Evelyn Lamb: Hello and welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, and I am joined, as always, by our other host. Will you introduce yourself?

Kevin Knudson: Hi, I’m Kevin Knudson. Yeah, I will. I'm Kevin Knudson, professor of mathematics at the University of Florida. It's been a while.

EL: Yeah.

KK: You know, I've actually gotten a few emails from our listeners saying, hey, where, where the hell is My Favorite Theorem? And I always have to reply, you know, we're trying, but everybody's busy.

EL: Yeah, and we're busy.

KK: And we're busy. But here we are. We are still committed. We're still into this. This is — we're going to go to year eight soon enough, which is kind of mind-blowing.

EL: Yes.

KK: I had less gray and more hair in those days. So here we are.

EL: You’re as lovely as ever.

KK: As are you, Evelyn, as are you. Yeah, although I kind of missed the green hair. I wish you would start coloring your hair again.

EL: Yeah. Honestly, like many people, during COVID, I just lost my ability to put forth more than minimal effort into my appearance.

KK: Yeah.

EL: Sorry, that sounds like a smear on other people. I just, I've heard this from other people. I'm not judging based on what I see from other people. But yes, it's just, like, the bleaching, the dyeing, it just, it's more maintenance than I'm willing to put forth right now.

KK: It’s a whole thing. And as one gets older, you just go, who cares?

EL: Yes. But anyway, we are delighted today to have Robin Wilson on the show. Robin, will you introduce yourself? Tell us where you're joining us from, and a little bit about yourself?

Robin Wilson: Yeah, hi everyone. So I am joining you from Los Angeles, currently in my office at Loyola Marymount University, where I'm a professor in the math department. And so I have been a professor since about 2007 and work in topology and math education. It's great to be here today.

KK: What part of town is Loyola in? I don't think I actually know where that is.

RW: Loyola Marymount is right on the coast, near LAX. So if you've ever visited us here and flown into that airport, then you've flown right over campus, and depending on which runway you land, you can actually, like, see the fountain.

KK: Okay. I'll be flying through LAX in December. I will try to take a look.

RW: Come say hello, yeah.

EL: And I have to say, if we were broadcasting a video of this, you have one of the best backgrounds, the beautiful bookshelf, and then the chalkboard behind you has the appropriate level of mathiness. So our listeners can't enjoy that. But I will say that, you know, it looks very math professor.

KK: It does. It could actually go in that book of math chalkboards. Have you seen this? What's it called? I can't remember. Anyway.

RW: I’m so honored.

EL: I think it’s something like Please Do Not Erase, or something. [Editor’s note: Very close! It’s just Do Not Erase by Jessica Wynne.]

KK: Please Do Not Erase. That’s right, yeah.

EL: But yes, we are so happy to have you here. And Robin and I have actually worked together before a few years ago, on a publication for — at at the time, known as MSRI. It has a new name now, the SL Math Institute now, but we worked together putting together a booklet for the math and racial justice workshops that they did in summer 2021, was it? And it was such a delight to work with you, and I'm glad that lo these many years later, we can get to chat about your favorite theorem. So what have you chosen for your favorite theorem today?

RW: Okay, so the theorem that I've chosen for my favorite theorem today, this was a tough one, and so I chose one that was sort of important for me on my journey. So the Poincare-Hopf index theorem.

KK: Oh, I love this theorem.

EL: All right!

RW: So should I tell you more about theorem?

KK: Please.

EL: Please.

RW: Okay, so the theorem, so I'll state the two-dimensional version of the theorem, which is the one that I can kind of see. So you take a surface and you add a smooth vector field on the surface, and there's an extra condition that the vector field has only finitely many critical points. And so it turns out that the sum of the indices of the critical points of the vector field is equal to the Euler characteristic of the surface. And so the theorem connects these two different areas of math, sort of analytical and topology. And so it was one that I encountered kind of at the beginning of my journey as a topologist. And so it's the one that I picked to share today.

EL: I love that, and I've got to say, I love how many people on our podcast come in and say, it connects this and this. And like mathematicians, we just love these theorems that connect, you know, calculus and topology, or, you know, algebraic geometry and topology, or something like that. It's just something, mathematicians just all love these bridges here. And so I, yeah, can you talk a little bit? You said you encountered it first as a kind of starting out as a beginning graduate student. You know? Can you tell us, bring us back to that moment, tell us about its importance in your life?

RW: Sure. So I was an undergraduate student, and there was a graduate student that was a TA for one of my classes at the time, and I asked him about what type of math he studied, and he drew this picture of a trefoil knot on the board. So shout to Aaron Abrams if you're out there, Aaron. And so I asked him, you know, how could I learn more about this? And he pointed me to a class, an algebraic topology class, that was being offered the next semester. So I signed up for it, and the class was real challenging, but I really enjoyed it. And then the following semester, I got a chance to do a senior thesis, and so I ended up going back to the instructor for that class, and he said yes. So also want to shout out Charlie Pugh for saying yes. And he chose the project that I would work on, the proof of this theorem. And so I'd never, I'd never heard of Poincare. I didn't know much about the historical context of the theorem, but it was — looking back, it was pretty significant that I got a chance to try and think about how to come up with my own proof of something that Poincare had proven, who was right there at the beginning of the field.

KK: And it's a surprisingly tricky theorem to prove. I mean, I was reading this actually, last year sometime. I was trying to remember how this proof goes, because what's remarkable about it is, there's lots of things involved, right? Because no matter which vector field — and it could have, like you said, it can only have finitely many zeros or singularities, but it might have 10 million of them, or it might only have two, but that number, the Euler characteristic, of course, is an invariant of the surface. So no matter how you wiggle this vector field, if you create something, some weird swirl on one side, well, some sort of opposite swirl has to happen somewhere else, effectively, to cancel it out. It’s really remarkable.

RW: Yeah, yeah, that's right. So I was trying to remember what we did to prove the theorem. We, you know, we really were wrestling with a lot of the tools that we used in the course, like, a lot of the details about triangulations of surfaces and trying to find the right ones and paths of vector fields and things that build mature for a very young mathematician back then.

KK: Yeah, yep, yep. So. And I guess one of my favorite corollaries is what, it's the hairy ball theorem, right?

RW: Yeah, that’s right. I was thinking about this. It's got to have, like, one of the worst names in all of mathematics.

KK: Yeah.

RW: So yeah, you can't comb a hairy ball flat without a cowlick, also known as the you can't comb a coconut theorem.

KK: Okay, that’s a little better.

EL: Yeah.

KK: A little little less innuendo, right?

RW: Yeah. That was a close, a close second for my favorite theorem. It's nice that they're connected.

EL: Yeah. We were talking earlier a little bit that, you know, it is hard for people to pick favorite theorems, and, you know, what does it mean if a theorem doesn't immediately leap to mind as a favorite? I just want everyone to know that we are, we might be mathematicians, but we are very not pedantic and mathematician-esque about definition of your favorite. We are very loose, and you know, it can be your favorite of the moment or your favorite for what it meant for your mathematical development. And we’re very imprecise with the definition of favorite on this podcast. All favorites are welcome.

RW: I must say that I had not thought about this theorem for years, until I was asked to find a favorite theorem.

KK: Well, it's sort of like on Instagram, there's this account we rate dogs. Do you know this one? So anyway, basically the guy rates dogs, but the lowest possible rating is 10.

EL: 10 out of 10.

KK: 10 out of 10. Theorems are sort of that way too.

EL: I don't know. I would say, I'm not going to name any theorem names. But I would say there are some theorems that I would put below 10 out of 10. Cancel me if you must. But you know, I’m going to put that out there.

KK: This is it. So we have to start our new Instagram account, clearly.

EL: We Rate Theorems.

RW: 10 out of 10.

KK: That’s right.

EL: Yeah. So another thing we like to do on this podcast is we ask our mathematicians, as if it weren't hard enough to choose a theorem, to choose a pairing for their theorem. You know, be it art, music, food, wine, any delight in life. What have you chosen to pair with the Poincare-Hopf [index] theorem?

RW: So I think I might have actually started with the food and then went back to the theorem. But there was this example that also really like captivated me, captured my attention as a student, and that's the hot fudge flow. So it's a vector field over a surface. And so the idea is to imagine a ball of ice cream, and you do what you do with ice cream. You take the hot fudge and you drizzle it on top of the ice cream, and you try and hit the center. And then what happens to the fudge? It sort of, you want it to expand and wrap around and then come back as a source and drip out of the bottom, if this was, you know, suspended in the air. So that's the hot fudge flow. And you can compute the sum of the indices of the critical points of that vector field, and it'll match of the Euler characteristic of the sphere. So the pairing is a hot fudge sundae.

KK: Okay.

EL: Excellent.

KK: That’s exactly perfect. Yeah.

EL: Of course we have to ask. What is your number one ice cream flavor for a hot fudge sundae?

