question how many ways are there to paint the faces of the cube with colors red, green, and blue.

KK: Right.

MO: You think, there are six faces, and the top face is either red, or green, or blue, and for every choice of color I use there, another face is red or green or blue, etc. So the number of colorings should be 3x3x3x3x3x3, 3^6.

EL: Right.

MO: But then, you know, you can put a little bit of a twist on this. You can say, how many ways are there to do this if you consider two colorings to be the same if you take the cube and rotate it, take one coloring, rotate the cube, and get another coloring.

EL: Right. If you had the red face on the left side, it could be on the top, and that would be the same.

MO: One naive approach that people tend to think works when they first are faced with this, is they think, OK, there are 6 faces, so maybe I can permute things 6 ways, so I divide the total number by 6.

KK: Wrong.

MO: Exactly. There are a lot of reasons. One is sort of the empirical reason. You said the answer was 3^6 if we’re not caring about symmetry. If you divide that by 6, there’s a little bit of a problem, right?

EL: Yeah.

MO: You can kind of see. If you have a painting where all the faces are red, no matter how you rotate that, you’re going to end up with the same coloring. But as you mentioned, if you color one face red and the rest green, for instance, then you get six different colorings when you rotate this cube around. So you’ve got to do something a little bit different. And Burnside's lemma essentially gives you a nice quick way to approach this by looking at something that’s completely different but easy to calculate. And so this is sort of why I love it a lot. It’s a really, really cool theorem that you can sort of explain at a maybe discrete math kind of level if you’re teaching at a university.

KK: So the actual statement, let’s see if I can remember this. It’s something like the number of colorings would be something like 1 over the order of the group of rotations times the sum of what is it the number of elements in each orbit, or something like that? I used to know this pretty well, and I’ve forgotten it now.

MO: Yeah, so something like that. So a way to think about it is, you have your object, and it has a bunch of symmetries. So if you took a square and you were coloring, say, the edges, this is an analogous situation to the faces of the cube. A square has 8 symmetries. There are the four rotations, but then you can also flip along axes that go through opposite edges, and then axes that go through opposite vertices.

So what Burnside's lemma says is something like this. If you want to know the number of ways to color, up to this rotational symmetry, you can look at every single one of these symmetries that you have. In the square it’s 8, in the cube it turns out to be 24. And for every single symmetry, you ask yourself how many ways are there to color with the three colors you have where the coloring does not change under that symmetry.

KK: The number of things fixed, essentially, right.

MO: Exactly. The number of things fixed by the symmetries. So like I mentioned, the cube has 24 symmetries. So let’s take an example of one. Let’s say you put a rod through the center of two opposite faces of the cube.

KK: Right.

MO: And you rotate 90 degrees along that. So you’re thinking about the top face and the bottom face and just rotating 90 degrees. Let’s just think about the colorings that would remain unchanged by that symmetry. So you’re free to choose whatever you’d like for the top and bottom face. But all the side faces will have to have the same color. Because as soon as you get another face. Whatever was in that face is now rotated 90 degrees as well. So if you count the number of colorings fixed by that rotation about the rod through the opposite faces, you get something like, well you have three choices for those side faces. As soon as you choose the color for one, you’re forced to use the same color for the rest. And then you have freedom in your top and bottom faces. So that’s just one of the symmetries. Now if you did that for every single symmetry and took the average of them, it turns out to be the number of ways to color the faces of the cube up to rotational symmetry in general.

So it’s kind of weird. There’s sort of two things that are going on. One is why in the world would looking at the symmetries and counting the number of colorings fixed under the symmetry have anything to do with the number of colorings in total up to symmetry in general? It’s not clear what the relationship there is at first. But the real cool part is that if you take every single symmetry and count the number of colorings, that’s a systematic thing you can do without having to think too hard. It’s a nice formula you can get at the answer quite quickly even though it seems like a complicated thing that you’re doing.

EL: Yeah. So I guess that naive way we were talking about to approach this where you just say, well I have three choices for this one, three choices for that one, you almost kind of look at it from the opposite side. Instead of thinking about how I’m painting things, I think about how I’m turning things. And then looking at it on a case by case basis rather than looking at the individual faces, maybe.

MO: Exactly. When I first saw this, I saw this as an undergrad, and I was like, “What?!” That was my initial reaction. It was a cool way to make some of this abstract math we were learning really come to life. And I could see what was happening in the mathematics physically, and that gave me a lot of intuition for a lot of the later things we were learning related to that theorem.

EL: Was that in a combinatorics class, or discrete math class?

MO: It was actually in a standalone combinatorics class that I learned this. And now another reason I really like this lemma is that I teach it in a discrete math course that I teach at Harvey Mudd, but then I revisit it in an abstract algebra course because really, you can prove this theorem using a theorem in abstract algebra called the orbit stabilizer theorem. So orbits are all of these different, you take one coloring, spin it around in all possible ways, you get a whole bunch of different ones, and stabilizers you can think of as taking one symmetry and asking what colorings are fixed under that symmetry. So that’s in our example what those two things are. In abstract algebra, there’s this orbit stabilizer theorem that has to do with more general objects: groups, like you mentioned. And then one of the things I really like about this theorem is that it sets the stage for even more advanced math like representation theory. I feel like a lot of the introductory concepts in a representation theory course really come back to things you play with in Burnside’s Lemma. It’s really cool in its versatility like that.

KK: That’s the context I know it in. So I haven’t taught group theory in 10 years or so, but it was in that course. Now I’m remembering all of this. It’s coming back. This is good. I’m glad we’re having this conversation. I’m with you. I think this is a really remarkable theorem. But I never took a combinatorics course that was deep enough where we got this far. I only know it from the groups acting on sets point of view, which is how you prove this thing, right? And as you say, it’s definitely leads into representation theory because, as you say, you can build representations of your groups. You just take a basis for a vector space and let it act this way, and a lot of those character formulas really drop out of this.

MO: Exactly.

KK: Very cool.

EL: So it sounds like you did not have a hard time choosing your favorite theorem. This was really, you sound very excited about this theorem.

MO: The way I tried to think about what my favorite theorem was what theorem to I constantly revisit in multiple different courses? If I do that, I must like it, right? And then I thought, hey, Burnside's Lemma is one that I teach in multiple courses because I like all the different perspectives that you can view it from. Then I had this thought: is Burnside's Lemma really a theorem?

KK: Yeah, it is.

MO: I felt justified in for the following reason, which is I think this lemma’s actually due to Frobenius, not Burnside. I thought, since the Burnside part is not really due to Burnside, then maybe the lemma part really is a theorem.

EL: I must say, Burnside sounds like it should be a Civil War general or something.

MO: Definitely.

EL: So what have you chosen to pair with your theorem?

MO: So I thought a chessboard marble cake would be perfect.

KK: Absolutely.

EL: OK!

MO: So first of all, I had a slice of one just about a few hours ago. It was my brother’s birthday recently, and I’m visiting family. There was leftover cake, and I indulged. But then I thought yeah, one of the prototypical problems when playing around with Burnside's Lemma is how many ways are there to color the cells of a chessboard up to rotational symmetry? So when I was eating the cake, I thought, hey, this is perfect!

EL: That’s great.

KK: How big of a chessboard was it?

MO: 8x8.

KK: Wow, that’s pretty remarkable.

MO: It was a big cake. I had a big piece.

KK: So when you sliced into it, was it 8x8 that way, or 8x8 across the top?

MO: Across the top.

KK: I’m sort of imagining, so my sister in law is a pastry chef, and she makes these remarkably interesting looking things, and it’s usually more like a 3x3, the standard if you go vertical.

EL: I’ve never tried to make a chessboard cake. I like to bake a lot, but anything that involves me being fussy about how something looks is just not for me. In baking. Eating I’m happy with.

MO: I’m the same. I really enjoy cooking a lot. I enjoy the cooking and the eating, not the design.

KK: Yeah, I’m right there with you. Well this has been fun. Thanks for joining us, Mohamed.

EL: Yeah.

MO: Thank you. This has been really enjoyable.

KK: Take care.

MO: Thank you.

[outro]

Show notes (click here)

Episode 9 - Ami Radunskaya

Evelyn Lamb: Welcome to My Favorite Theorem. I’m your host Evelyn Lamb. I’m a freelance math and science writer based in Salt Lake City. And today I am not joined by my cohost Kevin Knudson. Today I am solo for a very special episode of My Favorite Theorem because I am at MathFest, the annual summer meeting of the Mathematical Association of America. This year it’s in Chicago, a city I love. I lived here for a couple years, and it has been very fun to be back here with the big buildings and the lake and everything. There are about 2,000 other mathematicians here if I understand correctly. It’s a very busy few days with lots of talks to attend and friends to see, and I am very grateful that Ami Radunskaya has taken the time to record this podcast with me. So will you tell me a little bit about yourself?

Ami Radunskaya: Hi Evelyn. Thanks. I’m happy to be here at MathFest and talking to you. It’s a very fun conference for me. By way of introduction, I’m the current president for the Association for Women in Mathematics, and I’m a math professor at Pomona College in Claremont, which is a small liberal arts college in the Los Angeles County. My Ph.D. was in ergodic theory, something I am going to talk about a little bit. I went to Stanford for my doctorate, and before that I was an undergraduate at Berkeley. So I grew up in Berkeley, and it was very hard to leave.

EL: Yeah. You fall in love with the Bay Area if you go there.

AR: It’s a place dear to my heart, but I was actually born in Chicago.

EL: Oh really?

AR: So I used to visit my grandparents here, and it brings back memories of the Museum of Science and Industry and all those cool exhibits, so I’m loving being back here.

EL: Yeah, we lived in Hyde Park when we were here, so yeah, the Museum of Science and Industry.

AR: I think I was born there, Hyde Park.

EL: Oh? Good for you.

AR: My dad was one of the first Ph.D.s in statistics from the University of Chicago.

EL: Oh, nice.

AR: Although he later became an economist.

EL: Cool. So, what is your favorite theorem?

AR: I’m thinking today my favorite theorem is the Birkhoff ergodic theorem. I like it because it’s a very visual theorem. Can I kind of explain to you what it is?

EL: Yeah.

AR: So I’m not sure if you know what ergodic means. I actually first went into the area because I thought it was such a cool word, ergodic.

EL: Yeah, it is a cool word.

AR: I found out it comes from the Greek word ergod for path. So I’ve always loved the mathematics that describes change and structures evolving, so before I was a mathematician I was a professional cellist for about 10 years. Music and math are sort of as one in my mind, and that’s why I think I’m particularly attracted to the kinds of mathematics and the kinds of theory that describes how things change, what’s expected, what’s unexpected, what do we see coming out of a process, a dynamical process? So before I state the theorem, I need to tell you what ergodic means.

EL: Yeah.

AR: It’s an adjective. We’re talking about a function. We say a function is ergodic if it takes points: imagine you put a value into a function, you get out a new value. You put that value back in to the function, you get a new value. Repeat that over and over and over again, and now the function is ergodic if that set of points sort of visits everywhere in the space. So we say more technically a function is ergodic if the invariant sets, the sets it leaves alone, the sets that get mapped to themselves, are either the whole space or virtually nothing. A function is ergodic, a map is ergodic, if the invariant sets either have, we say, full measure or zero measure. So if you know anything about probability, it’s probability 1 or probability zero. I think that’s an easy way to think about measure.

EL: Yeah, and I think I’ve heard people describe ergodic as the time average is equal to the space average, so things are distributing very evenly when you look at long time scales. Is that right?

AR: Well that’s exactly the ergodic theorem. So that’s a theorem!

EL: Oh no!

AR: No, so that’s cool that you’ve heard of that. What I just said was that something is ergodic if the sets that it leaves unchanged are either everything or nothing, so these points, we call them the orbits, go everywhere around the set, but that doesn’t tell you how often they visit a particular piece of your space, whereas the ergodic theorem, so there are two versions of it. My favorite one is the one, they call it the pointwise ergodic theorem, because I think it’s easier to visualize. And it’s attributed to Birkhoff. So sometimes it’s called the Birkhoff ergodic theorem. And it’s exactly what you just said. So if you have an ergodic function, and then we start with a point and we sort of average it over many, many applications of the function, or iterations of the function, so that’s the time average. We think of applying this function once every time unit. The time average is the same as the integral of that function over the space. That’s the space average. So you can either take the function and see what it looks like over the entire space. And remember, that gives you, like, sizes of sets as well. So you might have your space, your function might be really big in the middle of the space, so when you integrate it over that piece, you’ll get a big hump. And it says that if I start iterating at any point, it’ll spend a lot more time in the space where the function is big. So the time average is equal to the space average. So that is the pointwise Birkhoff ergodic theorem. And I think it’s really cool because if you think about, say, if you’ve ever seen pictures of fractal attractors or something, so often these dynamical systems, these functions we’re looking at, are ergodic on their attractor. All the points get sucked into a certain subset, and then on that subset they stay on it forever and move around, so they’re ergodic on that attractor.

EL: Yeah.

AR: So if we just, say, take a computer and start with a number and plug it in our function and keep iterating, or maybe it’s a two-dimensional vector, or maybe it’s even a little shape, and you start iterating, you see a pattern appear because that point is visiting that set in exactly the right amount. Certain parts are darker, certain parts are lighter, and it’s as if, I don’t know in the old days, before digital cameras, we would actually develop photographs. Imagine you put that blank page in the developing fluid, and you sort of see it gradually appear. And it’s just like that. The ergodic theorem gives us that magical appearance of these shapes of these attractors.

EL: Yeah. That’s a fun image. I’m almost imagining a Polaroid picture, where it slowly, you know, you see that coming out.

AR: It’s the same idea. If you want to think about it another way, you’re sort of experiencing this process. You are the point, and you’re going around in your life. If your life is ergodic, and a lot of time it is, it says that you’ll keep bumping into certain things more often than others. What are those things you’ll bump into more often? Well the things that have higher measure for you, have higher meaning.

EL: Yeah. That’s a beautiful way to think about it. You kind of choose what you’re doing, but you’re guided.

AR: I call it, one measure I put on my life is the fun factor.

EL: That’s a good one.

AR: If your fun factor is higher, you’ll go there more often.

EL: Yeah. It also says something like, if you know what you value, you can choose to live your life so that you do visit those places more. That’s a good lesson. Let the ergodic theorem guide you in your life. OK, so what have you chosen to pair with this theorem?

AR: So the theorem has a lot of motion in it. A lot of motion, a lot of visualization. I think as far as music, it’s not so hard to think of an ergodic musical idea. Music is, after all, structures evolving through space.

EL: Exactly.

AR: I think I would pair Steve Reich’s Violin Phase. Do you know that piece?

EL: Yeah, yeah.

AR: So what it is, it’s a phrase on the violin, then you hear another copy of it playing at the same time. It’s a repetitive phrase, but one of them gets slightly out of phase with the other, and more and more and more and more. And what you hear are how those two combine in different ways as they get more and more and more and more out of phase. And if you think of that visually, you might think of rotating a circle bit by bit by bit, and in fact, we know irrational rotations of the circle are ergodic. You visit everywhere, so you hear all these different combinations of those patterns. So Steve Reich Violin Phase. He has a lot of pattern music. Some of it is less ergodic, I mean, you only hear certain things together. But I think that continuous phase thing is pretty cool.

EL: Yeah. And I think I’ve heard it as Piano Phase more often than Violin Phase.

AR: It’s a different piece. He wrote a bazillion of them.

EL: Yeah, but I guess the same idea. I really like your circle analogy. I almost imagine, maybe the notes are gears sticking out of the circle, and they line up sometimes. Because even when it’s not completely back in phase, sometimes the notes are playing at the same time but at a different part of the phrase. They almost lock in together for a little while, and then turn a little bit more and get out again and then lock in again at a different point in the phrase. Yeah, that’s a really neat visual. Have you performed much Steve Reich music?

