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.


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.


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.


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.

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.


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.

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.

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.


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.

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.


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.


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.

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.


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.