RW: I was actually hoping you wouldn't ask that I'm the most boring ice cream person. Vanilla is my favorite.

KK: Look, you can't go wrong.

RW: Yeah.

EL: I will say, it is very unfair to vanilla that it has become this word in in our our language, for something that's boring, or pedestrian, because, like, it is an incredibly complex flavor, like, if you get an actual vanilla bean, it's like, there's so much going on. And I don't, I don't know the the history of how vanilla became “boring,” but, you know it is, it is anything but boring. Justice for vanilla.

KK: And so complicated to grow, right? It only grows in very specific places.

EL: A few places. And it’s expensive. Isn’t it, like, the second or third most expensive spice after definitely saffron.

KK: Saffron, I think, is number one.

EL: Maybe something like cardamom. Cardamom is up there too, I think.

KK: It’s not cheap.

EL: No hate to vanilla.

KK: It’s not cheap, because one little pod of vanilla, one little pod at the store is like, $4 or something. You know, it's like, it's really, really absurd. But it's an orchid, right? I mean, so, I live in Florida. We can actually get orchids to grow here, but it's still not easy.

EL: Right. Do you know if the vanilla orchid can grow there?

KK: I doubt it. If it could, they would be cultivating it left and right. I actually think it's too hot here. It's not humid enough, somehow, yeah, so some orchids will work.

EL: Because I think, like Madagascar, Tahiti and maybe Mexican? Is it grown in Mexico also?

KK: I think there might be some spots in Mexico, yeah, like, maybe in southern Mexico, Oaxaca or something. But, yeah, anyway, okay, all right, this is not a vanilla podcast.

EL: Yeah, three mathematicians speak extemporaneously on vanilla cultivation. Tune in next week for the exciting conclusion.

KK: That’s right. Yeah, so Robin, we always like to give our guests a chance to plug anything they're doing. Where can we find you online, what sort of, any big projects you're working on that people might be interested in, or anything like that?

EL: Or have done recently?

RW: Yeah, so I have a really bad online presence right now. At the moment, the website could use some dusting off. But one of the projects that I'm working on that I'm excited about right now is in math education. So we've been making videos of Black mathematicians talking about their work, their educational experiences, and giving advice to young people. And so these are for K to 12 students, but also, I think they're going to be of interest to lots of folks. And so we do have a website, but the URL isn't in on my mind to pass on to you right now. Maybe I could share it with you afterwards.

EL: Yeah, we'll, we'll get that from you and put it in the show notes, so it’ll be easy for people to get.

RW: That’ll be fantastic, but thanks for letting me make that plug.

EL: Yeah, well, and I remember seeing recently, you did a talk at the Museum of mathematics, right with and was that a conversation with Ingrid Daubechies?

RW: It was so much fun. It was a conversation.

EL: Do you know if that is available in video form somewhere? I meant to look for that before we got on. But of course, I didn’t.

RW: You know, I had the same question cross my mind as I was approaching this as well. And I think it might be available, but it could be, like, for museum members.

EL: Okay.

RW: I need to check.

EL: Yeah, I remember seeing your saying your name in my inbox, and thought, well, that's cool. And you've also, do you mind talking a little bit about the Algebra Project and and Bob Moses?

RW: Sure.

EL: Because I know that's something that you've — I know I've talked with you about it before, and Bob Moses passed away around the time we were putting that book together.

KK: Yeah, it was a couple years ago.

EL: So, yeah, do you mind talking a little bit about it? I thought it was really interesting.

RW: Yeah, sure. That's something that I could talk about for a long time. So just just check me if I start going on too long. I met Bob Moses as a graduate student, and I think I was kind of wrestling with some identity issues about my interest in math, but also, you know, interest in social issues, and kind of wanted to make a difference in my community, and trying to figure out how these two things came together, and if I was doing one, did that mean that I couldn't do the other? And so I came across his book, Radical Equations. It was about math literacy and the civil rights movement, and he brought his work in the civil rights movement together with his work as a math teacher in in Boston, and it really kind of spoke to me. And so I got a chance to meet him, and ended up staying connected with him and Ben Moynihan at the Algebra Project, and so I worked with them in different ways, attending teacher professional development. We helped spearhead an effort in Los Angeles, where the Algebra Project curriculum was used in four different high schools supported by an NSF grant. We had a second effort here, where we've been running some summer programs for students through the Algebra Project. And recently I joined the board of directors, and so I’ve been involved with them since I was in my 20s, and so it was a real honor to be asked to kind of be a part of that, that part of the leadership for the project.

KK: Bob Moses really, really impressive man. And then, this idea that you know, that every you know, things are really important. You know, education is so important to advancing, you know, civil rights and things like that. I mean, Bob Moses was really spectacular. Our listeners, if they don't know much about him, should just look him up, because he was really impressive and influential, and by all accounts, a very kind man. Like I said, I've never met him, but just a really great human being.

RW: And I think what people, a lot of people don't know about him, is he was a math teacher first, he was teaching math and and the sit-in movements happened, and he got drawn into the sit-ins. And then when, when things kind of settled down, he went right back the math classroom. And so kind of think of him as one of us.

EL: Yeah. I think reading, reading radical equations a few years ago, I remember, you know, it's just like sometimes when you're a mathematician, especially if you're really involved in the academic math world you get so, you know, drawn into these very abstract questions that you feel like have nothing to do with, you know, anything resembling reality, or anything resembling social issues, and just the way that he writes about how access to good math education, like is so important for people to be prepared to, you know, have careers that they want, be able to have financial stability in their lives then, and just the, you know, the doors that it opens to have access to math at, you know, the middle school, high school level, really reminds you as a mathematician, like, oh, yeah, we are part of this society.

RW: Yeah, that's right, and we do have a really important role to play. That's one of my biggest takeaways from him that as mathematicians, we do have a really important role to play in how this whole thing turns out.

EL: Well, thank you so much for joining us. Really great to talk with you again.

RW: Thank you so much.

[outro]

On this episode of My Favorite Theorem, we had the pleasure of talking with Robin Wilson, a mathematician at Loyola Marymount University, about the Poincare-Hopf index theorem and the importance of math education. Below are some links you may enjoy after the episode.
An interview with Wilson for Meet a Mathematician
More on the Poincare-Hopf index theorem
The 2021 SLMath Workshop on Mathematics and Racial Justice and its follow-up, to be held in May 2025
Storytelling for Mathematics
The Algebra Project
The 2025 Critical Issues in Mathematics Education workshop, to be held in April 2025, focusing on mathematical literacy for citizenship

Evelyn Lamb: Hello and welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb, a freelance writer in Salt Lake City, Utah, where it is gorgeous spring weather, perfect weather to be sitting in my basement talking to people on Zoom. This is your other host.

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. I don't know, Evelyn, I saw the pictures on Instagram over the weekend and it looked cold in Utah. You wrote that you rode a century, right?

EL: Metric century.

KK: Okay. Metric.

EL: Just in case — you know, I don’t want people to think I'm quite that hardcore. Yeah, at least at this point in the season. Yeah, I hadn't managed to ride more than about 25 miles since last fall because weather, travel, just things conspiring against me. The week before I was like, I really need to get 30 or 40 miles in on Monday. And then it was, like, 20 mile an hour winds and sleet and I was like, well, I guess I'm just going into this cold, but it was fine. It was actually gorgeous weather. It was a little chilly at the start, but better than being too hot.

KK: Well, you know, the muscle memory takes over, right? So you can do — I mean, 62 miles isn't that much more than 25, really, once you have the legs, so congratulations.

EL: Yes, thank you. Well, we are delighted today to be joined by Kate Stange, who is in the Mountain Time Zone, something that I always feel thrilled about because I'm constantly converting time zones when I'm talking to people, and finally, someone I didn't have to do that for! So Kate, other than being in the Mountain Time Zone, what would you like to tell us about yourself?

Kate Stange: Oh, geez. Um, well, I'm also a cyclist. And so I'm jealous hearing about your rides.

EL: Wonderful!

KS: Here in Colorado we have we have this ride called the The Buff Classic. And so it has a 100 mile option where they close Boulder Canyon so that you can bike up the canyon without any cars.

EL: Oh, wow.

KS: Then you bike along the peak to peak highway. It's just wonderful.

EL: Yeah.

KK: Yeah. That sounds great.

EL: So you're at CU Boulder?

KS: Yes. And it's run from the campus. It starts right outside the math department.

EL: Oh, perfect. Yeah, just drop your stuff in your office and hop on and ride it?

KS: Yeah.

EL: Yeah, great. Well, we are thrilled to have you here today. And I guess we can just dive right in. What you're on what do you like to tell us about?

KS: My favorite theorem, at least for today, is the bijection between quadratic forms and ideal classes.

KK: That’s a lot of words.

EL: Yeah, and I'm so excited to hear about this, because I am honestly a little nervous about both quadratic forms and ideal classes, and a little embarrassed about being nervous about quadratic forms, not so much with ideal classes. So yeah, can you tell us a little bit about what that all means?