AR: I’ve performed some, mostly his ensemble pieces, which are really fun because you have to focus. One of my favorites of his is called Clapping Music because you can do it with just two people. It’s the same idea as the Violin Phase, but it’s a discrete shift each time, so a shift by an eighth note. So the pattern is [clapping]. One person claps that over and over and over, and the other person claps that same rhythm but shifts it by one eighth note each time. So since that pattern is 12 beats long, you come back to it after 12 beats. So it’s discretized. You do each one twice, so it’s 24, so it’s long enough.

EL: So that’s a non-ergodic one, a periodic transformation.

AR: Exactly. So that one I do a lot when I give talks about how we can describe mathematics with its musical manifestations, but we can also describe music mathematically.

EL: Just like you, music is one of my loves too. I played viola for a long time. I’ve never performed any Steve Reich, and I’m glad you didn’t ask me to spontaneously perform Clapping Music with you. I think that would be tough to do on the spot.

AR: We can do that offline.

EL: Yeah, we’ll do that once we hang up.

AR: As far as foods, I think there are some great pairings of foods with the ergodic theorem. In fact, I think we apply the ergodic theorem often in cooking. You know, you mix stuff up. So one thing I like to do sometimes is make noodles, with a roller thing.

EL: Oh, from scratch?

AR: Yeah. You just get some flour, get some eggs, or if you’re vegan, you get some water. That’s the ingredients. You mix it up and you put it through this roller thing, so you can imagine things are getting quite mixed up. What’s really cool, I don’t know if you’ve ever eaten something they call in Italy paglia e fieno, straw and hay.

EL: No.

AR: And all it is is pasta colored green, so they put a little spinach in one of them. So you’ve got white and green noodles. So when you cook some spinach, you’ve got your dough. You put some blobs of spinach in. You start mushing it around and cranking it through, and you see the blobs make these cool streaks, and the patterns are amazing, until it’s uniformly, more or less, green.

EL: Yeah.

AR: So I’d say, paglia e fieno, we put on some Steve Reich, and there you go.

EL: That’s great. A double pairing. I like it.

AR: You can think of a lot of other things.

EL: Yeah, but in the pasta, you can really see it, almost like taffy. When you see pulling taffy. You can almost see how it’s getting transformed.

AR: It’s getting all mushed around.

EL: Thank you so much for talking to me about the Birkhoff ergodic theorem. And I hope you have a good rest of MathFest.

AR: You too, Evelyn. Thank you.

[outro]

Show notes (click here)

Episode 8 - Justin Curry

Kevin Knudson: Welcome to MFT. I'm Kevin Knudson, your host, professor of mathematics at the University of Florida. I am without my cohost Evelyn Lamb in this episode because I'm on location at the Banff International Research Station about a mile high in the Canadian Rockies, and this place is spectacular. If you ever get a chance to come here, for math or not, you should definitely make your way up here. I'm joined by my longtime friend Justin Curry. Justin.

Justin Curry: Hey Kevin.

KK: Can you tell us a little about yourself?

JC: I'm Justin Curry. I'm a mathematician working in the area of applied topology. I'm finishing up a postdoc at Duke University and on my way to a professorship at U Albany, and that's part of the SUNY system.

KK: Contratulations.

JC: Thank you.

KK: Landing that first tenure-track job is always

JC: No easy feat.

KK: Especially these days. I know the answer to this already because we talked about it a bit ahead of time, but tell us about your favorite theorem.

JC: So the theorem I decided to choose was the classification of regular polyhedra into the five Platonic solids.

KK: Very cool.

JC: I really like this theorem for a lot of reasons. There are some very natural things that show up in one proof of it. You use Euler's theorem, the Euler characteristic of things that look like the sphere, R=2.

There's duality between some of the shapes, and also it appears when you classify finite subgroups of SO(3). You get the symmetry groups of each of the solids.

KK: Oh right. Are those the only finite subgroups of SO(3)?

JC: Well you also have the cyclic and dihedral groups.

KK: Well sure.

JC: They embed in, but yes. The funny thing is they collapse too because dual solids have the same symmetry groups.

KK: Did the ancient Greeks know this, that these were the only five? I'm sure they suspected, but did they know?

JC: That's a good question. I don't know to what extent they had a proof that the only five regular polyhedra were the Platonic solids. But they definitely knew the list, and they knew they were special.

KK: Yes, because Archimedes had his solids. The Archimedean ones, you are allowed different polygons.

JC: That's right.

KK: But there's still this sort of regularity condition. I can never remember the actual definition, but there's like 13 of them, and then there's 5 Platonics. So you mentioned the proof involving the Euler characteristic, which is the one I had in mind. Can we maybe tell our listeners how that might go, at least roughly? We're not going to do a case analysis.

JC: Yeah. I mean, the proof is actually really simple. You know for a fact that vertices minus edges plus faces has to equal 2. Then when you take polyhedra constructed out of faces, those faces have a different number of edges. Think about a triangle, it has 3 edges, a square has 4 edges, a pentagon is at 5. You just ask how many edges or faces meet at a given vertex? And you end up creating these two equations. One is something like if your faces have p sides, then p times the number of faces equals 2 times the number of edges.

KK: Yeah.

JC: Then you want to look at this condition of faces meeting at a given vertex. You end up getting the equation q times the number of vertices equals 2 times the number of edges. Then you plug that into Euler's theorem, V-E+F=2, and you end up getting very rigid counting. Only a few solutions work.

KK: And of course you can't get anything bigger than pentagons because you end up in hyperbolic space.

JC: Oh yeah, that's right.

KK: You can certainly do this, you can make a torus. I've done this with origami, you sort of do this modular thing. You can make tori with decagons and octagons and things like that. But once you get to hexagons, you introduce negative curvature. Well, flat for hexagons.

JC: That's one of the reasons I love this theorem. It quickly introduces and intersects with so many higher branches of mathematics.

KK: Right. So are there other proofs, do you know?

JC: So I don't know of any other proofs.

KK: That's the one I thought of too, so I was wondering if there was some other slick proof.

JC: So I was initially thinking of the finite subgroups of SO(3). Again, this kind of fails to distinguish the dual ones. But you do pick out these special symmetry groups. You can ask what are these symmetries of, and you can start coming up with polyhedra.

KK: Sure, sure. Maybe we should remind our readers about-readers-I read too much on the internet-our listeners about duality. Can you explain how you get the dual of a polyhedral surface?

JC: Yeah, it's really simple and beautiful. Let's start with something, imagine you have a cube in your mind. Take the center of every face and put a vertex in. If you have the cube, you have six sides. So this dual, this thing we're constructing, has six vertices. If you connect edges according to when there was an edge in the original solid, and then you end up having faces corresponding to vertices in the original solid. You can quickly imagine you have this sort of jewel growing inside of a cube. That ends up being the octahedron.

KK: You join two vertices when the corresponding dual faces meet along an edge. So the cube has the octahedron as its dual. Then there's the icosahedron and the dodecahedron. The icosahedron has 20 triangular faces, and the dodecahedron has 12 pentagonal faces. When you do the vertex counts on all of that you see that those two things are dual. Then there's the tetrahedron, the fifth one. You say, wait a minute, what's its dual?

JC: Yeah, and well it's self-dual.

KK: It's self-dual. Self-dual is a nice thing to think about. There are other things that are self-dual that aren't Platonic solids of course. It's this nice philosophical concept.

JC: Exactly.

KK: You sort of have two sides to your personality. We all have this weird duality. Are we self-dual?

JC: I almost like to think of them as partners. The cube determines, without even knowing about it, its soulmate the octahedron. The dodecahedron without knowing it determines its soulmate the icosahedron. And well, the tetrahedron is in love with itself.

KK: This sounds like an algorithm for match.com.

JC: Exactly.

KK: I can just see this now. They ask a question, “Choose a solid.” Maybe they leave out the tetrahedron?

JC: Yeah, who knows?

KK: You don't want to date yourself.

JC: Maybe you do?

KK: Right, yeah. On our show we like to ask our guests to pair their theorem with something.

JC: It's a little lame in that it's sort of obvious, but Platonic solids get their name from Plato's Timaeus. It's his description of how the world came to be, his source of cosmogeny. In that text he describes an association of every Platonic solid with an element. The cube is correspondent with the element earth. You want to think about why would that be the case? Well, the cube can tessellate three-space, and it's very stable. And Earth is supposed to be very stable, and unshakeable in a sense. I don't know if Plato actually knew about duality, but the dual solid to the cube is the octahedron, which he associated with air. So you have this earth-sky symbolic dualism as well.

Then unfortunately I think this kind of analogy starts to break down a bit. You have the icosahedron, the one made of triangle sides. This is associated to water. And if you look at it, this one sort of looks like a drop of water. You can imagine it rolling around and being fluid. But it's dual to the dodecahedron, this oddball shape. They only thought of four elements: earth, fire, wind, water. What do you do with this fifth one? Well that was for him ether.

KK: So the tetrahedron is fire?

JC: Yeah, the tetrahedron is fire.

KK: Because it's so pointy?

JC: Exactly.

KK: It's sort of rough and raw, or that They Might Be Giants Song “Triangle Man.” It's the pointiest one. Triangle wins every time.

JC: The other thing I like is that fire needs air to breathe. And if you put tetrahedra and octahedra together, they tessellate 3-space.

KK: So did they know that?

JC: I don't know. That's why this is fun to speculate about. They obviously had an understanding. It's unclear what was the depth or rigor, but they definitely knew something.

KK: Sure.

JC: We've known this for thousands of years.

KK: And these models, are they medieval, was it Ptolemy or somebody, with the nested?

JC: The way the solar system works.

KK: Nested Platonic solids. These things are endlessly fascinating. I like making all of them out of origami, out of various things. You can do them all with business cards, except the dodecahedron.

JC: OK.

KK: It's hard to make pentagons. You can take these business cards and you can make these. Cubes are easy. The other ones are all triangular faces, and you can make these triangular modules where you make two triangles out of business cards with a couple of flaps. And two of them will give you a tetrahedron. Four of them will give you an octahedron. The icosahedron is tricky because you need, what, 10 business cards. I have one on my desk. It's been there for 10 years. It's very stable once it's together, but you have to use tape along the way and then take the tape off. It's great fun. There's this great book by Thomas Hull, I forgot the name of it [Ed note: it's called Project Origami: Activities for Exploring Mathematics], a great origami book by Thomas Hull. I certainly recommend all of that.

Anything else you want to add? Anything else you want to tell us about these things? You have all these things tattooed on your body, so you must be

JC: I definitely feel pretty passionate. It's one of those things, if I have to live with this for 30 years, I'll know the Platonic solid won't change. There won't be suddenly a new one discovered.

KK: Right. It's not like someone's name, you might regret it later. But my tattoos are, this is man, woman, and son. My wife and I just had our 25th anniversary, so this is still good. I don't expect to have to get rid of that.

Anyway, well thanks, Justin. This has been great fun. Thanks for taking a few minutes out of your busy schedule. This is a really cool conference, by the way.

JC: I love it. We're bringing together some of the brightest minds in applied topology, and outside of applied topology, to see how topology can inform data science and how algebra interacts in this area, what new foundations we need and aspects of algebra.

KK: Yeah, it's very cool. Thanks again, and good luck in your new job.

JC: Thanks, Kevin.

[outro]

Show notes (click here)

Episode 7 - Henry Fowler

Evelyn Lamb: Welcome to My Favorite Theorem, the show where we ask mathematicians what their favorite theorem is. I’m your host 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. I had to wear a sweater yesterday.

EL: Oh my goodness! Yeah, I’ve had to wear a sweater for about a month and a half, so.

KK: Yeah, yeah, yeah.

EL: Maybe not quite that long.

KK: Well, it’ll be hot again tomorrow.

EL: Yeah. So today we’re very glad to have our guest Henry Fowler on. Henry, would you like to tell us a little bit about yourself?

Henry Fowler: I’m a Navajo Indian. I live on the Navajo reservation. I live by the Four Corners in a community, Tsaile, Arizona. It’s a small, rural area. We have a tribal college here on the Navajo Nation, and that’s what I work for, Diné College. I’m a math faculty. I’m also the chair for the math, physics, and technology. And my clan in Navajo is my maternal clan is Bitterwater and my paternal clan is Zuni Edge Water.

EL: Yeah, and we met at the SACNAS conference just a couple weeks ago in Salt Lake City, and you gave a really moving keynote address there. You talked a little bit about how you’re involved with the Navajo Math Circles.

HF: Yes. I’m passionate about promoting math education for my people, the Navajo people.

EL: Can you tell us a little bit about the Navajo Math Circles?

HF: The Navajo Math Circles started seven years ago with a mathematician from San Jose State University, and her name is Tatiana Shubin. She contacted me by email, and she wanted to introduce some projects that she was working on, and one of the projects was math circles, which is a collection of mathematicians that come together, and they integrate their way of mathematical thinking for grades K-12 working with students and teachers. Her and I, we got together, and we discussed one of the projects she was doing, which was math circles. And it was going to be here on the Navajo Nation, so we called it Navajo Math Circles. Through her project and myself here living on the Navajo Nation, we started the Navajo math circles.

KK: How many students are involved?

HF: We started first here at Diné College, we started first with a math summer camp, where we sent out applications, and these were for students who had a desire or engaged themselves to study mathematics, and it was overwhelming. Over 50 students applied for only 30 slots that were open because our grant could only sustain 30 students. So we screened the students and with the help of their regular teachers from junior high or high school, so they had recommendation letters that were also presented to us. So we selected the first 30 students. Following that we expanded our math circle to the Navajo Nation public school system, and there’s also contract schools and grant schools. Now we’re serving, I would say over 1,000 students now.

KK: Wow. That’s great. I assume these students have gone on to do pretty interesting things once they finish high school and the circle.

HF: Yes. We sort of strategized. We wanted to work with lower grades a little bit. We wanted to really promote a different way of thinking about math problems. We started off with the first summer math camp at the junior high or the middle school level, and also the students that were barely moving to high school, their freshman year or their 10th grade year. That cohort, the one that we started off with, they have a good rate of doing very well with their academic work, especially in math, at their high school and junior high school. We have four that have graduated recently from high school, and all four of them are now attending a university.

KK: That’s great.

EL: And some of our listeners may have seen there’s a documentary about Navajo math circles that has played on PBS some, and we’ll include a link to that for people to learn a little bit about that in the show notes for the episode. We invited you here to My Favorite Theorem, of course, because we like to hear about what theorems mathematicians enjoy. So what have you selected as your favorite theorem?

HF: I have quite a few of them, but something that is simple, something that has been an awe for mathematicians, the most famous theorem would be the Pythagorean theorem because it also relates to my cultural practices, to the Navajo.

KK: Really?

HF: The Pythagorean theorem is also how Navajo would construct their traditional home. We would call it a Navajo hogan. The Navajo would use the Pythagorean theorem charting how the sun travels in the sky, so they would open their hogan door, which is always constructed facing east. So once the sun comes out, it projects its energy, the light, into the hogan. The Navajo began to study that phenomenon, how that light travels in space in the hogan. They can predict the solstice, the equinox. They can project how the constellations are moving in the sky, so that’s just a little example.

EL: Oh, yeah. Mathematicians, we call it the Pythagorean theorem, but like many things in math, it’s not named after the first person ever to notice this relationship. The Pythagorean theorem is a2+b2=c2, the relationship between the lengths of the legs of a right triangle and the hypotenuse of a right triangle, but it was known in many civilizations before, well before Pythagoras was born, a long time ago. In China, India, the Middle East, and in North America as well.

HF: Yes, Navajo, we believe in a circle of life. There’s time that we go through our process of life and go back to the end of our circle, and it’s always about to give back, that’s our main cultural teaching, to give back as much as you can, back to the people, back to nature, back to your community, as well as what you want to promote, what you’re passionate about, to give back to the people that way. Our way is always interacting with circles, that phenomenon, and how the Navajo see the relationship to space, the relationship to sunlight, how it travels, how they capture it in their hogan. Also they can related it to defining distance, how they relate the Pythagorean theorem to distance as well as to a circle.

KK: What shape is the hogan? I’m sort of curious now. When the light comes in, what sort of shadows does it cast?

HF: The Navajo hogan is normally a nine-sided polygon, but it Navajo can also capture what a circle means by regular polygons, the more sides they have, drawing to become a circle. They understand that event, they understand that phenomenon. The nine sides is in relationship to when a child is conceived and then delivered, it’s nine months. The Navajo call it nine full moons because they only capture what’s going on within their environment, and they’re really observant to how the sky and constellations are moving. Their monthly calendar is by full moon. And so that’s how, when the light travels in, when they open that hogan door, it’s like a semi-circle. In that space they feel like they are also secure and safe, and that hogan is also a representation that they are the child of Mother Earth, and that they are the child of Father Sky. And so that hogan is structured in relationship to a mother’s womb, when a child is being conceived and that development begins to happen. Navajos say that the hogan is a structure where in relationship there are four seasons, four directions, and then there are four developments that happen until you enter old age. There will be the time of your birth, the time when you become an adult, mid-life, and eventually old age. So using that concept, when that door is open, they harvest that sunlight when it comes in. Now we are moving to the state of winter solstice. That happens, to western thinking, around December 22. To the Navajo, that would be the 13th full moon, so when that light comes in that day, it will be a repeated event. They will know where. When the light comes into the hogan, when the door is opened, it will project on the wall of the hogan. When it projects on that wall, they mark it off. Every time, each full moon, they capture that light to see where it hits on the wall. That’s how they understand the equinox, that’s how they understand the solstice, in relationship to how the light is happening.

KK: Wow, that’s more than my house can do.

HF: Then they also use a wood stove to heat the hogan. There’s an opening at the center of the hogan, they call the chimney. They capture that sunlight, and they do every full moon. Sometimes they do it at the middle of that calendar, they can even divide that calendar into quarters. When they divide it into quarters, they chart that light as it comes in through the chimney. They find out that the sun travels through the sky in a figure eight in one whole year. They understand that phenomenon too.

KK: Ancient mathematics was all about astronomy, right? Every culture has tried to figure this out, and this is a really ingenious solution, with the chimney and the light. That’s really very cool.

HF: The practice is beginning not to be learned by our next generation because now our homes are more standardized. We’re moving away from that traditional hogan. Our students and our young people are beginning not to interact with how that light travels in the hogan space.

KK: Did you live in a hogan growing up?

HF: Yes. People around probably my age, that was how they were raised, was in a traditional hogan. And that was home for us, that construction. Use the land, use nature to construct your home with whatever is nearby. That’s how you create your home. Now everything is standardized in relation to different building codes.

EL: So what have you chosen to pair with your theorem?

HF: I guess I pair my Pythagorean theorem to my identity, who I am, as a Navajo person. I really value my identity, who I am as an indigenous person. I’m very proud of my culture, my land, where I come from, my language, as well as I compare it to what I know, the ancient knowledge of my ancestors is that I always respect my Navajo elders.

KK: Very cool. Do you think that living in a hogan, growing up in a hogan, did that affect you mathematically? Do you think it sort of made you want to be a mathematician? Were you aware of it?

HF: I believe so. We did a lot of our own construction, nothing so much that would be store-bought. If you want to play with toys, you’d have to create that toy on your own. So that spatial thinking, driving our animals from different locations to different spots, and then bringing our sheep back at a certain time. You’d calculate this distance, you’d estimate distance. You’d do a lot of different relationships interacting with nature, how it releases patterns. You’d get to know the patterns, and the number sense, the relationships. I really, truly believe that my culture gave me that background to engage myself to study mathematics.

KK: Wow.

EL: Yeah, and now you’re making sure that you can pass on that knowledge and that love for mathematics to younger people from your community as well.

HF: That’s my whole passion, is to strengthen our math education for my Navajo people. Our Navajo reservation is as large as West Virginia.

EL: Oh, wow, I didn’t realize that.

HF: And there’s no leader that has stood up to say, “I’m going to promote math education.” Right now, in my people, I’m one of the leaders in promoting math eduction. It’s strengthening our math K-12 so we build our infrastructure, we build our economy, we build better lives for my Navajo people, and that we build our own scientists, we build our own doctors and nurses, and we want to promote our own students, to show interests or take the passion and have careers in STEM fields. We want to build our own Navajo professors, Navajo scholars, Navajo researchers. That all takes down to math education. If we strengthen the education, we can say we are a sovereign nation, a sovereign tribe, where we can begin to build our own nation using our own people to build that nation.

EL: Wow, that’s really important work, and I hope our listeners will go and learn a little bit more about the Navajo math circles and the work you do, and other teachers and everyone are doing there.

HF: It’s wonderful because we have so many social ills, social problems among my people. There’s so much poverty here. We have near 50 percent unemployment. And we want my people to have the same access to opportunity just like any other state out there. And the way, from my perspective, is to promote math education, to bring social justice and to have access to a fair education for my people. And it’s time that the Navajo people operate their own school system with their own indigenous view, create our own curriculum, create our own math curriculum, and standardize our math curriculum in line to our elders’ thinking, to our culture, to our language, and that’s just all for my Navajo people to understand their self-identity, so they truly know who they are, so they become better people, and they get that strength, so that that motivation comes. To me, that’s what my work is all about, to help my people as a way to combat the social problems that we’re having. I really believe that math kept me out of problems when I was growing up. I could have easily joined a gang group. I would not have finished my education, my western education, but math kept me out of problems, out of trouble growing up.

KK: You’re an inspiration. I feel like I’m slacking. I need to do something here.

EL: Yeah. Thank you so, so much for being on the podcast with us. I really enjoyed talking with you.

KK: Yeah, this was great, Henry. Thank you.

HF: You’re welcome.

[outro]

Show notes (click here)

Episode 6 - Eriko Hironaka

This transcript is provided as a courtesy and may contain errors.

EL: Welcome to My Favorite Theorem. I’m your host Evelyn Lamb. I’m a freelance math and science writer based in Paris for a few more days, but after that I’ll be based in Salt Lake City, Utah. And this is your other host.

KK: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida, where it’s raining. It’s been raining for a week. After a spring of no rain, now it’s raining. But that’s OK.

EL: I probably shouldn’t tell you that it’s absolutely gorgeous, sunny and 75 degrees in Paris right now.

KK: You really shouldn’t.

EL: OK, then I won’t. OK. Each episode we invite a mathematician on to find out about their favorite theorem. Today we’re very happy to welcome Eriko Hironaka onto the show. So would you like to tell us a little bit about yourself, Eriko?

EH: Yes, thank you, first of all, for having me on this show. It’s very flattering and exciting. I worked at Florida State University for almost twenty years. I was a professor there, and I recently moved to the American Mathematical Society. I’ve been working there for two years. One year, I guess, full time, so far. I work in the book program. I’m somebody who is a mathematician but is doing it from various angles.

EL: Yeah, I was really interested in having you on the podcast because I think that’s a cool perspective to have where you’ve been in the research world for a long time, but now you’re also seeing a broader view, maybe, of math, or kind of looking at it from a different angle than before. Do you mind telling us a little bit about what you do as a book person for the AMS?

EH: Yeah, what do I do? Actually I was thrown into this job, in a way. They said, OK, you’re going to work in the book program. Your job is basically to talk to people about books, and see if anybody wants to write a book, and if they do, you keep talking with them, and when they finally submit something, you prepare, maybe the real job part is to, once there’s a submission, start it through a review process, then also what’s kind of exciting is to convince the publishing group to actually publish the book. That part requires me to think about how this book fits into mathematics and mathematical literature and then also how much it’ll cost to produce the book and what’s involved in selling the book. Who is the audience and how can it be presented in the best possible way? I think of myself as sort of the connector between the author, who is thinking about the mathematics, and the publishers, who are thinking about. The AMS is a nonprofit, but to cover costs and make this a reasonable project.

EL: You see a lot of different aspects of this, then.

EH: Yeah, so I don’t know if I was more naive than most mathematicians, but I think most mathematicians don’t think beyond proving theorems and conveying and communicating their ideas to other people. Maybe they also think about what to write on their vita, and things like that. That kind of thing is very different. Right now I don’t really have a reason to keep up my vita in the same way that I used to. That was a big change for me.

KK: Right.

EH: I still do mathematics, I still do research, give talks, and things like that. I still write papers. But that’s really become something just for me. Not for me, it’s for math, I guess. But it’s not for an institute.

KK: It’s not for the dean.

EH: It’s not for the dean. Exactly.

KK: That’s really liberating, I would think, right?

EH: It’s super liberating, actually. It’s really great.

KK: Very cool. I dream about that. One of these days.

EH: I feel like I’m supporting mathematics kind of from the background. Now I think about professors as being on the battlefield. They’re directly communicating with people, with students, with individuals. And working with the deans. Making their curriculum and program and everything work.

EL: So what have you chosen as your favorite theorem?

EH: OK, Well, I thought about that question. It’s very interesting. I’ve even asked other people, just to get their reaction. It’s a very interesting question, and I’m curious to know what other people have answered on your podcast. When I think of a theorem I think about not just the statement, but more the proof. I might think of proofs I like, theorems whose proofs I like, or I might think about how this theorem helped me because I really needed something. It’s actually kind of utilitarian, but a favorite theorem should be more like what made you feel great. I have to say for that, it’s a theorem of my own.

KK: Cool, great.

EH: So I have a favorite theorem, or the theorem that made me so excited, and it was the first theorem I ever proved. The reason it’s my favorite theorem is because of a mixture not just of feeling great that I’d proved the theorem but also feeling like it was a big turning point in my life. I felt like I had proved myself in a way. That’s this theorem, I think of it as a polynomial periodicity theorem, and what it says. Do you want me to say what the theorem is?

KK: Yeah, throw it out there, and we’ll unpack it.

EH: So the theorem in most generality, it says that if you have a finite CW complex, if you have a sort of nice space, in my case I was looking at quasiprojective varieties, but any kind of reasonably nice space, you can take a sequence of coverings, of regular coverings, corresponding to a choice of map from the fundamental group of the space to some, say, free abelian group, and the way you get the sequence of coverings is you take that map and compose it with the map from that free abelian group to the free abelian group tensored with Z mod n. So if everything is finitely generated, that gives you a surjective map from the fundamental group of your space to a finite abelian group. And now by the general theorem of covering spaces gives you a sequence of finite coverings of your space. And then if you have that space having a natural completion you can talk about natural branch coverings associated to those for each n. My theorem was what happens to the first Betti numbers of these things? The rank of the first homology of these coverings. I showed that this sequence actually has a pattern. In fact, there is a polynomial for every set base space and map from the fundamental group of the base space to a free abelian group, there is a polynomial with possibly periodically changing coefficients so that the first Betti number is that polynomial evaluated at n.

KK: Wow.

EH: So n is the degree of the covering. The Betti numbers are changing periodically, the polynomials are changing periodically, but it’s a pattern, it’s a nice pattern, and there’s a single polynomial telling you what all of these Betti numbers are doing.

EL: So what was the motivation behind this theorem?

EH: This problem of understanding the first Betti number of coverings comes from work of Zariski back in the early 1900s. His goal was to understand moduli of plane curves with various kinds of singularities. Simply put, what he did was he tried to distinguish curves by looking at topology, blending topology with algebraic geometry. This was kind of a new idea. This is not very well known about Zariski, but one of his innovations was bringing in topology to the study of algebraic geometry.

KK: That’s why it’s called the Zariski topology, right? I don’t know. One assumes.

EH: In a way. Not really!

KK: I’m not a historian. My bad.

EH: He brought geometry topology in. The Zariski topology is more of an algebraic definition. What he did was he was interested, for example, he showed that, what he was interested in when you’re talking about moduli of plane curves is whether or not you can get from one plane curve with prescribed singularities, say a sextic curve, a degree six curve, in C2, in the complex plane, with exactly six simple cusps. So six points in the plane can either lie on a conic or not. General position means it doesn’t lie on lines, doesn’t lie on a conic. But if the six points lie on a conic, it turns out you cannot move within the space of sextics with six cusps to a sextic with six cusps not lying on a conic.

KK: OK

EH: They’re two distinct families. You’d have to leave that family to get from one to another. You can’t deform them in the algebraic category. To prove this, he said, well, basically, even though the idea of fundamental groups and studying fundamental groups was really new still and was just starting to be considered a tool for knot theory, for example, that came a little bit later. But he said, you can tell they’re different because their topology is different. For example, take coverings. Take your curve, and say it’s given by the equation F(x,y)=0. So F(x,y) is a polynomial. Take the polynomial z^n=F(x,y). You get a surface in three-dimensional space, and now look at the first Betti number of that. So the first Betti number, the first homology, can kind of be described algebraically in terms of other things, divisors and things like that. You can think of it as a very algebraic invariant, but you can also think of it as a topological invariant. Forget algebra, forget complex analysis, forget everything. And he showed that if you take the sextics with six cusps and you took z^n=F(x,y), you get things with first Betti number nontrivial, and by the way, periodically changing with n. In fact, when 6 divides n, it’s nontrivial. It jumps. Every time 6 divides n, it jumps. Otherwise I can’t remember, I think it’s zero. But in the case that the cusps are in general position, the first Betti numbers are always zero.

KK: OK.

EH: So that must mean that the topology is different. And if the topology is different, they can’t be algebraically equivalent. So that was the process of thinking, that topology can tell you something about algebraic geometry. And that kind of topology is what geometric topologists study now, fundamental groups, etc. But this was all a very new idea.

EL: So that’s kind of the environment that your theorem lives in, this intersection between topology and algebraic geometry.

EH: That’s right. So my theorem, Sarnak conjectured, I was working on fundamental groups of complements of plane curves, especially with multiple components, for my thesis. And Peter Sarnak was looking at certain problems coming from number theory that had to do with arithmetic subgroups of GL(n) and looking at what happens when you take your fields to be finite, and things like that. You get these finite fields. Somehow in his work, something coming from number theory, he wondered, hearing about what I was doing with fundamental groups and Alexander polynomials, which have to do with Betti numbers of coverings, he asked, “Can you show that the Betti numbers of coverings are periodic or polynomial periodic,” which is that other thing. I thought, OK, I’ll do this, and since I was already working topologically, I could get the topological part by looking at the unbranched coverings, and then I had to complete it. To understand the completion, the rest, there’s a difference between the Betti numbers of the unbranched coverings and the Betti numbers of the branched coverings, to understand that, I needed to understand intersections of curves on the surface, to sort of understand intersection theory of algebraic curves. And these have very special properties, nice properties coming from the fact that we’re talking about varieties.

KK: Right.

EH: And I used that to complete the proof. It was a real blend of topology and algebraic geometry. That’s what made it really fun.

KK: That’s a lot of mathematics going in. And I love your confidence. Peter Sarnak said, “Hey, can you do this?” and you said, “Yeah, I can do this.”

EH: Right, well I was feeling pretty desperate. It was really a time of: Should I do math? Should I not do math? Do I belong here? And I thought, “OK, I’ll try this. If it works, maybe that’s a sign.”

EL: So what have you chosen to pair with this theorem?

EH: As it happens, after I proved this theorem and I showed it to Sarnak, I basically wrote a three-page outline of the proof. I showed it to him, and he looked at it carefully and said, “Yeah, this looks right.” Also, you know, you can feel it when you have it. Suddenly everything has become so clear. I was glowing with this and driving from Stanford to Berkeley, which is about an hour drive, and I usually took a nicer route through the hills to the west, so you can imagine driving with these vales and woods, and it was beautiful sunshine and everything, and the Firebird suite starts out very quiet, and it just perfectly represented what it feels like to prove a theorem. It starts really quiet, and then it gets really choppy and frenzied.