KS: Yeah, sure. So quadratic forms is probably what sort of comes first in the story, at least sort of the way that the mathematics tells it, and also probably the historical way. And so a quadratic form is just a polynomial with an x2, a y2 and an xy. So it's like 3x2 + 7xy − y2. So that's quadratic form. And, as number theorists, one of the things that we're most interested in studying is what are the integer solutions to polynomial equations? And so first you start with linear equations. And there's a wonderful story to do with Euclidean algorithm and stuff there. And then you move on to quadratic. And really, these are sort of some of the first equations that you would start studying next, I guess. And so they go back to the classical days of number theory, Gauss and Euler and everybody. And, yeah, so they come in, what happens is that they come together in families. So different quadratic forms, you can actually just do a change of variables. And it'll look different, but it won't really be too different, particularly if you're interested in what numbers it can represent when you put in integers. So say I take x2 + y2, which is the simplest one, if I put in various different integers to that I get various different integers out. And then if I do a change of variables on that, just a little change of variables — like maybe I change x to x + y, but I leave y alone — the formula will look different after I do that change of variables, but as I put in all integers and look at all the stuff I get out, those two sets, the in and out, they're going to look the same. And so we kind of want to mod out, we want to ignore that difference. So I'm really thinking of equivalence classes of quadratic forms. So that's the first object.

EL: And that change of variables is kind of the only equivalence class thing that happens with them?

KS: Yeah. Yeah. Because they could really behave differently between the different classes.

KK: And you only allow a linear change of variables, right?

KS: Yes, exactly. Yes. Thank you.

EL: Yeah. Okay. So now, ideal classes.

KS: Now ideal classes. So this is an interesting one, to describe where it comes from, I think. So there's sort of — if you think about the history of math, I would say there are sort of two versions, there are sort of two histories of math at the same time. There's one, which is sort of the human history, which is fascinating and human and quirky. And then there's sort of the way that the math would like to unfold to human understanding the way that as any human coming to it, they might discover the pieces of the mathematics. And I don't know too much about the details of the human history. But in terms of how you might discover this, if you're just looking at the integers, you are interested in how they behave, you discover things like prime numbers, you've got addition, you've got multiplication, you've got powers, you might ask how these things interact. And at some point, maybe when complex numbers are discovered, you think about whether there are possibly other collections, or other number systems, other collections of numbers in which you could do the same kind of thing. And so one of the first examples of this would be what's called the Gaussian integers, which is where you take complex numbers, I don't know whether I should dive into complex numbers, but you take complex numbers with integer coefficients.

EL: So that means things like 1 + 2i or something. So the i and the 1 both have integers in front of them.

KK: Right.

KS: Yeah, exactly. And so this is a collection of integers, kind of, right? And we ask things like, okay, are there prime numbers? And so it turns out that there are in that system, there are Gaussian prime, so, like, 1 + i is a prime number. And so you kind of start to develop this whole theory that you have for the integers. But what you find is that in some of these systems, you lose unique factorization. So we love unique factorization in the integers, right? Every integer, up to reordering the factors and maybe putting on a minus sign, you have always a unique factorization into prime numbers. And in the Gaussian integers, that's true. But in some of these other systems, you lose that. And so what people tried to do is to try to fix it. And it turns out, the way to fix it is to add in what were I think, originally called ideal numbers. They were thought of as numbers that should be in the system that weren't in the system. And what they actually were were collections of numbers. They were sets of numbers instead of individual numbers. And the idea here is that, say, you were to take — like in the integers, if you took the number two, you could replace that idea, that idea of two-ness with just the collection of even integers. And so that's an ideal now. Instead of a number, it’s an ideal, and it's really carrying the same information. But now it's a subset. And so by moving from individual elements of the ring, of the collection of numbers, you move to subsets of them. Now you have more things, and so now you can recover unique factorization in that world. So those are ideals.

EL: Yeah. And so the Gaussian integers do have unique factorization.

KS: They do. Yeah.

EL: So this — actually, I kind of forgot, but recently, this came up in something I was writing where I wanted the example to be the Gaussian integers so bad because it doesn't have any square roots in it. But then it didn't work because it isn't true for that. I was trying to show how unique factorization could fail, but I didn't want to have to use square roots. But as far as I know, you can't do that. So then I fixed it by putting a square root of negative five in there and hoping that people would be patient with me about it. But yeah.

KK: So that's the example of one where you don't get unique factorization, right? So you take the integers and you join the square root of minus five. That's one example.

KS: That’s one of them. Yeah.

EL: And then it's like two and three are no longer primes.

KS: So if you multiply (1+ √ −5) × (1− √ −5)

KK: You get six. Yeah.

KS: You get six, which is also two times three. And those are two different prime factorizations of six.

KK: Right.

EL: Yeah, but it's so fun that you can do that, and I like your way of putting it where regardless of how these ideas actually formed, you can as a human, looking at some of these basic pieces like primes and then or primes and integers and square roots and things, you can kind of come up with this, like, what happens if I do this? And create this new thing where this this property that I know I always assumed — like unique factorization, when you're growing up, you know, when you take math classes in school and stuff, it just seems like so basic, like, how could you even prove that there's unique factorization? Because how else could you factor anything?

KS: Yeah, exactly.

EL: It feels so basic.

KS: Yeah. And this is what happened, I think, historically, too, is that people didn't expect it to fail. And so they were running into problems and it took a while to figure out that that's what was going wrong.

KK: Wasn’t this part of, was it Kummer who had a reported proof of Fermat's Last Theorem, and he just assumed unique factorization?

KS: That’s what I've heard, although I never trust my knowledge of history. Yeah.

KK: It’s probably true.

EL: Well, and there are a lot of good stories. And they may or may not be true sometimes. But yeah, okay. So we've got these, these two things.

KK: Yep.

EL: The quadratic forms and the ideal classes. So yeah, I guess either historically or mathematically, what is this connection? And how do you know that these two things are going to be related?

KS: Yeah, so they seem like different things. So I think quadratic forms were studied earlier. And at some point, people noticed that quadratic forms had an interesting property, which is that sometimes you could multiply them together and get another quadratic form, which is kind of hard to explain. But like, if you actually wrote out (x2 + y2) × (z2 + w2) and you multiplied that all out, you'd have a big jumble. But then you could factor it out. So it looked like, again, a square with some stuff inside, z’s and w's and whatever inside the brackets, plus a square. And so this meant that sometimes if you picked your forms correctly, and they had this sort of relationship, then if you looked at the values they represented, the numbers that can come out, when you're putting integers in, you would take that set of things the first one represents and the set of things the second one represents, and then you’d look at what the third one represents, and it would represent all of the products of those things. So there was this definite relationship, but the way I'm describing it to now is a little awkward, because it's a lot of algebra. But this is, I think, what was noticed first, somehow. And again, I might be mixing the human story with how math tends to want to unfold. I don't know exactly the history. But anyway, so you notice that there's this relationship. And that's kind of reminiscent of an operation, like a multiplication law. And what happens is that, in fact, that's coming from the fact that these ideal classes, each one of them — sorry, my mistake — so it's from the fact that each of these equivalence classes of binary quadratic forms, each one of them is associated to an ideal. And the ideals as the sort of generalization of the idea of number, they can be multiplied together to get new ones. And so on the ideal side, it makes sense that there's an operation because you're already living in a number ring where you've gotten an operation. But on the quadratic forms side, it's a surprise. And so that's one of things I like about this theorem is that you see some structure and you want to understand why. And the reason to understand why is just to change your perspective and realize these objects can be viewed as a different kind of object where that behavior is completely natural. Yeah, so that's one thing that I like about it.

EL: And does this theorem have a name or an attribution that you know?

KS: Oh, it's such a classical theorem that no, I don't know.

KK: Right. It's just the air you breathe, right? So what's the actual explicit bijection? So you've taken a quadratic form. What's the corresponding ideal?

KS: Well, actually, the other way is a little bit easier to figure it out.

KK: Yeah, let's go that way.

KS: So let's take the Gaussian integers, okay. And in the Gaussian integers, you've got — for your ideal, so think of it as a subset of the Gaussian integers. But because it's an ideal, it has the property that it has the same shape as the Gaussian integers. I actually usually like to draw a picture. So I'm going to try to draw a picture just out loud. So if you think of the Gaussian integers in the complex plane, they fill out a grid, right? It's all the integer coordinates in that plane. So that's a grid. And if you want to see what the ideals are, they’re subsets that are square grids as well, but fit inside that grid that we started with, maybe rotated or scaled out.

EL: Okay.

KS: But they're square again.

EL: Okay.

KS: And so, what you can do is with this example, specifically, you can take the norm of each of these elements in the Gaussian integers. So the norm of a complex number, usually I think of it as the length from the origin. But I don't want to do the square root part. So if I have a Gaussian integer x + iy, I'm going to take x2 + y2, and that's the norm.