KK: And scary.

EH: Exactly, scary. The struggling bird, he’s anxious and frightened, really, really unsettling. And then there’s this real gentleness, feeling like it’s going to be OK, it’s going to be OK. But that also is a bit disturbing. There’s something about it that’s disturbing. So it keeps you listening, even though it’s very sweet and the themes are developed, it’s a very beautiful theme. Then there’s this bang and then it becomes really frenzied again, super frenzied, but excited. And then it becomes bolder and bolder. And then that melody comes in, and it starts to really come together. And it starts to feel like you’re running, like there’s a direction, and then finally it gets quiet again. There’s this serenity. And this time the serenity is real. All this stuff has built up to it, and that starts to build and the beautiful theme comes out in the end. It’s just this glorious wonder at the very end. It was like all my excitement was just exemplified in this piece of music.

EL: I love that picture of you driving through California, blasting Firebird.

EH: Yes, exactly.

EL: With this triumphant proof that you’d just done. That’s really a great picture.

KK: So my son just finished high school, and he wants to be a composer. He’s going to go to college and study composition. And I actually sort of credit that piece, Firebird suite, as one of the pieces that really motivated him to become a composer. That and Rhapsody in Blue.

EH: It really tells a story.

KK: Yes, it does. It’s really spectacular. So I think maybe a lot of our listeners don’t know that you have a rather famous father.

EH: Yes.

KK: Your father won the Fields medal for proving resolution of singularities in characteristic zero, right?

EH: Yes.

KK: What was that like?

EH: Yeah, so I had a really strange relationship with mathematics. Because I grew up with a mathematician father, I avoided math like the plague, you know. Partly because my father was a mathematician, and I thought that was kind of strange, that it didn’t fit in with the rest of the world that I knew. I grew up in the suburbs. It wasn’t a particularly intellectual background. For me, the challenge to my life was to figure out how to fit in, which I was failing at miserably. But I thought that was my challenge. Doing well in math was not the way to fit in in school. I would kind of deliberately add in mistakes to make sure that I didn’t get a good grade.

KK: Really? Wow.

EH: I would kick myself if I forgot and I would get a high grade and everybody would say, “How did she do that?” You know what I mean? I thought of math as this embarrassment, in a way, to tell the truth, strangely enough. But on the other hand, through my father and his friends and colleagues, I knew that mathematics also had this very beautiful side, and the people who did it were very happy people, it seemed. I saw that other side as well. And I think that was an advantage because I knew that math was really cool. It’s just that that wasn’t my thing. I didn’t want to do that. Also, my teachers were not very exciting. The math teachers seemed to make math as boring as possible. So I had this kind of split personality when it came to math, or split feeling about what math was.

EL: Yeah.

EH: But then when I started to do math, I started somehow accidentally to do math in college, and I actually got much more attracted to it. It was after vaguely stumbling through calculus and things like that. So I never really learned calculus, but I started skipping through calculus, and I took more advanced classes, and it just really clicked, and I got hooked. I learned calculus in graduate school, as some people do, by teaching it.

KK: Well that’s when you really learn it anyway, that’s right.

EH: Some people have this impression that lots of mathematicians had the advantage of having access to mathematics from a young age, but I think it’s not obvious how that’s an advantage. In some cases it could be that they were nurtured in mathematics. I mean, I talk to my kids about mathematics, and it’s a fun thing we do together. But I don’t think that’s necessarily the case of people with mathematical parents. In my case it certainly wasn’t the case for me. But still it was an advantage because I knew that there was this thing called mathematics, and many people don’t know that.

EL: Yeah. And, like you said, you knew that mathematicians were happy with their work, and just even knowing that there’s still math to prove. That was something, when I started to do math, I didn’t really understand that there was still more math to do, it wasn’t just learning calculus really well. But going and finding and exploring these new things.

KK: I had that same experience. I remember when I was in high school, telling people I was going to go to graduate school and be a math professor, and they said, “Well, what do you do?” I said, “I don’t know, I guess you write another calculus book.” Which we certainly do not need, right?

EH: Or we need different kinds.

KK: So, I say that, but I’m actually writing one, so you know.

EH: Oh, are you?

KK: Just in my spare time, right? I have so much of it these days.

EH: I think there is a need for calculus books, it’s just maybe different kinds.

KK: Well now that I know someone at the publishing house at the AMS…

EH: Absolutely. I’m going to follow up on this.

KK: Oh, wow. Well this has been fun.

EL: Yeah, thank you so much.

EH: Well thank you for asking me. It gave me a chance to think about different things, and it’s been fun talking to people about, “What’s you’re favorite theorem?”

EL: Good math conversation starter.

EH: Yeah, absolutely.

KK: Thanks for joining us, Eko.

EH: Thank you.

KK: Thanks for listening to My Favorite Theorem, hosted by Kevin Knudson and Evelyn Lamb. The music you’re hearing is a piece called Fractalia, a percussion quartet performed by four high school students from Gainesville, Florida. They are Blake Crawford, Gus Knudson, Del Mitchell, and Bao-xian Lin. You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at Kevin’s website, kpknudson.com, and Evelyn’s blog, Roots of Unity, on the Scientific American blog network. We love to hear from our listeners, so please drop us a line at myfavoritetheorem@gmail.com. Or you can find us on Facebook and Twitter. Kevin’s handle on Twitter is @niveknosdunk, and Evelyn’s is @evelynjlamb. The show itself also has a Twitter feed. The handle is @myfavethm. Join us next time to learn another fascinating piece of mathematics.

Show notes (click here)

Episode 5 - Dusa McDuff

This transcript is provided as a courtesy and may contain errors.

Evelyn Lamb: Hello and welcome to My Favorite Theorem. I’m your host Evelyn Lamb. I’m a freelance math and science writer based in Salt Lake City, but I’m currently recording in Chicago at the Mathematical Association of America’s annual summer meeting MathFest. Because I am on location here, I am not joined by our cohost Kevin Knudson, but I’m very honored to be in the same room as today’s guest, Dusa McDuff. I’m very grateful she took the time to talk with me today because she’s pretty busy at this meeting. She’s been giving the Hendrick Lecture Series and been organizing some research talk sessions. So I’m very grateful that she can be here. The introductions at these talks have been very long and full of honors and accomplishments, and I’m not going to try to go through all that, but maybe you can just tell us a little bit about yourself.

Dusa McDuff: OK. Well, I’m British, originally. I was born in London and grew up in Edinburgh, where I spent the first twenty years or so of my life. I was an undergraduate at Edinburgh and went to graduate study at Cambridge, where I was working in some very specialized area, but I happened to go to Moscow in my third year of graduate study and studied with a brilliant mathematician called Gelfand [spelling], who opened my eyes to lots of interesting mathematics, and when I came back, he advised that I become a topologist, so I tried to become a topologist. So that’s more what I’ve been doing recently, gradually moving my area of study. And now I study something called symplectic topology, or symplectic geometry, which is the study of space with a particular structure on it which comes out of physics called a symplectic structure.

EL: OK. And what is your favorite theorem?

DM: My favorite theorem at the moment has got to do with symplectic geometry, and it’s called the nonsqueezing theorem. This is a theorem that was discovered in the mid-80s by a brilliant mathematician called Gromov who was trying to understand. A symplectic structure is a strange structure you can put on space that really groups coordinates in pairs. You take two coordinates (x1,y1) and another two coordinates (x2,y2), and you measure an area with respect to the first pair, an area with respect to the second pair, and add them. You get this very strange measurement in four-dimensional space, and the question is what are you actually measuring? The way to understand that is to try to see it visually. He tried to explore it visually by saying, “Well, let’s take a round ball in four-dimensional space. Let’s move it so we preserve this strange structure, and see what we end up with.” Can we end up with arbitrary curly shapes? What happens? One thing you do know is that you have to preserve volume, but apart from that, nothing else was known.

So his nonsqueezing theorem says that if you took a round ball, say the radii were 1 in every direction, it’s not possible to move it so that in two directions the radii are less than 1 and in the other directions it’s arbitrary, as big as you want. The two directions where you’re trying to squeeze are these paired directions. It’s saying you can’t move it in such a way.

I’ve always liked this theorem. For one thing, it’s very important. It characterizes the structure in a way that’s very surprising. And for another thing, it’s so concrete. It’s just about shapes in four dimensions. Now four dimensions is not so easy to understand.

EL: No, not for me, at least!

DM: Thinking in four dimensions is tricky, and I’ve spent many, many years trying to understand how you might think about moving things in four dimensions, because you can’t do that.

EL: And to back up a little bit, when you say a round ball, are you talking about a two-dimensional ball that’s embedded in four-dimensional space, or a four-dimensional ball?

DM: I’m talking about a four-dimensional ball.

EL: OK.

DM: It’s got radius 1 in all directions. You’ve got a center point and move in distance 1 in every direction, that gives you a four-dimensional shape, it’s boundary is a three-dimensional sphere, in fact.

EL: Right, OK.

DM: Then you’re trying to move that, preserving this rather strange structure, and trying to see what happens.

EL: Yeah, so this is saying that the round ball is very rigid in some way.

DM: It’s very round and rigid, and you can’t squeeze it in these two related directions.

EL: At least to preserve the symplectic structure. Of course, you can do this and preserve the volume.

DM: Exactly.

EL: This is saying that symplectic structures are

DM: Different, intrinsically different, in a very direct way.

EL: I remember one of the pictures in your talk kind of shows this symplectic idea, where you’re basically projecting some four-dimensional thing onto two different two-dimensional axes. It does seem like a very strange way to get a volume on something.

DM: It’s a strange measurement. Why you have that, why are you interested in two directions? It’s because they’re related. This structure came from physics, elementary physics. You’re looking at the movement, say, of particles, or the earth around the sun. Each particle has got a position coordinate and a velocity coordinate. It’s a pairing of position and velocity for each degree of freedom that gives this measurement.

EL: And somehow this is a very sensible thing to do, I guess.

DM: It’s a very sensible thing to do, and people have used the idea that the symplectic form is fundamental in order to calculate trajectories, say, of rockets flying off. You want to send a probe to Mars, you want to calculate what happens. You want to have accurate numerical approximations. If you make your numerical approximations preserve the underlying symplectic structure, they just do much better than if you just take other approximation methods.

EL: OK.

DM: That was another talk, that was a fascinating talk at this year’s MathFest telling us about this, showing even if you’re trying to approximate something simple like a pendulum, standard methods don’t do it very well. If you use these other methods, they do it much better.

EL: Oh wow, that’s really interesting. So when did you first learn about the nonsqueezing theorem?

DM: Well I learned about it essentially when it was discovered in the mid-1980s.

EL: OK.

DM: I happened to be thinking about some other problem, but I needed to move these balls around preserving the symplectic structure. I just realized there was this question and I couldn’t necessarily do this when Gromov showed that one really could not do this, that there’s a strict limit. So I’ve always been interested in questions, many other questions coming from that.

EL: Another part of this podcast is that we like to ask our guests to pair their theorem with another delight in life, a food, beverage, piece of art or music, so what have you chosen to pair with the nonsqueezing theorem?

DM: Well you asked me this, and I decided I’d pair it with an avocado because I like avocados, and they have a sort of round, pretty spherical big seed in the middle. The seed is sort of inside the avocado, which surrounds it.

EL: OK. I like that. And the seed can’t be squeezed. The avocado’s seed cannot be squeezed. Is there anything else you’d like to say about the nonsqueezing theorem?

DM: Only that it’s an amazing theorem, that it really does underlie the whole of symplectic geometry. It’s led to many, many interesting questions. It seems to be a simple-minded thing, but it means that you can define what it means to preserve a symplectic structure without using derivatives, which means you can try and understand much more general kinds of motions, which are not differentiable but which preserve the symplectic structure. That’s a very little-understood area that people are trying to explore. What’s the difference between having a derivative and not having a derivative? It’s a sort of geometric thing. You actually see surprising differences. That’s amazing to me.

EL: Yeah. That’s a really interesting aspect to this that I hadn’t thought about. In the talk that you gave today was that the ball can’t be squeezed but the ellipsoids can. It’s this really interesting difference, also, between the ellipsoids and the ball.

DM: Right. So you have to think that somehow an ellipsoid, which is like a ball, but one direction is stretched, it’s got certain planes, there are certain discrete things you can do. You can slice it and then fold it along that slice. It’s a discrete operation somehow. That gives these amazing results about bending these ellipsoids.

EL: That’s another fascinating aspect to it. You I’m sure don’t remember this, but we actually met nine years ago when I was at the Institute for Advanced Study’s summer program for women in math. I’m pretty sure you don’t remember because I was too shy to actually introduce myself, but I remember you gave a series of lectures there about symplectic geometry. I studied Teichmüller theory, something pretty far away from that, and so I didn’t know if I was going to be interested in those. I remember that you really got me very interested in doing that many years ago. I was really excited when I saw that you were here and I’d be able to not be quite so shy this year and actually get to talk to you.

DM: That’s the thing, overcoming shyness. I used to be very shy and didn’t talk to people at all. But now I’m too old, I’ve given it all up.

EL: Well thank you very much for being on this podcast, and I hope you have a good rest of MathFest.

DM: Thank you.

Show notes (click here)

Episode 4 - Jordan Ellenberg

This transcript is provided as a courtesy and may contain errors.

Kevin Knudson: Welcome to My Favorite Theorem. I’m Kevin Knudson, a mathematician at the University of Florida. I’m joined by my other cohost.

Evelyn Lamb: Hi. I’m Evelyn Lamb. I’m a freelance writer currently based in Paris.

KK: Currently based in Paris. For how much longer?

EL: Three weeks. We’re down to the final countdown here. And luckily our bank just closed our account without telling us, so that’s been a fun adventure.

KK: Well, who needs money, right?

EL: Exactly.

KK: You’ve got pastries and coffee, right? So in this episode we are pleased to welcome Jordan Ellenberg, professor of mathematics at the University of Wisconsin. Jordan, want to tell everyone about yourself?

Jordan Ellenberg: Hi. Yes, this is Jordan Ellenberg. I’m talking to you from Madison, Wisconsin today, where we are enjoying the somewhat chilly, drizzly weather we call spring.

KK: Nice. I’ve been to Madison. It’s a lovely place. It’ll be spring for real in a little while, right?

JE: It’ll be lovely. It’s going to be warm this afternoon, and I’m going to be down at the Little League field watching my son play, and it’s as nice as can be.

KK: What position does he play?

JE: He’s 11, so they mix it up. They don’t have defined positions.

KK: I have an 11-year-old nephew who’s a lefty, and they want him to pitch all the time. He’s actually pretty good.

JE: It’s same thing as asking a first-year graduate student what their field is. They should move around a little bit.

KK: That’s absolutely true.

JE: 11 is to baseball as the first year of grad school is to math, I think. Roughly.

KK: That’s about right. Well now they start them so young. We’re getting off track. Never mind. So we’re here to talk about math, not baseball, even though there’s a pretty good overlap there. So Jordan, you’re going to surprise us. We don’t actually know what your favorite theorem is. So why don’t you lay it on us. What’s your favorite theorem?

JE: It is hard to pick your favorite theorem. I think it’s like trying to pick your favorite kind of cheese, though I think in Wisconsin you’re almost required to have one. I’m going to go with Fermat’s Little Theorem.

KK: OK.

EL: This is a good theorem. Can you tell us what that is?

JE: I’m not even going to talk about the whole theorem. I’m going to talk about one special case, which I find very beautiful, which is that if you take a prime number, p, and raise 2 to that power, and then you divide by p, then the remainder is 2. In compact terms, you would say 2 to the p is congruent to 2 mod p. Shall we do a couple?

KK: Sure.