KK: Okay.

KS: All right. And so if I take the whole Gaussian integers, which is itself an ideal, that's one of the subsets that is valid as an ideal, then if I take all of the values, all the norms of all those elements, that's all the values of x2 + y2. So from my collection of integers, I take all of the values and that's actually a quadratic form.

KK: Okay.

KS: Okay?

KK: Okay.

KS: And so you can do this with the other ideals as well. So for each one, you look at the norms of all of its elements, and that is a quadratic form and the values of that quadratic form?

KK: Right. So the Gaussian integers are Euclidean, right? So it's PID, right?

KS: It is. It’s a principal ideal domain.

KK: So everything's generated by one element, basically every ideal?

KS: That’s right.

KK: So that makes your life a little simpler, I suppose.

KS: Yeah. So the ideals, in that case, really, they're not so different than the numbers themselves. This is one of those ones where you don't have to go to ideals. But by going to it, you think about instead of just, say, 1+i the number, you think about all the multiples of 1+i and you take all of those, and you take their norms.

EL: Okay. And I told you, when we were emailing earlier, that you'd have to hold my hand a little bit on this. So yeah, sorry, if this is a too simple question or something. But like, what is the quadratic form like the x2 + whatever xy +whatever y2 that you get from the the Gaussian integers that you just said?

KS: Right. So if we take the Gaussian integers, if I take x+iy as a Gaussian integer, its norm is x2 + y2. That’s the form right there.

EL: Okay. Yeah. All right.

KS: And then if I were to take a subset, like all the multiples of 1+i, I'm not plugging in all x's and y's. I'm plugging in only multiples of 1+i, so you end up with a slightly different form popping out.

EL: Yeah, so I guess it's kind of like x+x then.

KK: 2x2 squared basically, right?

KS: Yeah. Yeah. You could have Yeah, various things in various different situations, but yeah.

EL: Okay, thank you. Yeah. And so, yeah, can you talk a little bit about how you encountered this theorem? If it was something that like you really loved to start out with? Or if your appreciation has grown as you have continued as a mathematician?

KS: Yeah, well, it's one of these things, so I think everybody has things that they're attracted to mathematically, they all have a mathematical personality. And there's some sort of particular kinds of things that attract you. And for me, one of the things is the sort of projection theorems that tell you that a particular structure, if you look at it a different way, has a whole different personality. And it's actually the same thing, but it has just become totally different. So I really love those things. And I've always loved number theory, because it has such simple questions. But then when you dig into them, you always run into such fascinating, complex structure hidden. And so this is one of those things that if you have that kind of personality thing you, just keep bumping into. And so for me, and all of the research that I've done and things I've been interested in, I keep coming back to this theorem and bumping into it in different places. It shows up when you study complex multiplication of elliptic curves, it shows up when you study continued fractions, it shows up all over the place. And it just seems so fundamental. And it's sort of like maybe one of the most fundamental examples of this special kind of theorem that I really enjoy.

EL: Okay.

KK: Cool.

EL: All right, well, then the next portion of the podcast is the pairing. So yeah, as you know, we like to ask our guests to pair their theorem with something that helps you appreciate the theorem even more. What have you chosen for that?

KS: So, when I think about this theorem, I just it's a treat. So I think the only thing that comes to mind really over and over again is just chocolate. I love chocolate. And that's what you should enjoy this theorem with because maybe you should just be happy enjoying it.

KK: I mean, chocolate pairs with everything.

KS: That’s true. It's a bit of a cop out.

KK: No, no, that’s okay. So our most recent favorite chocolate is Trader Joe's has this stuff. And it's got, I don’t even know what’s in it, pretzels and something else crunched up in these like bark of chocolate. And it’s a dark chocolate I really recommend it. So you must have a Trader Joe's in Boulder, right?

EL: Are you a dark, milk, or white chocolate person?

KS: Oh, definitely dark. Yeah, I have a dark chocolate problem, actually.

EL: Yeah, the Trader Joe's. For me the dark chocolate peanut butter cups are are always purchased when I go to Trader Joe’s.

KK: Dark chocolate feels healthier, right? It's got more antioxidants and a little less sugar. So you're like this is fine, less milk. Okay. All right. It's actually it's a fruit, right?

EL: It’s a bean. You’re having a black bean pate right there.

KK: That’s right.

EL: Yeah, well, Salt Lake is actually a hub of craft chocolate. We have some really wonderful chocolate makers here, like single origin, super fancy kind of stuff. So if either of you are here, we'll have to pick up some and enjoy together. And yeah, along with quadratic forms and ideal classes.

KS: Sounds wonderful.

EL: Yeah. So something I meant to talk about this earlier in the episode, but you mentioned that you'd like to illustrate things, and that is how we first met is, through mathematical illustration. So I don't know, maybe it's a failure of imagination on my part, but I always, I'm always fascinated by like, number theorists who are really into illustration as well, because I think of, like, geometry, as you know, it shapes it as the more naturally illustrate-y parts of math. But would you talk a little bit about it, you know, illustrating number theory? And if if you've done anything related to this particular theorem, or if there's something else you want to talk about with your mathematical illustration?

KS: Oh, yeah, that's a that's a great idea. Yeah. So there's actually building up gradually a wonderful community of people who are interested in illustrating mathematics. And so that's maybe one of the things that you could add a link for is the website for the community.

EL: Definitely.

KS: Yeah. And so I've always found that the way I think about mathematics is very visual. I mean, I think as human beings, we have access to this whole facility for visual thinking, because we're embedded in this three-dimensional world that we're living in. And another way that we think about mathematics, I think often, is we're using another one of our natural facilities, which is our sort of social understanding facility, where we imagine characters interacting with each other and having motivations and stuff like that. But for me, it was always a very visual thing. And so even though it wasn't taught in that way, in my mind, somehow these things were always very visual things. And so I've always been really attracted to situations where you can see some hidden geometry in number theory. And with this particular theorem, there is a little bit of nice hidden geometry. I mean, the first hint of this is that when I talked about ideals in the Gaussian integers, I visualized them as a lattice.

EL: Yeah.

KS: And in all of these number rings, you can do this, you can you can think about lattices. And you're really talking about lattices, and lattices have things like shape. And you know, there's lengths and angles and stuff like that to talk about. And so one of the really cool things that you can do is you can think about, for example, with the Gaussian integers or with some other ring of interest that you can put in the plane like this, into the complex plane, then you can ask this question, it's a natural question that people ask: how can I study the collection of objects instead of the individual objects themselves? So if you want to study the collection of lattices, say, two-dimensional lattices in the plane, then one way to do it would be okay, how do I decide on a lattice? Well, I have one vector that's generating it, and then another vector that's generating it. So let's put the first one, let's sort of ignore issues of scaling and rotation, let's put the first one down pointing from like zero to one. And then the other one is somewhere, but now you don't have any choice anymore. No more freedom. And so you can think of the plane itself as a sort of moduli space, as a parameter space for the collection of lattices. And this space has a lot of beautiful properties. So you might as well order your vectors so that we're just talking about the upper half plane. So the first vector is from zero to one, and the other one is an angle less than 180 degrees from that. And so when you start looking at the geometry of this plane, and you want to talk about well, okay, I want to look at lattices, but maybe I don't care so much about what basis I'm using for the lattice, you start to divide the plane up in certain ways. And what you discover is that the natural way to talk about this plane is using hyperbolic geometry, actually. And so all of a sudden, you're doing hyperbolic geometry. And I find myself doing hyperbolic geometry sometimes when I'm doing number theory, because when I want to look at these these objects and stuff, that's just the natural world in which they live. I mean, the mathematics kind of tells you what you have to do you. You know?

EL: Yeah.

KS: And so those are moments that I really enjoy, because you're doing something that you think is just some algebra, but all of a sudden, it turns out it's geometry.

EL: Very cool. Yeah. So we will include a link to your website, which I know has some of the cool illustrations that you've done available there. And to the illustrating math, there's an online seminar that meets monthly that is really nice to go to, if you can. And, yeah, it's a lot of fun. And I, yeah, just so many different fields of math represented with that in ways that I never would have guessed.

KS: That’s true. That's one of the nice things about that community is that there's people from all different areas that you wouldn't normally interact with, because usually you have a pretty narrow research circle, if you're doing research in mathematics. But there, you're talking to everybody. And it has a much more creative feel for that reason, you get surprised by people's ideas, because they come from just a little bit farther from your home base, you know?

EL: Yeah. And I think it also kind of pushes people to really think about how they're explaining things, where you have a shorthand when you're working with someone who is in, or talking to someone who's in such a close field, and since you don't necessarily have that same common background, people, I think, it seems like are very thoughtful about how they describe things and what they assume that you already know.

KS: Yeah, exactly. It's just good to get out of your little corner.

EL: Yeah.

KK: All right. Well, this has been great. I definitely learned something today. I did not know this connection between ideal classes and quadratic forms.

KS: Oh, I thought of one more, one book I'd like to plug.

KK: Okay. Please do.

EL: Great. Yes.

KS: Yeah, so Martin Weissman has written a book called — I'm going to get the title slightly wrong. It's An Illustrated Theory of Numbers, maybe? Oh, you have it. Oh, I got it right.

EL: Yeah. An Illustrated Theory of Numbers. It's been holding up my laptop, after I read it, I will say.

KS: Yeah, and so you were asking about illustrating number theory, and this is just a beautiful book. It's completely accessible. I used it when I was teaching an introduction in number theory for undergraduates. But it doesn't require any particular background because it starts from, you know, we’ve got the integers, we’ve got addition, we’ve got multiplication, let's do some stuff. And and he really looks hard for ways to turn theorems which are usually completely algebraic into something visual, and they're just lovely.

EL: Yeah, and really amazing illustrations, and full color, like everywhere, which I know is more expensive to make books, and that's why books so often have the color in the middle, like in a little section and not the whole thing. But I do think this is just much more pleasurable to read because it it is does use that aspect, too. And it's not as stark as every page being black and white.

KS: Yeah, it's so inviting. It's a wonderful book.

EL: Yeah. Great recommendation. Thank you so much for joining us. I really enjoyed talking with you.

KS: Yeah, me too. Thank you so much for having me on.

[outro]

For this episode, we were excited to talk to Kate Stange from the University of Colorado, Boulder about the bijection between quadratic forms and ideal classes. Below are some links you might find interesting as you listen.
Stange's website
The Illustrating Mathematics website and seminar, which meets monthly on the second Friday
An Illustrated Theory of Numbers by Martin Weissman
The Buff Classic bike ride in Boulder

Kevin Knudson: Welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I am joined, as always, by my fabulous co-host.

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, trying to remember how to do this. It's been a minute since we've recorded one of these. We kind of went dormant for the winter.

KK: Yeah, a little bit, a little bit. Yeah. But Punxsutawney Phil told us — I don’t, what did he say? Let's pretend he said six more weeks of winter.

EL: I think he usually does. I don’t know.

KK: I mean, objectively, there are always six more weeks of winter. Like, the calendar says so, right?

EL: Yeah.

KK: Anyway, yeah.

EL: And, you know, he probably is pretty good at seeing shadows if he's a prey animal because he'd be used to seeing, like, a bird coming overhead.

KK: That’s an interesting question.

EL: Do birds eat groundhogs?

KK: That’s what I was going to wonder. I mean, like, eagles, maybe, but groundhogs are pretty large, right? I mean,

EL: Yeah. What eats groundhogs?

KK: Well, that's something to investigate later.

EL: Yeah.

KK: So it is Pi Day, right?

EL: It is! Well…

KK: We’re actually, we're recording this on Pi Day. When our listeners hear this, it won't be, but we're recording.

EL: And, I always have to put in a plug for my calendar.

KK: That’s right.

EL: The AMS math page-a-day calendar on which Pi Day does not occur on this day.

KK: That’s right.

EL: There are other Pi days on this calendar, none of which is this day, my little joke here. So you can find that in the AMS bookstore.

KK: Right. Are you Team Pi or Team Tau?

EL: I’m Team whichever one works for the calculation that you’re doing. It’s not that big a deal.

KK: That’s right. That's right. Okay. All right. Enough of us, enough of our useless banter, although we did discuss what's the ratio of banter to actual talk, right, that there's, there's like a perfect ratio. But we are pleased today to welcome Karen Saxe. Karen, why don't you introduce yourself and let us know all about you?

Karen Saxe: Hi, there, everybody. So first of all, happy Pi Day. If listeners know who I am, I was a professor at Macalester College for about for over 25 years. And then about seven years ago came to work at the American Mathematical Society, where I am very happy to be the director of the Government Relations Office. So I work in DC with Congress and federal agencies. And could quite a bit about this. I'm also happy to be here because it's Women's History Month. And it will be appropriate that it is Pi Day when you hear what my favorite theorem is.

KK: Okay, good to know. So, I'm curious to know more about this government relations business. So I mean, I know that the AMS does a lot of work on Capitol Hill, but maybe some of our listeners don’t. Can you explain a little more about what your office does?

KS: Yeah, so we do a lot of things. So first of all, we communicate — I sort of view the work of our office as going two ways. One is to communicate to Congress why mathematics is important to almost everything they make decisions about, you know, our national security, health care, you know, modeling epidemics, thinking, like you’re in Florida, thinking about how to model severe weather and things they care about, and then why they should fund fundamental research in mathematics and all sciences. And then also you know, how they make decisions about education. So we tell Congress, we give them advice and feedback on our view about what they should do in those realms. And then on the sort of flip side, I tell the AMS community, the whole math community about what Congress is doing and what's happening at the agencies like the NSF, and Department of Defense and Department of Energy, that that they might care about things, things that would affect their lives. So that’s sort of it in a nutshell. I spend a lot of time on the hill. I just came this morning, I went to a briefing put on by the National Science Board, which is the presidentially-appointed board that oversees the NSF. And they put out a congressionally mandated report every few years on the state of, it's called the indicators report. I'm sure I found it more interesting than everybody else, but it's pretty fascinating. You know, it covers everything from publications around the world, like which countries are are putting out the most science publications, what the collaborator network looks like around the world, and that to sort of US demographic information about education, you know, who's getting undergraduate degrees? Who's getting two year degrees? Who's getting PhDs, that that sort of thing. It covers a lot, actually. Pretty interesting.

KK: Yeah, yeah. All that in like two hours, right, and then it's over.

KS: Yeah, all that in two hours. And then they give you the big report that you can. And I've got them sitting in front of me. But given that this is a podcast, showing things doesn't work.

KK: Well, we do it all the time.

KS: Here’s one of the reports I picked up this morning. Actually, one really, so they're, you know, they're one thing. And you might end up cutting this, but one thing that's sort of fascinating to me is they always list barriers for getting into STEM degrees. And you know, there are things listed, like college accessibility, things that — and even going back. So like, you know, school kids who say they don't have science teachers in their schools, they don't have math teachers, but they've added to this list. “I can't support my family on a graduate student stipend.” So this is something.

EL: Yeah.

KK: That’s real.

KS: And we are, we've endorsed a bill in Congress that would look that would help to improve the financial stability, I guess, you would say, or the ability to be a grad student or a postdoc. So it's looking at stipends, it's looking at benefits, you know, leave time, all that sort of stuff, making it a job that you can choose to take when you're 23, and have a family to support and could make a hell of a lot more money doing something else with a math undergraduate degree.

EL: Yeah, and not see it as something where it's like, you're kind of putting off real life for a little longer, which I think maybe in the past was more of the model, like, oh, yeah, you'll have a real career later. But you know, in your mid-20s, you'll just keep being a student and not have kids or, you know, things, you know, not have parents to support or things like that.

KS: Exactly.

KK: Yeah. Okay. That's, that's good to know. Thank you for all that hard work you do, Karen. So but this is a math podcast.

KS: Right.

KK: So what’s your favorite theorem?

KS: Okay, so first, I'm going to tell you about the three theorems that I didn't choose.

KK: Cool.

EL: Great.

KS: So — I'm sure everybody goes through this — and thinking about my research, it would probably have to be the Riesz-Thorin interpolation theorem, which basically tells you that if you've got a bounded linear operator on two Lp spaces, then it's bounded on every Lp space in between those two values of p, so I used that all the time when I did research on that sort of thing. Then, but I was primarily a teacher of undergraduates, and kind of my two favorite theorems to teach are always Liouville’s theorem and, and then the uncountability of the real numbers.

EL: Yeah.

KS: And Liouville, they’re the one that says, you know, that there's a bounded — if you have a bounded entire function function, it's got to be constant. And the result is so stunning, and it gives a great proof of the fundamental theorem of algebra, that every non-constant polynomial has a root. So I always love teaching that. And then of course, like, Cantor’s diagonal argument about the real numbers, nothing beats that proof in terms of like, cool proof, in my opinion.

EL: Yeah. All-time great.

KS: Yeah, all-time great, right. And I think it's been mentioned on your podcast before. But what I picked was this theorem that says that if you have a given fixed perimeter, then the circle maximizes the two-dimensional shape you can make, so the isoperimetric theorem.

EL: Nice! And as you said, very appropriate for Pi Day.

KS: Yeah, which, I hadn’t even thought about that, which is sort of also embarrassing. But until we started acknowledging Pi Day, I hadn't thought about that. So another way to say it, or the way you might see it in a textbook, is if you have a perimeter P and an area A, then P2−4πA is greater than or equal to 0, with equality if and only if you have a circle. So this theorem has a very long, fascinating history. Lots of great applications. And for all those reasons, I love it. I love history.

KK: Yeah.