JE: For instance, 2^5 is 32. Computing the remainder when you divide by 5 is easy because you can just look at the last digit. 32 is 2 more than 30, which is a multiple of 5. This persists, and you can do it. Should we do one more? Let’s try. 2 to the 7th is 128, and 126 is a multiple of 7, so 128 is 2 mod 7.

KK: Your multiplication tables are excellent.

JE: Thank you.

KK: I guess being a number theorist, this is right up your alley. Is this why you chose it? How far back does this theorem go?

JE: Well, it goes back to Fermat, which is a long time ago. It goes back very early in number theory. It also goes back for me very early in my own life, which is why I have a special feeling for it. One thing I like about it is that there are some theorems in number theory where you’re not going to figure out how to prove this theorem by yourself, or even observe it by yourself. The way to get to the theorem, and this is true for many theorems in number theory, which is a very old, a very deep subject, is you’re going to study and you’re going to marvel at the ingenuity of whoever could have come up with it. Fermat’s Little Theorem is not like that. I think Fermat’s Little Theorem is something that you can, and many people do, and I did, discover at least that it’s true on your own, for instance by messing with Pascal’s Triangle, for example. It’s something you can kind of discover. At least for me, that was a very formative experience, to be like, I learned about Pascal’s triangle, I was probably a teenager or something. I was messing around and sort of observed this pattern and then was able to prove that 2 to the p was congruent to 2 mod p, and I thought this was great. I sort of told a teacher who knew much more than me, and he said, yeah, that’s Fermat’s Little Theorem.

I was like, “little theorem?” No, this was a lot of work! It took me a couple days to work this out. I felt a little bit diminished. But to give some context, it’s called that because of course there’s the famous Fermat’s Last Theorem, poorly named because he didn’t prove it, so it wasn’t really his theorem. Now I think nowadays we call this theorem, which you could argue is substantially more foundational and important, we call it the little theorem by contrast with the last theorem.

EL: Going back to Pascal’s triangle, I’m not really aware of the connection between Fermat’s Little Theorem and Pascal’s triangle. This is an audio medium. It might be a little hard to go through, but can you maybe explain a little bit about how those are connected?

JE: Sure, and I’m going to gesticulate wildly with my hands to make the shape.

EL: Perfect.

JE: You can imagine a triangle man dance sort of thing with my hands as I do this. So there’s all kinds of crazy stuff you can do with Pascal’s triangle, and of course one thing you can do, which is sort of fundamental to what Pascal’s triangle is, is that you can add up the rows. When you add up the rows, you get powers of two.

EL: Right.

JE: So for instance, the third row of Pascal’s triangle is 1-3-3-1, and if you add those up, you get 8, which is a power of 2, it’s 2^3. The fifth row of Pascal’s triangle is 1-5-10-10-5-1. I don’t know, actually. Every number theorist can sort of rattle off the first few rows of Pascal’s triangle. Is that true of topologists too, or is that sort of a number theory thing? I don’t even know.

KK: I’m pretty good.

JE: I don’t want to put you on the spot.

EL: No, I mean, I could if I wrote them down, but they aren’t at the tip of my brain that way.

JE: We use those binomial coefficients a lot, so they’re just like right there. Anyway, 1-5-10-10-5-1. If you add those up, you’ll get 32, which is 2^5. OK, great. Actually looking at it in terms of Pascal’s triangle, why is it the case that you get something congruent to 2 mod 5? And you notice that actually most of those summands, 1-5-10-10-5-1, I’m going to say it a few times like a mantra, most of those summands are multiples of 5, right? If you’re like, what is this number mod 5, the 5 doesn’t matter, the 10 doesn’t matter, the 10 doesn’t matter, the 5 doesn’t matter. All that matters is the 1 at the beginning and the 1 at the end. In some sense Fermat’s Little Theorem is an even littler theorem, it’s the theorem that 1+1=2. That’s the 2. You’ve got the 1 on the far left and the 1 on the far right, and when the far left and the far right come together, you either get the 2016 US Presidential election, or you get 2.

KK: And the reason they add up to powers of 2, I guess, is because you’re just counting the number of subsets, right? The number of ways of choosing k things out of n things, and that’s basically the order of the power set, right?

JE: Exactly. It’s one of those things that’s overdetermined. Pascal’s triangle is a place where so many strands of mathematics meet. For the combinatorists in the room, we can sort of say it in terms of subsets of a set. This is equivalent, but I like to think of it as this is the vertices of a cube, except by cube maybe I mean hypercube or some high-dimensional thing. Here’s the way I like to think about how this works for the case p=3, right, 1-3-3-1. I like to think of those 8 things as the 8 vertices of a cube. Is everybody imagining their cube right now? We’re going to do this in audio. OK. Now this cube that you’re imagining, you’re going to grab it by two opposite corners, and kind of hold it up and look at it. And you’ll notice that there’s one corner in one finger, there’s one corner on your opposite finger, and then the other six vertices that remain are sort of in 2 groups of 3. If you sort of move from one finger to the other and go from left to right and look at how many vertices you have, there’s your Pascal’s triangle, right? There’s your 1-3-3-1.

One very lovely way to prove Fermat’s Little Theorem is to imagine spinning that cube. You’ve got it held with the opposite corners in both fingers. What you can see is that you can sort of spin that cube 1/3 of a rotation and that’s going to group your vertices into groups of 3, except for the ones that are fixed. This is my topologist way. It’s sort of a fixed point theorem. You sort of rotate the sphere, and it’s going to have two fixed points.

EL: Right. That’s a neat connection there. I had never seen Pascal’s triangle coming into Fermat’s little theorem here.

JE: And if you held up a five-dimensional cube with your five-dimensional fingers and held opposite corners of it, you would indeed see as you sort of when along from the corner a group of 5, and then a group of 10, and then a group of 10, and then a group of 5, and then the last one, which you’re holding in your opposite finger.

EL: Right.

JE: And you could spin, you could spin the same way, a fifth of a rotation around. Of course the real truth, as you guys know, as we talk about, you imagine a five-dimensional cube, I think everyone just imagines a 3-dimensional cube.

KK: Right. We think of some projection, right?

JE: Exactly.

KK: Right. So you figured out a proof on your own in the case of p=2?

JE: My memory is that I don’t think I knew the slick cube-spinning proof. I think I was thinking of the Pascal’s triangle. This thing I said, I didn’t prove, as we were just discussing, I mean, you can look at any individual row and see that all those interior numbers in the triangle are divisible by 5. But that’s something that you can prove if you know that the elements of Pascal’s triangle are the binomial coefficients, the formula is n!/k!(n-k)!. It’s not so hard to prove in that case that if n is prime, then those binomial coefficients are all divisible by p, except for the first and last. So that was probably how I proved it. That would be my guess.

KK: Just by observation, I guess. Cool.

EL: We like to enjoy the great things in life together. So along with theorems, we like to ask our guests to pair something with this theorem that they think complements the theorem particularly well. It could be a wine or beer, favorite flavor of chocolate…

JE: Since you invited somebody in Wisconsin to do this show, you know that I’m going to tell you what cheese goes with this theorem.

EL: Yes, please.

KK: Yes, absolutely. Which one?

JE: The cheese I’ve chosen to pair with this, and I may pronounce it poorly, is a cheese called gjetost.

EL: Gjetost.

JE: Which is a Norwegian cheese. I don’t know if you’ve had it. It almost doesn’t look like cheese. If you saw it, you wouldn’t quite know what it was because it’s a rather dark toasty brown. You might think it was a piece of taffy or something like that.

EL: Yeah, yeah. It looks like caramel.

JE: Yes, it’s caramel colored. It’s very sweet. I chose it because a, because like Fermat’s Little Theorem, I just really like it, and I’ve liked it for a long time; b, because it usually comes in the shape of a cube, and so it sort of goes with my imagined proof. You could, if you wanted to, label the vertices of your cheese with the subsets of a 3-element set and use the gjetost to actually illustrate a proof of Fermat’s Little Theorem in the case p=3. And third, of course, the cheese is Norwegian, and so it honors Niels Henrik Abel, who was a great Norwegian mathematician, and Fermat’s Little Theorem is in some sense the very beginning of what we would now call Abelian group theory. Fermat certainly didn’t have those words. It would be hundreds of years before the general apparatus was developed, but it was one of the earliest theorems proved about Abelian groups, and so in that sense I think it goes with a nice, sweet Norwegian cheese.

EL: Wow, you really thought this pairing through. I’m impressed.

JE: For about 45 seconds before we talked.

EL: I’ve actually made this cheese, or at least some approximation of this. I think it’s made with whey, rather than milk.

JE: On purpose? What happened?

EL: Yeah, yeah. I had some whey left over from making paneer, and so I looked up a recipe for this cheese, and I had never tried the real version of it. After I made my version, then, I went to the store and got the real one. My version stood up OK to it. It didn’t taste exactly the same, but it wasn’t too bad.

JE: Wow!

KK: Experiments in cheesemaking.

JE: In twelve years, I’ve never made my own cheese. I just buy it from the local dairy farmers.

EL: Well it was kind of a pain, honestly. It stuck to everything. Yeah.

JE: Someone who lives in Paris should not be reduced to making their own cheese, by the way. I feel like that’s wrong.

EL: Yes.

KK: I’m not surprised you came up with such a good pairing, Jordan. You’ve written a novel, right, years ago, and so you’re actually a pretty creative type. You want to plug your famous popular math book? We like to let people plug stuff.

JE: Yes. My book, which came out here a few years ago, it’s called How Not to Be Wrong. It’ll be out in Paris in two weeks in French. I just got to look at the French cover, which is beautiful. In French it’s called, I’m not going to be able to pronounce it well, like “L’art de ne dire n’importe pas”, [L’art de ne pas dire n’importe quoi] which is “The art of not saying whatever nonsense,” or something like this. It’s actually hard work to translate the phrase “How not to be wrong” in French. I was told that any literal translation of it sounds appallingly bad in French.

This book is kind of a big compendium of all kinds of things I had to say with a math angle. Some of it is about pure math, and insights I think regular people can glean from things that pure mathematicians think about, and some are more on the “statistical news you can use” side. It’s a big melange of stuff.

KK: I’ve read it.

JE: I’m a bit surprised people like it and have purchased it. I guess the publishing house knew that because they wouldn’t have published it, but I didn’t know that. I’m surprised people wanted it.

KK: I own it in hardback. I’ll say it. It’s really well done. How many languages is it into now?

JE: They come out pretty slowly. I think we’ve sold 14 or 15. I think the number that are physically out is maybe []. I think I made the book hard to translate by having a lot of baseball material and references to US cultural figures and stuff like that. I got a lot of really good questions from the Hungarian translator. That one’s not out, or that one is out, but I don’t have a copy of it. It just came out.

KK: Very cool.

JE: The Brazilian edition is very, very rich in translator’s notes about what the baseball words mean. They really went the extra mile to be like, what the hell is this guy talking about?

KK: Is it out in Klingon yet?

JE: No, I think that will have to be a volunteer translator because I think the commercial market for Klingon popular math books is not there. I’m holding out for Esperanto. If you want my sentimental favorite, that’s what I would really like. I tried to learn Esperanto when I was kid. I took a correspondence course, and I have a lifelong fascination for it. But I don’t think they publish very many books in Esperanto. There was a math journal in Esperanto.

EL: Oh wow.

KK: That’s right, that’s right. I sort of remember that.

JE: That was in Poland. I think Poland is one of the places where Esperanto had the biggest popularity. I think the guy who founded it, Zamenhof, was Polish.

KK: Cool. This has been fun. Thanks, Jordan.

JE: Thank you guys.

EL: Thanks a lot for being here.

KK: Thanks a lot.

KK: Thanks for listening to My Favorite Theorem, hosted by Kevin Knudson and Evelyn Lamb. The music you’re hearing is a piece called Fractalia, a percussion quartet performed by four high school students from Gainesville, Florida. They are Blake Crawford, Gus Knudson, Dell Mitchell, and Baochau Nguyen. You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at Kevin’s website, kpknudson.com, and Evelyn’s blog, Roots of Unity, on the Scientific American blog network. We love to hear from our listeners, so please drop us a line at myfavoritetheorem@gmail.com. Or you can find us on Facebook and Twitter. Kevin’s handle on Twitter is @niveknosdunk, and Evelyn’s is @evelynjlamb. The show itself also has a Twitter feed. The handle is @myfavethm. Join us next time to learn another fascinating piece of mathematics.

Show notes (click here)

Episode 3 - Emille Davie Lawrence

This transcript is provided as a courtesy and may contain errors.

EL: Welcome to My Favorite Theorem. I’m one of your hosts, Evelyn Lamb. I’m a freelance math and science writer currently based in Paris. And this is my cohost.

KK: Hi, I’m Kevin Knudson, professor of mathematics at the University of very, very hot Florida.

EL: Yeah. Not so bad in Paris yet.

KK: It’s going to be a 96-er tomorrow.

EL: Wow. So each episode, we invite a mathematician to come on and tell us about their favorite theorem. Today we’re delighted to welcome Emille Davie Lawrence to the show. Hi, Emille.

EDL: Hello, Evelyn.

EL: So can you tell us a little bit about yourself?

EDL: Sure! So I am a term assistant professor at the University of San Francisco. I’m in the mathematics and physics department. I’ve been here since 2011, so I guess that’s six years now. I love the city of San Francisco. I have two children, ages two and almost four.

EL: Who are adorable, if your Facebook is anything to go by.

EDL: Thank you so much. You’ll get no arguments from me. I’ve been doing math for quite a while now. I’m a topologist, and my mathematical interests have always been in topology, but they’ve evolved within topology. I started doing braid groups, and right now, I’m thinking about spatial graphs a lot. So lots of low-dimensional topology ideas.

EL: Cool. So what is your favorite theorem?

EDL: My favorite theorem is the classification theorem for compact surfaces. It basically says that no matter how weird the surface you think you have on your hands, if it’s a compact surface, it’s only one of a few things. It’s either a sphere, or the connected sum of a bunch of tori, or the connected sum of a bunch of projective planes. That’s it.

EL: Can you tell us a little bit more about what projective planes are?

EDL: Obviously a sphere, well, I don’t know how obvious, but a sphere is like the surface of a ball, and a torus looks like the surface of a donut, and a projective plane is a little bit stranger. I think anyone who would be listening may have run into a Möbius band at some point. Basically you take a strip of paper and glue the two ends of your strip together with a half-twist. This is a Möbius band. It’s a non-orientable half-surface. I think sometimes kids do this. They pop up in different contexts. One way to describe a projective plane is to take a Möbius band and add a disc to the Möbius band. It gives you a compact surface without boundary because you’ve identified the boundary circle of the Möbius band to the boundary of the disc.

EL: Right, OK.

EDL: Now you’ve got this non-orientable thing called a projective plane. Another way to think about a projective plane is to take a disc and glue one half of the boundary to the other half of the boundary in opposite directions. It’s a really weird little surface.

KK: One of those things we can’t visualize in three dimensions, unfortunately.

EDL: Right, right. It’s actually hard to explain. I don’t think I’ve ever tried to explain it without drawing a picture.

EL: Right. That’s where the blackboard comes in hand.

KK: Limitations of audio.

EL: Have you ever actually tried to make a projective plane with paper or cloth or anything?

EDL: Huh! I am going to disappoint you there. I have not. The Möbius bands are easy to make. All you need is a piece of paper and one little strip of tape. But I haven’t. Have you, Evelyn?

EL: I’ve seen these at the Joint Meetings, I think somebody brought this one that they had made. And I haven’t really tried. I’d imagine if you tried with paper, it would probably just be a crumpled mess.

EDL: Right, yeah.

EL: This one I think was with fabric and a bunch of zippers and stuff. It seemed pretty cool. I’m blanking now on who is was.

KK: That sounds like something sarah-marie belcastro would do.

EL: It might have been. It might have been someone else. There are lots of cool people doing cool things with that. I should get one for myself.