KS: I love math.

KK: Yeah. Do you have a favorite proof of this theorem?

KS: I do, actually. Yeah. Well, I didn't know you'd ask that. So there are a lot of proofs. And the one that I like, and this comes from being an analyst probably, is in the early 1900s. Hurwitz gave a proof using Fourier series. I love that proof. And proofs are quite old, going back thousands of years to the Greeks. And then in 1995, Peter Lax actually gave a new short calculus-based proof. But I like the Fourier series proof, just because I like Fourier series.

EL: Yeah, that's a topic that I wish I understood better. Somehow I kind of missed really, ever feeling like I've really got my teeth into Fourier series. Maybe that's a little embarrassing to admit on a math podcast.

KK: I don’t know. I took that one PDEs class as an undergrad and, like, that's where you see it, you know, doing the — whichever, the wave or the heat equation, whichever one it is — maybe both? I don't know. And then that’s it, that shows you how much I remember, too.

KS: Yeah. Good. So you're not gonna dare ask me to give you that proof or anything?

EL: Yeah, generally, a proof like that on audio is not the ideal medium.

KS: It doesn’t work.

EL: Actually, you brought up these ancient proofs. So yeah. Yeah, I guess how long has humanity known this fact, do you think, or do you know?

KS: So it's considered that the Greeks knew the proof. And then it was proved around 200 BCE. It even features in Virgil's version of the tale of Dido, Queen Dido.

EL: Oh, that’s right.

KS: So yeah, I think that was around 50 or 100 BCE, after the Greeks knew the theorem. So can I say what that story is?

EL: Yeah.

KK: Yeah, please.

KS: So she apparently fled her home after her brother had killed her husband. Okay, so we're already in an interesting phase. She somehow ended up on the north coast of Africa after that, and she was bargaining to get some land. And they told her, oddly, that that somehow she could get as much land as she could enclose with an oxhide.

KK: Okay.

KS: And so she took this oxide and cut it into very thin strips, and then enclosed an area, that was the largest she could conceive of, with the given per perimeter.

KK: Okay.

KS: So there's that. So it appeared, like, 2000 years ago, or more, and then you sort of we sort of jumped into the early 1800s when Steiner gave geometric proofs. But what's kind of fascinating is his proofs all assumed that a solution existed. And I haven't looked at these proofs, at least not in a long time. But then later in that century, Weierstrass is credited with giving a proof that, well, first, he proves that a solution does in fact exist. And he did use the calculus of variations to get this proof. So that's, that's sort of the story of the, of the theorem.

EL: Yeah, this actually — you know, we say the Greeks knew this, but I kind of wonder if this is one of those things that humans would kind of intuitively know, even if they're not in a framework where they have language about proving mathematical theorems, even if that's not an aspect of, of their culture, but it seems like you're trying to get into the mentality of like, what is really intuitive or innate about mathematics for humans? And I wonder if that, you know, we kind of would understand, well, if I took a square or something, I could sort of bow it out a little bit, and get a little more area with the same string.

KS: Actually, I mean, one reason I love this theorem is you can give string to kids, and I used to do this, like in elementary schools, and tell them make the biggest shape. And you have to tell them what closed is, no, you have to describe that the string has to come back to where it started. And they all come up with a circle. And this is, you know, second, third grade kids. So it is really intuitive. Yeah. So what it's meant by the Greeks knew this theorem is not 100 percent clear.

KK: Because they didn’t even use pi, right?

KS: And then actually, Evelyn to what you just said, you know, there's something that's quite interesting to me, which is that, you know, if you think about, you know, shapes of constant width, you know what I'm talking about?

EL: Yeah.

KS: So, if you take the fixed perimeter, there's an infinite number of these, the circle’s the largest one and those Reuleaux, I think that's how you say his name, those triangles are the ones of smallest area.

EL: Okay.

KS: And you were just kind of alluding to that, like take a triangle and go puff out the sides, or something.

KK: And you can push in.

KS: Yeah. Right. And you can do it for any regular polygon.

EL: Yeah. Well, British money has a couple of these that are I think heptagons, Reuleaux heptagons? Are they all called Reuleaux? Or just the triangles? I don't know.

KS: No, but you’re right about that, they do. And so it's kind of funny, I saw something that was talking about these points, like, what possessed them to make those points? And if you have a machine that has a hole size, and you know, it could fit a circle, it has a diameter, right, but it can also obviously fit one of these other shapes. Yeah. So that works. And I think you're right. It's a heptagon, heptagonal version of those.

EL: Yeah. The first time I went to the UK, this was, I think, the most exciting things on my trip to me, was these coins. Like, who thought to make these? And I actually, I remember, I wrote a blog post about it and discovered that it was a little hard to figure out if I had the rights to use a picture because all the images of these coins are like, technically property of the Crown.

KS: That’s funny.

EL: Abolish the monarchy, man.

KK: Her Majesty relented in the end?

EL: Yeah, so strange. I was like, well, I'm not gonna beg the queen for the right to post this on my math blog. So I don't remember what happened with that. Hopefully, I'm not opening myself to takedown.

KS: I think you’re probably okay.

EL: Hopefully the statute of limitations has run out on that. Anyway.

KK: I recently came across, I was going through an old notebook, and I found — I don't know why I tucked it in there — from the late 90s. I had one of these 10 Deutsche Mark notes that had Carl Gauss on it.

EL: Oh, nice.

KK: And so I put it on Instagram. And I'm now starting to worry. Wait a minute. Will the German government come after me? Although it's not really legal tender anymore.

EL: Yeah, the pre-2000, whenever they went to the Euro, government.

KK: It was pre-Euro. Yeah, I think I’m safe too.

EL: But anyway, getting back to the math, Karen. So, has this been a favorite of yours for a long time? I guess to me, this is one that I don't think the first time I saw it, I would have been super impressed by it. So what was your experience? What's your history with this theorem?

KS: Right. So like, why did I decide I liked it? Because yeah, it's sort of like, okay, I mean, it's appealing, because everybody can understand it, it’s very intuitive. It's got this, the proof has this interesting history. But why I like it is because you probably know that I'm pretty engaged with congressional redistricting. And when they do measures of compactness of districts, this is the theorem that kind of motivates all their measures.

KK: The Polsby-Popper metric, right?

KS: Yes, exactly. And so you take the Polsby-Popper measure, which was come up in 1991. So like, different states, should I say something about redistricting?

KK: Sure, yeah.

KS: So yeah, I mean, just like the very brief thing is every 10 years, we have to do the census. This is mandated in our Constitution, for the purposes of reapportionment of the House of Representative seats to the state so then after the census is done the seats, which we now have 435 of them, they're doled out to the states. And how that's done is a whole nother you know, interesting math problem, more interesting, probably. But then once the states get their number of seats, like how many in Florida?

KK: We’re up to 27? [Editor’s note: It’s actually 28.]

KS: So let's pretend there's 27 for a minute.

KK: I think that’s right. [Ron Howard voice: It wasn’t.]

KS: Okay. Then, you know, the Florida Legislature, probably, I don't know who does it in Florida, but somebody.

KK: Let’s not talk about that.

KS: Yeah, let’s not talk about that. Whoever’s in charge has to carve up the state geographically into 27 districts, one for each representative, and how they do that geographic carving up is extremely complicated. And to answer the question, “Has this been gerrymandered?” there are certain measures of what's called compactness, and this is like a whole nother thing I could talk for hours on. And compactness sort of measures the lack of convexity, sort of, so like, are there long skinny arms going out? And this is where obviously, like a podcast is, is not the best. But in any case, you know, are there long skinny arms going out, or does the thing look like a circle? So the Polsby-Popper measure tells you how close to a circle, or a disk because it's filled in, but in any case, your district is. Well, that's kind of weird, because if you think about tiling any state with circles, it’s just not going to happen.

EL: Right.

KS: Yeah. So just to sort of fetishize circles is bizarre. But I guess, like, what are your other options? Well, there are lots of other options. But the Polsby-Popper is the most common. There's a handful of states that require specific compactness measures in their process, and many other states that require compactness, but they don't specify the actual measure. In any case, the Polsby-Popper is the most common. And the other common measure is called the Reock measure, and that also fetishizes circles. It's a similar type thing. So with the Polsby-Popper, it's kind of interesting, because they they first published it in a law journal in 1991, in this context for redistricting, but it has actually been mentioned, as far back as the late ‘20s. And can I read you a funny a funny opening line?

EL: Yeah, sure.

KS: So the it first appeared, as far as I know, in a 1927 paper in the Journal of Paleontology. Okay. And how's this for the start of a paper? In quotes: “How round is a rock? This is a question that the geologist is often forced to ask himself.” Okay.

EL: Nice.

KS: So that's a great opening sentence. And then it kind of carries on: “when he wishes to consider the amount of erosion that a stone has received.” And then the paper is actually about measuring the roundness of grains of sand.

EL: Oh, cool.

KS: So there's a lot to say here that the paper is filled with hilarious hand drawings, you know, but also, of course, that geologists seem to be male is another observation.

EL: Yeah, well, and the grammar rules of the time.

KS: Yeah, exactly. But even just this past January, I ran into a paper that was published, and uses this to measure the aggressiveness. It's in, like, a cancer journal. I can't remember which one. And I wrote it down, but of course, what do you know, I can't see it. Anyways — oh, Cancer Medicine is the name of the journal — and it used the Polsby-Popper measure to measure aggressiveness of tumor growth. So you know, it has a life.

EL: That's so so interesting. When you said Journal of Paleontology, I was just like, how is that going to come up in paleontology? But what do you say? Yeah, how round is a rock? It's like, yeah, you do need to measure that. I actually, just the other day watched this interesting video about sand grains and like, certain beaches, or, and certain dunes have different acoustical properties. Due to, like, if they've got a lot of the same sized sand grains and if they pack really well, or if they don't, sometimes there can be the squeaking effect, like when you walk on it, or in a dune, like when there's wind, there can be these like deep, deep resonances, like almost a thunder sound that happens.

KS: Oh, that is interesting.

EL: And this this video went and looked under the microscope at the sand on these different beaches, and kind of showed how some of them packed together better or worse, and some of them are more uniform. So they might secretly be using that metric.

KK: They might.

KS: That’s fascinating. I mean, I heard I've heard that squeaky sound on beaches.

EL: I never have I'm not a huge beach person. So I guess, yeah, but I'm curious about going to one of these beaches someday now.

KS: Yeah. And when you said that I was thinking of the packing, like how they pack, but that would have to do with their shape, and their size. Well, I don't know.

KK: So this is a sphere packing question now. And it's yes.

EL: Or a “how sphere-y is your sphere”-packing question.

KS: How spherey is your sphere?

EL: Not quite as catchy.

KK: Right. So the other part of this podcast is we like to ask our guests to pair their theorem with something, so what pairs well with the isoperimetric inequality?

KS: So naturally, you know, a mathematician would ask, are there analogs in higher dimensions? Right? And then back to how spherey is your sphere, so I play tennis quite a bit. So I'm going to pair it with tennis.

EL: Excellent.

KS: The shape of the ball abides by the theorem.

KK: Yes. Right.

KS: And works for so many reasons.

EL: Yeah. Well, and you are not the the first My Favorite Theorem guest to pick tennis, actually.

KK: That’s right. Yeah.

EL: Yeah, we've had Dr. Curto.

KK: Carina.

EL: Yeah. Carina Curto, paired paired hers with tennis. It was it was about linear algebra. That's right. Yeah. Yeah. Hers was about how this thing kind of goes back and forth. When you're doing this thing in linear algebra. So you picked different aspects of tennis to pair with your theorem.

KK: Yep. Do you play much do you, you play, you play a lot?

KS: I play — it’s embarrassing to put on a very well listened-to podcast — that I do play a lot, because I don't know how good I am.

KK: That doesn’t matter.

KS: But I play a couple times a week.

KK: I used to play quite a bit. So as a teenager, certainly. And then in my 30s I played a lot. I played a little league tennis. This is when I lived in Mississippi. And actually, my team won the state championship two years running at our level.

KS: Oh, wow.

KK: But I'm not any good. This was like, you know, I'm like a 3.5. Like, you know, just a very intermediate sort of player.

KS: Yeah, that's what I am.

KK: Yeah, my shoulder won't take it anymore.

KS: I still, I feel lucky. Because physically, I can do it. Right now. I'm in a 40+ league, and that's good. But next season, whatever you call it, or next season, I guess, I'm in an 18+ League, and I've done this before. It means the other players are allowed to be as young as 18. It’s a little humbling, even if we can serve, you know, we have the technical skills, like they’re, you know, like the shots you use in the 40s, like, lobbing is not a good strategy in 18+ because they can run.

KK: Back when I was in my 30s and played, I played a lot of singles still, and I could still do it. But when I would come up against the 20-year-olds, it'd be a lot harder. But then I also learned, I used to play a lot of doubles with with these guys in their 70s. And they destroyed me every time. They were just —because they knew where to be. They had such skill and good instincts for where the ball was going to be. It was humbling in that way.

KS: Yeah, it's it's fun. And I prefer playing doubles these days. It's just more fun and different strategy.

KK: Yeah, and less court to cover. That helps.

KS: Less court to cover. And it’s more social. It's a lot of fun.

KK: Yeah, so you haven't succumbed to pickleball, have you?

KS: I played once, on my 60th birthday. Because no one would play tennis with me. And I got invited to a pickleball thing. And I was like, Okay, we're gonna do it. And, you know, it was fun, but I haven't really. It’s a challenge in Minnesota playing pickleball because it's so windy and the balls are so light, and it’s like whiffle ball.

KK: That’s what they are, basically.

KS: The ball kind of blows around all over the place. So yeah, I haven't I succumbed to doing that. In DC I'm lucky to have enough people to play tennis with. There's a lot of them.

KK: Cool. All right.

EL: Yeah. Great pairing.

KK: Yeah, yeah. So we also give our guests a chance to plug anything they're working on. You sort of already did that. I mean, you're doing all the work. Anything else you want to pitch?

KS: I mean, back to what I do, one reason I love this new job is I get to go in and and make connections to any Congressperson. You know, they have their own interests motivated by their own history, their own life, their own constituents. And this can be — there are obvious things we think about, like people, congressional members who are interested in their electric grid, or ocean modeling for the Hawaii delegation. But it's fun. And it's a fun challenge to think of things. So there's one newish member who was a truck driver before he was elected to Congress. And, we went in and their office was like, we can't make a connection to math. And we started talking about logistics, you know, truck routing. And it was great. It turned into a great conversation where they hadn't really thought about that. So this is what I really love about my job, trying to connect math to anything they’ve got. What they’re interested in, I'm gonna I'm gonna try to connect math, and there are very few issues that that can't be connected.

EL: Yeah, well I actually have a question, something that our listeners might be interested in is like if a mathematician is listening to this, and wonders, how can I get more connected to what's happening? How can I understand what math and science, you know, representatives do on the hill? Is there a newsletter or a website or something that you have that they could look at? And, you know, maybe find ways to get more involved? Or at least more informed?

KS: Yeah, definitely. So first of all, I used to write a blog, but I don't do that anymore for the AMS. The AMS Government Relations page — so my office is the Office of Government Relations. And I believe if you search, AMS government relations, you'll get to my webpage, you know, the one that I call mine, and you'll see a lot of different things there. There are ways to get engaged. We offer felt three fellowships. Two are for graduate students, one is for a person with a PhD in mathematics to come and to come here physically and do things. One is a boot camp for graduate students, a three-day graduate boot camp to come learn about legislative policy. And then the the biggest one is a year long fellowship and working in Congress. I do hill visits with people. And you know, I'm pretty willing to bring almost any mathematician to the hill, and that can be virtual these days. So we have volunteer members through our committee work who fly in and do these hill visits. We did this last Wednesday, we had about 25 AFS, volunteers fly in, and that was a fantastic day. But I can do them virtually. I've done them with big groups of grad students from departments, and people can email me if they want. And I think you guys have my email.

EL: Yeah. Thanks.

KS: So those are the big ways. And then for AMS members who are a little more advanced in their careers, you can volunteer for AMS committees. And there's the Committee on Science Policy, which really focuses on this one. And then I'm also in charge of the Human Rights Committee for the AMS, which can be of interest to a lot of people.

KK: Sure.

EL: For sure.

KK: Lots going on there.

KS: Yeah, lots going on.

KK: Well, Karen, this is terrific. Thanks so much for taking time out of your day, and thanks for joining us.

KS: Thank you.

[outro]

In this episode, we enjoyed talking with Karen Saxe about her work as the director of the American Mathematical Society's Office of Government Relations and her favorite theorem, the isoperimetric theorem. Below are a few links you might find relevant as you listen:
Saxe's website and the homepage of the AMS Office of Government Relations

A survey of the history of the isoperimetric problem by Richard Tapia 
The 1995 proof by Peter Lax
Evelyn's blog post about 50 pence coins and other British objects of constant width
The Polsby-Popper test to measure gerrymandering
A public lecture by mathematician Moon Duchin about mathematics and redistricting
The 1927 Journal of Paleontology article that first uses the Polsby-Popper metric (though not with that name)
An Atomic Frontier video about squeaky sand
Our episode with fellow tennis-enjoyer Carina Curto

The 10 Deutsche Mark note

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

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. How's it going?

EL: All right. Yeah, I was I was trying to think about what to say. And I was like, well, the most exciting thing in my life right now is that our city is starting a pilot program of food waste, like a specific food waste bin.

KK: Okay.

EL: But then I realized I also did an 80 mile bike ride last Saturday, and that's the first time I've biked that far. And that might be slightly more exciting than compost.