EDL: Yeah, yeah. I can see cloth and zippers working out a lot better than a piece of paper.

EL: So back to the theorem. Do you know what makes you love this theorem?

EDL: Yeah. I think just the fact that it is a complete classification of all compact surfaces. It’s really beautiful. Surfaces can get weird, right? And no matter what you have on your hands, you know that it’s somewhere on this list. That makes a person like me who likes order very happy. I also like teaching about it in a topology class. I’ve only taught undergraduate topology a few times, but the last time was last spring, a year ago, spring of 2016, and the students seemed to really love it. You can play these “What surface am I?” games. Part of the proof of the theorem is that you can triangulate any surface and cut it open and lay it flat. So basically any surface has a polygonal representation where you’re just some polygon in the plane with edges identified in pairs. I like to have this game in my class where I just draw a polygon and identify some of the edges in pairs and say, “What surface is this?” And they kind of get into it. They know what the answers, what the possibilities are for the answers. You can sort of just triangulate it and find the Euler characteristic, see if you can find a Möbius band, and you’re off to the races.

KK: That’s great. I taught the graduate topology course here at Florida last year. I’m ashamed to admit I didn’t actually prove the classification.

EDL: You should not be ashamed to admit that. It’s something at an undergraduate level you get to at the end, depending on how you structure things. We did get to it at the end of the course, so I don’t know how rigorously I proved it for them. The combinatorial step that goes from: you can take this polygonal representation, and you can put it in this polygonal form, always, that takes a lot of work and time.

EL: There are delicacies in there that you don’t really know about until you try to teach it. I taught it also in class a couple years ago, and when I got there, I was like, “This seemed a little easier when I saw it as a student.” Now that I was trying to teach it, it seemed a little harder. Oh, there are all of these t’s I have to cross and i’s I have to dot.

KK: That’s always the way, right?

EDL: Right.

KK: I assigned as a homework assignment that my students should just compute the homology of these surfaces, and even puncture them. Genus g, r punctures, just as a homework exercise. From there you can sort of see that homology tells you that genus classifies things, at least up to homotopy invariants, but this combinatorial business is tricky.

EDL: It is.

EL: Was this a love at first sight kind of theorem, or is this a theorem that’s grown on you?

EDL: I have to say it’s grown on me. I probably saw it my first year of graduate school, and like all of topology, I didn’t love it at first when I saw it as a first-year graduate student. I did not see any topology as an undergrad. I went to a small, liberal arts college that didn’t have it. So yeah, I have matured in my appreciation for the classification theorem of surfaces. It’s definitely something I love now.

KK: You’re talking to a couple of topologists, so you don’t have to convince us very much.

EDL: Right.

KK: I had a professor as an undergrad who always said, “Topology is analysis done right.”

EDL: I like that.

KK: I know I just infuriated all the analysts who are listening. I always took that to heart. I always took that to heart because I always felt that way too. All those epsilons and deltas, who wants all that?

EDL: Who needs it?

KK: Draw me a picture.

EL: I was so surprised in the first, I guess advanced calculus class I had, a broader approach to calculus, and I learned that all these open sets and closed sets and things actually had to do with topology not necessarily with epsilons and deltas. That was really a revelation.

KK: So you’re interested in braids, too, or you were? You moved on?

EDL: I would say I’m still interested in braids, although that is not the focus of my research right now.

KK: Those are hard questions too, so much interesting combinatorics there.

EDL: That’s right. I think that’s sort of what made me like braid groups in the first place. I thought it was really neat that a group could have that geometric representation. Groups, I don’t know, when you learn about groups, I guess the symmetric group is one of the first groups that you learn about, but then it starts to wander off into abstract land. Braid groups really appealed to me, maybe just the fact that I liked learning visually.

EL: It’s not quite as in the clouds as some abstract algebra.

KK: And they’re tied up with surfaces, right, because braid groups are just the mapping class group of the punctured disc.

EDL: There you go.

KK: And Evelyn being the local Teichmüller theorist can tell us all about the mapping class groups on surfaces.

EL: Oh no! We’re getting way too far from the classification of surfaces here.

KK: This is my fault. I like to go off on tangents.

EDL: Let’s reel it back in.

EL: You mentioned that you’ve matured into true appreciation of this lovely theorem, which kind of brings me to the next part of the show. The best things in life are better together. Can you recommend a pairing for your theorem? This could be a fine wine or a flavor of ice cream or a favorite piece of music or art that you think really enhances the beauty of this classification theorem.

EDL: I hate to do this, but I’m going to have to say coffee and donuts.

KK: Of course.

EDL: I really tried to say something else, but I couldn’t make myself do it. A donut and cup of coffee go great with the classification of compact surfaces theorem.

EL: That’s fair.

KK: San Francisco coffee, right? Really good dark, walk down to Blue Bottle and stand in line for a while?

EDL: That’s right. Vietnamese coffee.

KK: There you go. That’s good.

EL: Is there a particular flavor of donut that you recommend?

EDL: Well you know, the maple bacon. Who can say no to bacon on a donut?

KK: Or on anything for that matter.

EDL: Or on anything.

KK: That’s just a genus one surface. Can we get higher-genus donuts? Have we seen these anywhere, or is it just one?

EDL: There are some twisted little pastry type things. I’m wondering if there’s some higher genus donuts out there.

EL: If nothing else there’s a little bit of Dehn twisting going on with that.

EDL: There’s definitely some twisting.

EL: I guess we could move all the way over into pretzels, but that doesn’t go quite as well with a cup of coffee.

EDL: Or if you’re in San Francisco, you can get one of these cronuts that have been all the rage lately.

EL: What is a cronut? I have not quite understood this concept.

EDL: It is a cross between a croissant and a donut. And it’s flakier than your average donut. It is quite good. And if you want one, you’re probably going to have to stand on line for about an hour. Maybe the rage has died down by now, maybe. But that’s what was happening when they were first introduced.

EL: I’m a little scared of the cronut. That sounds intense but also intriguing.

EDL: You’ve got to try everything once, Evelyn. Live on the edge.

EL: The edge of the cronut.

KK: You’re in Paris. We’re not too concerned about your ability to get pastry.

EL: I have been putting away some butter.

KK: The French have it right. They understand that butter does the heavy lifting.

EDL: It’s probably a sin to have a cronut in Paris.

EL: Probably. But if they made one, it would be the best cronut that existed.

EDL: Absolutely.

KK: Well I think this has been fun. Anything else you want to add about your favorite theorem?

EDL: It’s a theorem that everyone should dig into, even if you aren’t into topology. I think it’s one of those foundational theorems that everyone should see at least once, and look at the proof at least once, just for a well-rounded mathematical education.

KK: Maybe I should look at the proof sometime.

EL: Thanks so much for joining us, Emille. We really enjoyed having you. And this has been My Favorite Theorem.

EDL: Thank you so much.

KK: Thanks for listening to My Favorite Theorem, hosted by Kevin Knudson and Evelyn Lamb. The music you’re hearing is a piece called Fractalia, a percussion quartet performed by four high school students from Gainesville, Florida. They are Blake Crawford, Gus Knudson, Del Mitchell, and Bao-xian Lin. You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at Kevin’s website, kpknudson.com, and Evelyn’s blog, Roots of Unity, on the Scientific American blog network. We love to hear from our listeners, so please drop us a line at myfavoritetheorem@gmail.com. Or you can find us on Facebook and Twitter. Kevin’s handle on Twitter is @niveknosdunk, and Evelyn’s is @evelynjlamb. The show itself also has a Twitter feed. The handle is @myfavethm. Join us next time to learn another fascinating piece of mathematics.

Show notes (click here)

Episode 2 - Dave Richeson

This transcript is provided as a courtesy and may contain errors.

Evelyn Lamb: Welcome to My Favorite Theorem. I’m your host Evelyn Lamb. I am a freelance math writer usually based in Salt Lake City but currently based in Paris. And this is your other host.

KK: I’m Kevin Knudson, professor of mathematics at the University of Florida.

EL: Every episode we invite a mathematician on to tell us about their favorite theorem. This week our guest is Dave Richeson. Can you tell us a little about yourself, Dave?

Dave Richeson: Sure. I’m a professor of mathematics at Dickinson College, which is in Carlisle, Pennsylvania. I’m also currently the editor of Math Horizons, which is the undergraduate magazine of the Mathematical Association of America.

EL: Great. And so how did you get from wherever you started to Carlisle, Pennsylvania?

DR: The way things usually work in academia. I applied to a bunch of schools. Actually, seriously, my wife knew someone in Carlisle, Pennsylvania. My girlfriend at the time, wife now, and she saw the list of schools that I was applying to and said, “You should get a job at Dickinson because I know someone there.” And I did.

KK: That never happens!

EL: Wow.

DR: That never happens.

KK: That never happens. Dave and I actually go back a long way. He was finishing his Ph.D. at Northwestern when I was a postdoc there.

DR: That’s right.

KK: That’s how old-timey we are. Hey, Dave, why don’t you plug your excellent book.

DR: A few years ago I wrote a book called Euler’s Gem: The Polyhedron Formula and the Birth of Topology. It’s at Princeton University Press. I could have chosen Euler’s Formula as my favorite theorem, but I decided to choose something different instead.

KK: That’s very cool. I really recommend Dave’s book. It’s great. I have it on my shelf. It’s a good read.

DR: Thank you.

EL: Yeah. So you’ve told us what your favorite theorem isn’t. So what is your favorite theorem?

DR: We have a family joke. My kids are always saying, “What’s your favorite ice cream? What’s your favorite color?” And I don’t really rank things that way. This was a really challenging assignment to come up with a theorem. I have recently been interested in π and Greek mathematics, so currently I’m fascinated by this theorem of Archimedes, so that is what I’m giving you as my favorite theorem. Favorite theorem of the moment.

The theorem says that if you take a circle, the area of that circle is the same as the area of a right triangle that has one leg equal to the radius and one leg equal to the circumference of the circle. Area equals 1/2 c x r, and hopefully we can spend the rest of the podcast talking about why I think this is such a fascinating theorem.

KK: I really like this theorem because I think in grade school you memorize this formula, that area is π r2, and if you translate what you said into modern terminology, or notation, that is what it would say. It’s always been a mystery, right? It just gets presented to you in grade school. Hey, this is the formula of a circle. Just take it.

DR: Really, we have these two circle formulas, right? The area equals π r2, and the circumference is 2πr, or the way it’s often presented is that π is the circumference divided by the diameter. As you said, you could convince yourself that Archimedes theorem is true by using those formulas. Really it’s sort of the reverse. We have those formulas because of what Archimedes did. Pi has a long and fascinating history. It was discovered and rediscovered in many, many cultures: the Babylonians, the Egyptians, Chinese, Indians, and so forth. But no one, until the Greeks, really looked at it in a rigorous way and started proving theorems about π and relationships between the circumference, the diameter, and the area of the circle.

EL: Right, and something you had said in one of your emails to us was about how it’s not even, if you ask a mathematician who proved that π was a constant, that’s a hard question.

DR: Yes, exactly. I mean, in a way, it seems easy. Pi is usually defined as the circumference divided by the diameter for any circle. And in a way, it seems kind of obvious. If you take a circle and you blow it up or shrink it down by some factor of k, let’s say, then the circumference is going to increase by a factor of k, the diameter is going to increase by a factor of k. When you do that division you would get the same number. That seems sort of obvious, and in a way it kind of is. What’s really tricky about this is that you have to have a way of talking about the length of the circumference. That is a curve, and it’s not obvious how to talk about lengths of curves. In fact, if you ask a mathematician who proved that the circumference over the diameter was the same value of π, most mathematicians don’t know the answer to that. I’d put money on it that most people would think it was in Euclid’s Elements, which is sort of the Bible of geometry. But it isn’t. There’s nothing about the circumference divided by the diameter, or anything equivalent to it, in Euclid’s Elements.

Just to put things in context here, a quick primer on Greek mathematics. Euclid wrote Elements sometime around 300 BCE. Pythagoras was before that, maybe 150 years before that. Archimedes was probably born after Euclid’s Elements was written. This is relatively late in this Greek period of mathematics.

KK: Getting back to that question of proportionality, the idea that all circles are similar and that’s why everybody thinks π is a constant, why is that obvious, though? I mean, I agree that all circles are similar. But this idea that if you scale a circle by a factor of k, its length scales by k, I agree if you take a polygon, that it’s clear, but why does that work for curves? That’s the crux of the matter in some sense, right?

DR: Yeah, that’s it. I think one mathematician I read called this “inherited knowledge.” This is something that was known for a long time, and it was rediscovered in many places. I think “obvious” is sort of, as we all know from doing math, obvious is a tricky word in math. It’s obvious meaning lots of people have thought of it, but if you actually have to make it rigorous and give a proof of this fact, it’s tricky. And so it is obvious in a sense that it seems pretty clear, but if you actually have to connect the dots, it’s tricky. In fact, Euclid could not have proved it in his Elements. He begins the Elements with his famous five postulates that sort of set the stage, and from those he proves everything in the book. And it turns out that those five postulates aren’t enough to prove this theorem. So one of Archimedes’ contributions was to recognize that we needed more than just Euclid’s postulates, and so he added two new postulates to those. From that, he was able to give a satisfactory proof that area=1/2 circumference times radius.

KK: So what were the new postulates? DR: One of them was essentially that if you have two points, then the shortest distance between them is a straight line, which again seems sort of obvious, and actually Euclid did prove that for polygonal lines, but Archimedes is including curves as well. And the other one is that if you have, it would be easier to draw a picture. If you had two points and you connected them by a straight line and then connected them by two curves that he calls “concave in the same direction,” then the one that’s in between the straight line and the other curve is shorter than the second curve. The way he uses both of those theorems is to say that if you take a circle and inscribe a polygon, like a regular polygon, and you circumscribe a regular polygon, then the inscribed polygon has the shortest perimeter, then the circle, then the circumscribed polygon. That’s the key fact that he needs, and he uses those two axioms to justify that.

EL: OK. And so this sounds like it’s also very related to his some more famous work on actually bounding the value of π.

DR: Yeah, exactly. We have some writings of his that goes by the name “Measurement of a Circle.” Unfortunately it’s incomplete, and it’s clearly not come down to us very well through history. The two main results in that are the theorem I just talked about and his famous bounds on π, that π is between 223/71 and 22/7. 22/7 is a very famous approximation of π. Yes, so these are all tied together, and they’re in the same treatise that he wrote. In both cases, he uses this idea of approximating a circle by inscribed and circumscribed polygons, which turned out to be extremely fruitful. Really for 2,000 years, people were trying to get better and better approximations, and really until calculus they basically used Archimedes’ techniques and just used polygons with more and more and more and more sides to try to get better approximations of π.

KK: Yeah, it takes a lot too, right? Weren’t his bounds something like a 96-gon?

DR: Yeah, that’s right. Exactly.

KK: I once wrote a Geogebra applet thing to run to the calculations like that. It takes it a while for it to even get to 3.14. It’s a pretty slow convergence.

DR: I should also plug another mathematician from the Greek era who is not that well known, and that is Eudoxus. He did work before Euclid, and big chunks of Euclid’s Elements are based on the work of Eudoxus. He was the one who really set this in motion. It’s become known as the method of exhaustion, but really it’s the ideas of calculus and limiting in disguise. This idea of proving these theorems about shapes with curved boundaries using polygons, better and better approximations of polygons. So Eudoxus is one of my favorite mathematicians that most people don’t really know about.

KK: That’s exactly it, right? They almost had calculus.

DR: Right.

KK: Almost. It’s really pretty amazing.

DR: Yes, exactly. The Greeks were pretty afraid of infinity.

KK: I’m sort of surprised that they let the method of exhaustion go, that they were OK with it. It is sort of getting at a limiting process, and as you say, they don’t like infinity.

DR: Yeah.

KK: You’d think they might not have accepted it as a proof technique.

DR: Really, and maybe this is talking too much for the mathematicians in the audience, but really the way they present this is a proof by contradiction. They show that it can’t be done, and then they get these polygons that are close enough that it can be done, and that gives them a contradiction. The final style of the proof would, I think, be comfortable to them. They don’t really take a limit, they don’t pass to infinity, anything like that.

EL: So something we like to do on this podcast is ask our guest to pair their theorem with something. Great things in life are often better paired: wine and cheese, beer and pizza, so what’s best with your theorem?

DR: I have to go with the obvious: pie, maybe pizza.

KK: Just pizza? OK?

EL: What flavor? What toppings?

KK: What goes on it?

DR: That’s a good question. I’m a fan of black olives on my pizza.

KK: OK. Just black olives?

DR: Maybe some pepperoni too.

KK: There you go.

EL: Deep dish? Thin crust? We want specifics.

DR: I’d say thin crust pizza, pepperoni and black olives. That sounds great.

EL: You’d say this is the best way to properly appreciate this theorem of Archimedes, is over a slice of pizza.

DR: I think I would enjoy going to a good pizza joint and talking to some mathematicians and telling them about who first proved that circumference over diameter is π, that it was Archimedes.

Actually, I was saying to Kevin before we started recording that I actually have a funny story about this, that I started investigating this. I wanted to know who first proved that circumference over diameter is a constant. I did some looking and did some asking around and couldn’t really get a satisfactory answer. I sheepishly at a conference went up to a pretty well-known math historian, and said, “I have this question about π I’m embarrassed to ask.” And he said, “Who first proved that circumference over diameter is a constant?” I said, “Yes!” He’s like, “I don’t know. I’d guess Archimedes, but I really don’t know.” And that’s when I realized it was an interesting question and something to look at a little more deeply.

EL: That’s a good life lesson, too. Don’t be afraid to ask that question that you are a little afraid to ask.

KK: And also that most answers to ancient Greek mathematics involve Archimedes.

DR: Yeah. Actually through this whole investigation, I’ve gained an unbelievable appreciation of Archimedes. I think Euclid and Pythagoras probably have more name recognition, but the more I read about Archimedes and things that he’s done, the more I realize that he is one of the great, top 5 mathematicians.

KK: All right, so that’s it. What’s the top 5?

DR: Gosh. Let’s see here.

KK: Unordered. DR: I already have Archimedes. Euler, Newton, Gauss, and who would number 5 be?

KK: Somebody modern, come on.

DR: How about Poincaré, that’s not exactly modern, but more modern than the rest. While we’re talking about Archimedes, I also want to make a plug. There’s all this talk about tau vs. pi. I don’t really want to weigh in on that one, but I do think we should call π Archimedes’ number. We talk about π is the circumference constant, π is the area constant. Archimedes was involved with both of those. People may not know he was also involved in attaching π to the volume of the sphere and π to the surface area of the sphere. Here I’m being a little historically inaccurate. Pi as a number didn’t exist for a long time after that. But basically recognizing that all four of these things that we now recognize as π, the circumference of a circle, the area of a circle, the volume of a sphere, and the surface area of a sphere. In fact, he famously asked that this be represented on his tombstone when he died. He had this lovely way to put all four of these together, and he said that if you take a sphere and then you enclose it in a cylinder, so that’s a cylinder that’s touching the sphere on the sides, think of a can of soda or something that’s touching on the top as well, that the volume of the cylinder to the sphere is in the ratio 3:2, and the surface area of the cylinder to the sphere is also the ratio 3:2. If you work out the math, all four of these versions of π appear in the calculation. We do have some evidence that this was actually carried out. Years later, the Roman Cicero found Archimedes’ tomb, and it was covered in brambles and so forth, and he talks about seeing the sphere and the cylinder on Archimedes’ tombstone, which is kind of cool.

EL: Oh wow.

DR: Yeah, he wrote about it.

KK: Of course, how Archimedes died is another good story. It’s really too bad.

DR: Yeah, I was just reading about that this week. The Roman siege of Syracuse, and Archimedes, in addition to being a great mathematician and physicist, was a great engineer, and he built all these war devices to help keep the Romans at bay, and he ended up being killed by a Roman soldier. The story goes that he was doing math at the time, and the Roman general was apparently upset that they killed Archimedes. But that was his end.

KK: Then on Mythbusters, they actually tried the deal with the mirrors to see if they could get a sail to catch on fire.

DR: I did see that! Some of these stories have more evidence than others. Apparently the story of using the burning mirrors to catch ships on fire, that appeared much, much later, so the historical connection to Archimedes is pretty flimsy. As you said, it was debunked by Mythbusters on TV, or they weren’t able to match Archimedes, I should say.

KK: Well few of us can, right?

DR: Right. The other thing that is historically interesting about this is that one of the most famous problems in the history of math is the problem of squaring the circle. This is a famous Greek problem which says that if you have a circle and only a compass and straightedge, can you construct a square that has the same area as the circle? This was a challenging and difficult problem. Reading Archimedes’ writings, it’s pretty clear that he was working on this pretty hard. That’s part of the context, I think, of this work he did on π, was trying to tackle the problem of squaring the circle. It turns out that this was impossible, it is impossible to square the circle, but that wasn’t discovered until 1882. At the time it was still an interesting open problem, and Archimedes made various contributions that were related to this famous problem.

EL: Yeah.

KK: Very cool.

DR: I can go on and on. So today, that is my favorite theorem.

KK: We could have you on again, and it might be different?

DR: Sure. I’d love to.

KK: Well, thanks, Dave, we certainly appreciate you being here.

DR: I should say if people would like to read about this, I did write an article, “Circular Reasoning: Who first proved that c/d is a constant?” Some of the things I talked about are in that article. Mathematicians can find it in the College Math Journal, and it just recently was included in Princeton University Press’s book The Best Writing on Mathematics, 2016 edition. You can find that wherever, your local bookstore.

EL: And where else can our loyal listeners find you online, Dave?

DR: I spend a lot of time on Twitter. I’m @divbyzero. I blog occasionally at divisbyzero.com.

EL: OK.

DR: That’s where I’d recommend finding me.

KK: Cool.

EL: All right. Well, thanks for being here.

DR: Thank you for asking me. It was a pleasure talking to you.

Show notes (click here)

Episode 1 - Amie Wilkinson

Kevin Knudson: Welcome to My Favorite Theorem. I’m Kevin Knudson, professor of mathematics at the University of Florida, and I’m joined by my cohost.

Evelyn Lamb: I’m Evelyn Lamb. I’m a freelance writer currently based in Paris.

KK: Yeah, Paris. Paris is better than Gainesville. I mean, Gainesville’s nice and everything.

EL: Depends on how much you like alligators.

KK: I don’t like alligators that much.

EL: OK.

KK: This episode, we’re thrilled to welcome Amie Wilkinson of the University of Chicago. Amie’s a fantastic mathematician. Say hi, Amie, and tell everyone about yourself.

AW: Hi, everyone. So Kevin and I go way back. I’m a professor at the University of Chicago. Kevin and I first met when we were pretty fresh out of graduate school. We were postdocs at Northwestern, and now we’ve kind of gone our separate ways but have stayed in touch over the years.

KK: And, let’s see, my son and your daughter were born the same very hot summer in Chicago.

AW: Yeah, that’s right.

KK: That’s a long time ago.

AW: Right. And they’re both pretty hot kids.

KK: They are, yes. So, Amie, you haven’t shared what your favorite theorem is with Evelyn and me, so this will be a complete surprise for us, and we’ll try to keep up. So what’s your favorite theorem?

AW: Fundamental theorem of calculus.

KK: Yes.

EL: It’s a good theorem.

KK: I like that theorem. I just taught calc one, so this is fresh in my mind. I can work with this.

AW: Excellent. Probably fresher than it is in my mind.

EL: Can you tell us, remind our listeners, or tell our listeners what the fundamental theorem of calculus is?

AW: The fundamental theorem of calculus is a magic theorem as far as I’m concerned, that relates two different concepts: differentiation and integration.

So integration roughly is the computation of area, like the area of a square, area of the inside of a triangle, and so on. But you can make much more general computations of area like Archimedes did a long time ago, the area inside of a curve, like the area inside of a circle. There’s long been built up, going back to the Greeks, this notion of area, and even ways to compute it. That’s called integration.

Differentiation, on the other hand, it has to do with motion. In its earliest forms, to differentiate a function means to compute its slope, or speed, velocity. It’s a computation of velocity. It’s a way of measuring instantaneous motion. Both of these notions go way back, to the Greeks in the case of area, back to the 15th century and the people at Oxford for the computation of speed, and it wasn’t until the 17th century that the two were connected. First by someone named James Gregory, and not long after, sort of concurrently, by Isaac Barrow, who was the advisor of Isaac Newton. Newton was the one who really formalized the connection between the two.

EL: Right, but this wasn’t just a lightning bolt that suddenly came from Newton, but it had been building up for a while.

AW: Building up, actually in some sense I think it was a lightning bolt, in the sense that all of the progress happened within maybe a 30-year period, so in the world of mathematics, that’s sort of, you could even say that’s a fad or a trend. Someone does something, and you’re like, oh my god, let’s see what we can do with this. It’s an amazing insight that the two are connected.

The most concrete illustration of this is actually one I read on Wikipedia, which says that suppose you’re in a car, and you’re not the driver because otherwise this would be a very scary application. You can’t see outside of the car, but you can see the odometer. Sorry, you can’t see the odometer either. Someone’s put tape over it. But you can see the speedometer. And that’s telling you your velocity at every second. Every instance there’s a number. And what the fundamental theorem of calculus says is that if you add up all of those numbers over a given interval of time, it’s going to tell you how far you’ve traveled.

KK: Right.

AW: You could just take the speed that you see on the odometer the minute you start driving the car and then multiply by the amount of time that you travel, and that’ll give you kind of an approximate idea, but you instead could break the time into two pieces and take the velocity that you see at the time and the velocity that you see at the midpoint, and take the average of those two velocities, multiplied by the amount of time, and that’ll give you a better sense. And basically it says to compute the average velocity multiplied by the time, and you’re going to get how far you’ve gone. That’s basically what the fundamental theorem of calculus means.

KK: So here’s my own hot take on the fundamental theorem: I think it’s actually named incorrectly. I think the mean value theorem is the real fundamental theorem of calculus.

AW: Ah-ha.

KK: If you think about the fundamental theorem, it’s actually a pretty quick corollary to the fundamental theorem.

AW: Right.

KK: Which essentially just describes, well, the version of the fundamental theorem that calculus remember, namely that to compute a definite integral, “all you have to do” — and our listeners can’t see me doing the air quotes—but“all you have to do” is find the antiderivative of the function, we know how hard that problem is. That’s a pretty quick corollary of the mean value theorem, basically by the process you just described, right? You’ve got your function, and you’re trying to compute the definite integral, so what do you do? Well, you take a Riemann sum, chop it into pieces. Then the mean value theorem says over each subinterval, there’s some point in there where the derivative equals the average rate of change over that little subinterval. And so you replace with all that, and that’s how you see the fundamental theorem just drop out. This Riemann sum is essentially just saying, OK, you find the antiderivative and that’s the story. So I used to sort of joke, I always joke with my students, that one of these days I’m going to write an advanced calculus book sort of like “Where’s Waldo,” but it’s going to be “Where’s the Mean Value Theorem?”

AW: I like that.

KK: Whenever you teach advanced calculus for real, not just that first course, you start to see the mean value theorem everywhere.

AW: See, I think of the mean value theorem as being the flip side of the fundamental theorem of calculus. To me, what is the mean value theorem? The mean value theorem is a movie that I saw in high school calculus that was probably filmed in, like, 1960-something.

KK: Right. On a movie projector?

AW: Yeah, on a movie projector.

KK: A lot of our listeners won’t know what that is.

AW: It’s a very simple little story. A guy’s driving, again it’s a driving analogy.

KK: Sure, I use these all the time too.

AW: And he stops at a toll booth to get his ticket, and the ticket is stamped with the time that he crosses the tollbooth, and then he’s driving and driving, and he gets to the other tollbooth and hands the ticket to the toll-taker, and the toll-taker says, “You’ve been speedin.’ The reason I know this is the mean value theorem.” He says it just like that, “The mean value theorem.” I wish I could find that movie. I’m sure I could. It’s so brilliant. What that’s saying is if I know the distance I’ve traveled from A to B, I could calculate what the average speed is by just taking, OK, I know how much time it took. So that second toll-taker knows (a) how much time it took, and (b), the distance because he knows the other tollbooth, right? And so he computes the average speed, and what the mean value theorem says is somewhere during that trip, you had to be traveling the average speed.

KK: Right.

AW: So, it’s sort of like I can do speed from distance, so if you took too little time to travel the distance, you had to be speeding at some point, which is so beautiful. That’s sort of the flip side. If you know the distance and the amount of time, then you know the average speed. Whereas the first illustration I gave is you’re in this car, and you can’t see outside or the odometer, but you know the average speed, and that tells you the distance.

KK: So maybe they’re the same theorem.

EL: They’re all the same.

AW: In some sense, right.

KK: I think this is why I still love teaching calculus. I’ve been doing it for, like, 25 years, but I never get tired of it. It’s endlessly fascinating.

AW: That’s wonderful. We need more calculus teachers like you.

KK: I don’t know about that, but I do still love it.

AW: Or at least with your attitude.

KK: Right. There we go. So this is actually, the fundamental theorem is just sort of a one-dimensional version. There are generalizations, yes?

AW: Yes, there are. That gets to my favorite generalization of the fundamental theorem of calculus, which is Stokes' Theorem.

KK: Yeah.

AW: So what does Stokes’ Theorem do? Well, for one thing, it explains why π appears both in the formula for the circumference of a circle and in the formula for the area of the circle, inside of the circle.

KK: That’s cool.

AW: Right? One is πr2, and the other is 2πr, and roughly speaking, suppose you differentiate with respect to r. This is sort of bogus, but it’s correct.

KK: Let’s go with it.

AW: You differentiate πr2, you get 2πr. The point is that Stokes'’ Theorem, like the fundamental theorem of calculus, relates two quantities of a geometric object, in this case a circle. One is an integral inside the object, and the other is an integral on the boundary of the object. And what are you integrating? So Stokes' Theorem says if you have something called a form, and it’s defined on the boundary of an object, and you differentiate the form, then the integral of the derivative of the form on the inside is the integral of the original form on the boundary.

EL: Yeah.

AW: And the best way to illustrate this is with a picture, I’m afraid. It’s a beautiful, the formula itself has this beautiful symmetry to it.

EL: Yeah. Well, our listeners will be able to see that online when we post this, so we’ll have a visual aid.

AW: OK. So Stokes' Theorem establishes the duality of differentiation on the one hand, which is like analysis-calculus, right, and taking the boundary of an object on the other hand.

KK: That’s geometry, right.

AW: And boundary we denote by something that looks like a d, but it’s sort of curly, and we call it del. And differentiation we denote by d. The point is that those two operations can be switched and you get the same thing. You switch those operations in two different places, you get the same thing. That duality leads to differential topology. I mean, it’s just… The next theorem that’s amazing is De Rahm’s theorem that comes out of that.

KK: Let’s not go that far.

AW: OK.

KK: It’s remarkable. You think, in calculus 3, at the very end we teach students Stokes’ Theorem, but we sort of get there incrementally, right? We teach Green’s theorem in the plane, and then we give them the divergence theorem, right, which is still the same. They’re all the same theorem, and we never really tie it together really well, and we never go, oh, by the way, if we would unify this idea, we’d say, by the way, this is really just the fundamental theorem of calculus.

AW: Right.

KK: If you take your manifold to be a closed interval in the plane. So this makes me wonder if we need to start modernizing the calculus curriculum. On the other hand, then that gets a little New Math-y, right?

AW: No, no, I think we should totally normalize the curriculum in this way.

KK: Do you?

AW: Yeah, sure. It depends on what level we’re talking about, obviously, but I’ve always found that, OK, so, I’m going to confess the one time I taught multivariable calculus to “regular” students — granted, this was ages ago — I was so irritated by the current curriculum I couldn’t hide it.

KK: Oh, I see.

AW: But I’ve taught, lots and lots of times, multivariable calculus to somewhat more advanced students, to honors students who might become math majors, might not. And I always adopt this viewpoint, that the fundamental theorem of calculus is relating your object — your geometric object is just an interval, and it’s boundary is just two points, and differentiation-integration connects the difference of values of functions at two points with the integral over the interval.

KK: Then that gets to the question of, is that the right message for everyone? I could imagine this does work well with students who might want to be math majors. But in an engineering school, for example. I haven’t taught multivariable in maybe 15 years, but I’m tending to aim at engineers. But engineers, they don’t work outside of three dimensions, for the most part. Would this really be the right way to go? I don’t know.

AW: First of all, it’s good for turning students who are interested in calculus, who are interested in math, into math majors. So for me, that’s an effective tool.

KK: I absolutely believe that.

AW: Yeah, I don’t know about engineering students. They really have a distinct set of needs.

KK: Right.

AW: I mean, social scientists, for example, work regularly in very high dimensions, and I have taught this material to social scientists back at Northwestern, and that was also, I think, pretty successful.

KK: Interesting. Well, that’s a good theorem. We love the fundamental theorem around here.

EL: The best things in life are often better together. So one of the things we like to do on My Favorite Theorem is to ask our guests to pick a pairing for their theorem, a fine wine or tea, beer, ice cream, piece of music, so what would you like to pair with the Fundamental Theorem of Caclulus?

AW: Something like a mango, maybe.

EL: A mango!

AW: Something where you have this organic, beautiful shape that, if you wanted to understand it analytically, you would have to use calculus. So first of all, mango is literally my favorite.

KK: I love them too. Oh, man.

AW: Ripe mango. It has to be good. Bad mango is torture.

KK: This is one of the perks of living in Florida. We have good mangoes here.

AW: What I love about the mango is it’s a natural form that is truly not spherical. It’s a fruit that has this clearly organic and very smooth shape. But to describe it, I don’t even know.

KK: It’s not a solid of revolution.

AW: I don’t know why it grows like that.

KK: Well the pit is weird, right? The pit’s sort of flat.

AW: Yeah.

KK: Why does it grow like that? That’s interesting. Because most things, like an avocado, for example, it’s sort of pear-shaped, and the pit is round.

AW: An avocado is another example of a beautiful organic shape that is not perfectly spherical. So yeah, and I love avocado as well, so maybe I could have a mango-avocado salad.

EL: Oh, yeah. Really getting quite gourmet.

KK: And this goes to the fundamental theorem, right? Because you have to chop that up into pieces, which, I mean.

AW: Right?

KK: It’s sort of the Riemann sum of your two things.

AW: And they’re very hard, both of them are very hard to get the fruit out, reasonably difficult to get the fruit out of the shell.

KK: You know the deal, right? You cut it in half first and then you dice it and scoop it out, right?

AW: You mean with the mango, right?

KK: You do with an avocado, too. Yeah.

AW: You know, I’ve never thought to do that with an avocado.

KK: Yeah, you cut the avocado, take a big knife and just cut it and then split it open, pop the pit out, and then just dice it and scoop it out.

AW: Oh. I usually just scoop and dice, but you’re right. In the mango you do the same, but then you start turning it inside out, and it looks like a hand grenade. So beautiful.

KK: You do the same thing with the avocado, and just scoop it. See?

AW: That’s a really interesting illustration, too, because when you turn inside out the mango, you can see these cubes of fruit that are spreading apart. You sort of can see how by changing the shape of the boundary, you change radically the sort of volume enclosed by the boundary. Because those things spread apart because of the reversed curvature.

EL: Yeah.

KK: Now I’m getting hungry.

AW: Yeah.

EL: Yeah, that’s the problem with these pairings, right? We record an episode, and then we all have to go out to eat.

AW: Of course a more provincial kind of thing, a more everyday object, piece of fruit, would be, as you said, pear. That’s more connected to Isaac Newton.

EL: True, yeah.

AW: Apples.

KK: The apples falling on his head, yeah. Cool. Well, this was fun, Amie. Thanks for joining us. Anything else you want to add? Any projects you want to plug? We try to give everybody a chance to do that. What are you working on these days?

AW: My area is dynamical systems, which..

KK: Is hard!

AW: It’s hard, but it’s also connected very closely. It’s not that hard.

KK: Smale said it’s hard.

AW: It’s connected very closely to the fundamental theorem. I study how things change over time.

KK: Right.

AW: So I’ve been helping out, or I don’t know if I’ve actually been helping, but I’ve been talking a lot with some physicists who build particle accelerators, and we’re trying to use tools from pure mathematics to design these accelerators more effectively.

EL: Oh wow.

AW: To keep the particles inside the accelerator, moving in a focused beam.

EL: Nice.

AW: It’s a direct application of certain areas of smooth dynamical systems.

KK: Very cool. You never know where your career is going to take you.

AW: It’s very fun.

KK: That’s part of the beauty of mathematics, you never know where it’s going to lead you.

EL: Thanks so much for joining us on My Favorite Theorem.

AW: Thank you for having me. It’s been a lot of fun.

KK: Thanks for listening to My Favorite Theorem, hosted by Kevin Knudson and Evelyn Lamb. The music you’re hearing is a piece called Fractalia, a percussion quartet performed by four high school students from Gainesville, Florida. They are Blake Crawford, Gus Knudson, Dell Mitchell, and Baochau Nguyen. You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at Kevin’s website, kpknudson.com, and Evelyn’s blog, Roots of Unity, on the Scientific American blog network. We love to hear from our listeners, so please drop us a line at myfavoritetheorem@gmail.com. Or you can find us on Facebook and Twitter. Kevin’s handle on Twitter is @niveknosdunk, and Evelyn’s is @evelynjlamb. The show itself also has a Twitter feed. The handle is @myfavethm. Join us next time to learn another fascinating piece of mathematics.

Show notes (click here)

Episode 0 - Your Hosts' Favorite Theorems

KK: Welcome to My Favorite Theorem. I’m Kevin Knudson, and I’m joined by my cohost.

EL: I’m Evelyn Lamb.

KK: This is Episode 0, in which we’ll lay out our ground rules for what we’re going to do. The idea is every week we’ll have a guest, and that guest will tell us what his or her favorite theorem is, and they’ll tell us some fun things about themselves, and Evelyn had good ideas here. What else are we going to do?

EL: Yeah, well, with any great thing in life, pairings are important. So we’ll find the perfect wine, or ice cream, or work of 19th century German romanticism to include with the theorem. We’ll ask our guests to help us with that.

KK: Since this is episode 0, we thought we should probably set the tone and let you know what our favorite theorems are. I’m going to defer. I’m going to let Evelyn go first here. What’s your favorite theorem?

EL: OK, so we’re recording this on March 23rd, which is Emmy Noether’s birthday, her 135th, to be precise. I feel like I should say Noether’s theorem. It’s a theorem in physics that relates, that says basically conserved quantities in physics come from symmetries in nature. So time translation symmetry yields conservation of energy and things like that. But I’m not going to say that one. I’m sorry, physics, I just like math more.

So I’m going to pick the uniformization theorem as my favorite theorem.

KK: I don’t think I know that theorem. Which one is this?

EL: It’s a great theorem. When I was doing math research, I was working in Teichmüller theory, which is related to hyperbolic geometry. This is a theorem about two-dimensional surfaces. The upshot of this theorem is that every two-dimensional surface can be given geometry that is either spherical, flat — so, Euclidean, like the flat plane — or hyperbolic. The uniformization itself is related to simply connected Riemann surfaces, the ones with no holes, but using this theorem you can show that 2-d surfaces with any number of holes have one of these kinds of 2-d geometry. This is a great theorem. I just love that part of topology where you’re classifying surfaces and everything. I think it’s nice A little of the history is that it was conjectured by Poincaré in 1882 and Klein in 1883. I think the first proof was by Poincaré in the early 1900s. There are a lot of proofs of it that come from different approaches.

KK: Now that you tell me what the theorem is, of course I know what it is. Being a topologist, I know how to classify surfaces, I think.

That is a great theorem. There’s so much going on there. You can think about Riemann surfaces as quotients of hyperbolic space, and you have all this fun geometry going. I love that theorem. In fact, I’m teaching our graduate topology course this year, and I didn’t do this. I’m sorry. I had to get through homology and cohomology. So yeah, surfaces are classified. We know surfaces.

So what are you going to pair this with?

EL: So my pairing is Neapolitan ice cream. I’m going a bit literal with this. Neapolitan ice cream is the ice cream that has part of it vanilla, part of it chocolate, and part of it strawberry. So this theorem says that surfaces come in three flavors.

KK: Nice.

EL: When I was a little kid, when we had our birthday parties at home, my mom always let us pick what ice cream we wanted to have, and I always picked Neapolitan so that if my friends liked one of the flavors but not the others, they could have whichever flavor they wanted.

KK: You’re too kind.

EL: Really, I’m just such a good-hearted person.

KK: Clearly.

EL: Yeah, Neapolitan. Three flavors of surfaces, three flavors of ice cream.

KK: Nice. Although nobody ever eats the strawberry, right?

EL: Yeah, I love strawberry ice cream now, but yeah, when I was a little kid chocolate and vanilla were a little more my thing.

KK: I remember my mother would sometimes buy the Neapolitan, and I remember the strawberry would just sit there, uneaten, until it got freezer burn, and we just threw it away at that point.

EL: I guess the question is, which of the kinds of geometry is strawberry?

KK: Well, vanilla is clearly flat, right?

EL: Yeah, that’s good. I guess that means strawberry must be spherical.

KK: That seems right. It’s pretty unique, right? Spherical geometry is kind of dull, right? There’s just the sphere. There’s a lot more variation in hyperbolic geometry, right?

EL: Yeah, I guess so. I feel like there are more different kinds of chocolate-flavored ice cream, and hyperbolic, there are so many different hyperbolic surfaces.

KK: Right. Here in Gainesville, we have a really wonderful local ice cream place, and twice a year they have chocolate night, and they have 32 different varieties of chocolate.

EL: Oh my gosh.

KK: So you can go and you can get a ginormous bowl of all 36 flavors if you want, but we usually get a little sample of eight different flavors and try them out. It’s really wonderful. I think that’s the right classification.

EL: OK. So Kevin?

KK: Yes?

EL: What is your favorite theorem?

KK: Well, yeah, I thought about this for a long time, and what I came up with was that my favorite theorem is the ham sandwich theorem. I think it’s largely because it’s got a fun name, right?

EL: Yeah.

KK: And I remember hearing about this theorem as an undergrad for the first time. This was a general topology course, and you don’t prove it in that, I think. You need some algebraic topology to prove this well. I thought, wow, what a cool thing! There’s something called the ham sandwich theorem. So what is the ham sandwich theorem? It says: say you have a ham sandwich, which consists of two pieces of bread and a chunk of ham. And maybe you got a little nuts and you put one piece of bread on top of the fridge, and one on the floor, and your ham is sitting on the counter, and the theorem is that if you have a long enough knife, you can make one cut and cut all of those things in half. Mathematically what that means is that you have three blobs in space, and there is a single plane that cuts each of those blobs in half exactly. I just thought that was a pretty remarkable theorem, and I still think it’s kind of remarkable theorem because it’s kind of hard to picture, right? Your blobs could be anywhere. They could be really far apart, as long as they have positive measure, so as long as they’re not some flat thing, they actually have some 3-d-ness to them, then you can actually find a plane that does this. What’s even more fun, I think, is that this is a consequence of the Borsuk-Ulam theorem, which in this case would say that if you have a continuous function from the 2-sphere to the plane, then two antipodal points have to go to the same place. And that’s always a fun theorem to explain to people who don’t know any mathematics, because you can say, somewhere, right now, there are two opposite points on the surface of the earth where the temperature and the humidity are the same, for example.

EL: Yeah.

KK: I love that kind of theorem, where there’s a good physical interpretation for it. And of course there are higher-dimensional analogues, but the idea of the ham sandwich theorem is great. Everybody’s had a ham sandwich, probably, or some kind of sandwich. It doesn’t have to be ham. Maybe we should be more politically correct. What’s a good sandwich?

EL: A peanut butter sandwich is a great sandwich.

KK: A peanut butter sandwich. But the peanut butter is kind of hard to get going, right? You don’t really want that anywhere except in the middle of the sandwich. You don’t want to imagine this blob of peanut butter. The ham you can kind of imagine.

EL: It’s really saying that you don’t even have to remove the peanut butter from the jar. You can leave the peanut butter in the jar.

KK: There you go.

EL: You can cut this sandwich in half.

KK: Your knife’s going to have to cut through the whole jar. It’s gotta be a pretty strong knife.

EL: Yeah. We’re already asking for an arbitrarily long knife.

KK: Yes.

EL: You don’t think our arbitrarily long knife can cut through glass? Come on.

KK: It probably can, you’re right. How silly of me. If we’re being so silly and hyperbolic, we might as well.

EL: We’re mathematicians, after all.

KK: You’re right, we are. So I thought about the pairing, too. Basically, I’ve got a croque monsieur, right?

EL: Right.

KK: You’re in France. You probably eat these all the time. So what does one have with a croque monsieur? It’s not really fancy food. So I think you’ve got to go with a beer for this, and if I’m getting to choose any beer, we have a wonderful local brewery here, First Magnitude brewery, it’s owned by a good friend of mine. They have a really nice pale ale. It’s called 72 Pale Ale. I invite everyone to look up First Magnitude Brewing on the internet there and check them out. It’s a good beer. Not too hoppy.

EL: OK.

KK: It’s hoppy enough, but it’s not one of those West Coast IPA’s that makes your mouth shrivel up.

EL: Yeah, socks you in the face with the hops.

KK: Yeah, you don’t need all of that.

EL: So actually, if you think of the two pieces of bread as one mass of bread and the ham as its own thing, then you could also bisect the bread, the ham, and the beer with one knife.

KK: That’s right, we could do that.

EL: Yeah, if you really wanted to make sure to eat your meal in two identical halves.

KK: Right. So you have vanilla donuts and balls of chocolate, no, no, the donuts, wait a minute. The hyperbolic spaces were chocolate. This is starting to break down. But the flat geometry is the plane. But there’s a flat torus too, right? So you could have a flat donut, or a flat plane. Very cool. This is fun. I think we’re going to have a good time doing this.

EL: I think so too. And I think we’re going to end each episode hungry.

KK: It sounds that way, yeah. In the weeks to come, we have a pretty good lineup of interesting people from all areas of mathematics and all parts of the world, hopefully. I’m excited about this project. So thanks, Evelyn, for coming along with me on this.

EL: Yeah. Thank you for inviting me. I’m looking forward to this.

KK: Until next time, this has been My Favorite Theorem.

KK: Thanks for listening to My Favorite Theorem, hosted by Kevin Knudson and Evelyn Lamb. The music you’re hearing is a piece called Fractalia, a percussion quartet performed by four high school students from Gainesville, Florida. They are Blake Crawford, Gus Knudson, Dell Mitchell, and Baochau Nguyen. You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at Kevin’s website, kpknudson.com, and Evelyn’s blog, Roots of Unity, on the Scientific American blog network. We love to hear from our listeners, so please drop us a line at myfavoritetheorem@gmail.com. Or you can find us on Facebook and Twitter. Kevin’s handle on Twitter is @niveknosdunk, and Evelyn’s is @evelynjlamb. The show itself also has a Twitter feed. The handle is @myfavethm. Join us next time to learn another fascinating piece of mathematics.

Show notes (click here)