KK: Are you working up the centuries? Are you are you heading for?

EL: We’ll see. I felt pretty fine after 80. I also don't feel like I wanted to do 20 more miles. So we'll see. Someday, maybe

KK: The last century I did was, wow, it was 2003. It was 20 years ago. This is one called the six gap century in Georgia. And it goes over six mountain passes in the mountains of North Georgia, one of which has, like, a 15% grade, which is quite steep. It took me about eight hours. And then I hung up my bike and didn't ride it for like three months.

EL: Well, I mean, it would probably take me at least eight hours to do a flat century.

KK: Yeah, but back in my youth I could do a flat century in about five, but not anymore. Not anymore. So let's keep this banter going because I — so Ben Orlin, a former guest on our podcast, I saw Math with Bad Drawings today had various golden ratios. One of which was the golden ratio of hot fudge to ice cream in a hot fudge sundae, which he argues is one to one, but that's way too much fudge.

EL: That is so much fudge!

KK: But the podcast, like, substance to banter golden ratio, he claims is like two to one. So like a third of this should just be like us, you know, just shooting it.

EL: Saying nothing.

KK: Yeah.

EL: Well, I must admit, that's why I listen to fewer podcasts that maybe I would want to because I have a low banter tolerance. Which brings us to our guest today. Yeah, so we are very happy to welcome Corrine Yap today. Would you like to tell us a little bit about yourself?

Corrine Yap: Yes. So I am currently a visiting assistant professor at the Georgia Institute of Technology, Georgia Tech, in the math department. I'm also a postdoc affiliated with the Algorithms and Randomness Center. But I just got my PhD in the spring from Rutgers University.

KK: Congratulations!

CY: Thank you! I do a lot of, like, probabilistic combinatorics, and stuff around that. So that's sort of my main research work. I also do some performing and some playwriting as well. I actually just got back from a performance yesterday Worcester Polytechnic Institute in Massachusetts.

EL: Oh, wow.

CY: So very busy this time.

EL: Yeah. That’s actually one of the reasons that I've been wanting to invite you for a while. And I was like, well, I should wait until I've seen one of her shows. And then it just has not aligned to work out. Because I know you've done them at the Joint Meetings and things, and the times that I have been there and you have been there, it’s just not been a good time. So it's like, well, I'm not going to put this off forever. So even though I have not yet seen one of your shows, I'm very glad that that we could invite you and have you here and yeah, well, can you talk a little bit about the kinds of theater that you do, or kinds of — I don't know if it's mostly theater or more, like other? I don't know, speaking performances?

CY: Yeah. So it really started when I was a lot younger. And also in college, I primarily studied both mathematics and theatre, with no sort of vision as to what that would turn into in terms of a job or career or anything. I just really enjoyed doing both of them. And as an undergraduate I thought I was mainly interested in acting, but I started studying playwriting while at Sarah Lawrence College in Westchester, New York. And I started writing this play, which is the play that I continue to perform. It's called Uniform Convergence. And it's a one-woman play that's about math. It tells the story of Sofia Kovalevskaya, who is a historical Russian mathematician. She was born in 1850. And it tells a little bit about her life and how she faced a lot of obstacles to be successful as one of the first few women in academia. But it also has a portion that is sort of inspired by my experiences being Asian American, and also being a woman pursuing mathematics. And the setting is that of a real analysis classroom, a lecture where the character Professor….

EL: Hence, uniform convergence.

CY: Yeah, and she is lecturing to her students. So at one point, they do reach the point of the class where they do uniform convergence as a topic. So, you know, in the past, I did a lot more — like, in college, I did, you know, the auditioning for plays and being involved in rehearsals, and all this sort of stuff. But since going to graduate school, and now having an actual job, this one play is sort of the main way that I keep my ties to doing theater and the theater world.

KK: Very cool.

EL: Yeah. Well, that's cool. I didn't realize that Kovalevskaya was the subject of this. I actually just read Alice Munro's short story, Too Much Happiness, which is based on her life. And actually was not my favorite short story in the collection that it’s in, but it, you know, she is such a compelling figure and another woman who was interested in math, you know, at a time when it was a lot harder for a woman to have an academic career in any field, and was interested in literature. Wrote, I think both memoirs and fiction?

CY: She also wrote a play.

EL: Oh, wow.

CY: Yeah. But it wasn't about math. But yeah, she was very much also in both of these worlds in, you know, sort of a more artistic, creative mindset as well as a mathematical one.

EL: Yeah. Fascinating person. So yeah, that's really interesting. And hopefully someday I'll get to see it.

CY: Yeah, I'm still performing. I didn't think I necessarily would be. But it's been since 2017. I've been performing it at different college campuses, and sometimes at conferences at different parts of the country. And I still get invited places. So as long as that keeps happening, I'll keep going.

EL: Yeah, when I was still in academia and doing a postdoc, I did, you know, I'd started doing writing. And sometimes I would get invited to do both like a research seminar talk and a public engagement kind of talk. And so that that might be in your future as well.

CY: Yeah, maybe.

KK: Yeah. Broader impacts.

EL: Wearing both hats on one trip.

CY: Yeah, I actually, I forgot I am doing that. I think this is the first time I'm doing it. At Duke in October, when one day I'll be giving a seminar talk, and then the next day, I'll be performing the play.

EL: Yeah, cool. Well, we invited you on here to talk about your plays, but also to talk about your favorite theorem. So what have you chosen?

CY: Yeah, so I've chosen Mantel’s theorem as my favorite theorem. So this is a theorem that is in the area called extremal combinatorics. And I'll explain what that means. But the statement of the theorem is pretty straightforward. It says that if you have a graph, which I’m a combinatorialist, so for me graphs mean, collections of vertices with edges connecting pairs of vertices. If you have a graph on N vertices, then the maximum number of edges you can have without forming any triangles — so just three edges and three vertices connected to each other — the maximum number of edges you can have with no triangles is N squared over four with appropriate floor.

KK: Yeah, sure.

CY: And this seems like, okay, this is this is just a statement, maximum number of edges. What's so cool about that? You actually, we actually also know where the N squared over four comes from. It’s, the extremal example is the complete bipartite graph on parts of size N over two. So what that means is, you split your vertices up into two sets, each of size half the total universe. And all of your edges go between the two parts. So from one part to the other, not inside the vertices of the parts. So complete means you have all the possible edges crossing between the parts, and then bipartite because you have the two parts of the vertices, and that has N squared over four edges. And it has no triangles in it.

KK: Not even any cycles.

CY: Yes. Yeah, no odd cycles. Yeah.

KK: Okay, all right.

CY: Yeah. So, one reason I really liked this is because when I first learned it, I didn't really think much of it, I learned it in an undergraduate class in combinatorics. And there are, like, three, maybe four proofs that we learned that were all pretty short and straightforward. One of the most basic proofs is just via induction on the number of vertices, and there's nothing, there's no really heavy machinery that's needed at all. And I didn't think much of it. And I didn't have any context as to like, why do we care about this sort of thing. But every year, I learn more and more things that make me appreciate this theory, more and more, because it really was the foundation for this whole field that we call extremal combinatorics, which is really centered on these questions of, like, what are the maxima and minima of certain things that we want to count when we put certain constraints on the problem? So this is an example we want the maximum number of edges. And our constraint is we have no triangles. And you can, there are a lot of different directions you can go with this sort of theorem. One of the most sort of classical foundational ones is just to replace triangle with a different type of graph. Like you could say, Okay, if I want the maximum number of edges with no cycle of length four, or cycle of length 10, right, what can I say? Or if I want the maximum number of edges with no complete graph of size five, where complete means you know, the vertices, you have every possible edge between every pair of vertices. And this type of problem, sort of replacing triangle with other things. It's called a Turán type problem, because there's Turán's theorem that generalizes mantle's theorem to complete graphs of higher orders. And we basically know the answer of what the extremal number is, and the extremal constructions for almost every graph, except for when you consider a bipartite graph as your, instead of triangles.

EL: As the thing you're trying to avoid?

CY: Exactly. And there's a reason for this, there's a theorem where it basically fails, or it's trivial in the case that your forbidden graph is bipartite. And so there's been a lot of study, it's still a very active area of research. And what people are doing is sort of taking different flavors of this Turán type problem that sort of started with Mantel’s theorem. And my first paper in graduate school was on a topological version of this theorem, where we were looking at these higher-dimensional structures called hyper-graphs, which you can think of as a higher dimensional version of a graph, and looking at a more geometric or topological viewpoint on these hyper-graphs by making them into simplicial, abstract simplicial complexes. So we don't have to go into the details of that. But I found it, you know, when I did that project, I found it very cool that that we could take this seemingly purely combinatorial, graph theoretic statement about just counting edges, and somehow turn it into something that requires a little bit more of a geometric or topological point of view, which is not something I had spent much time with before. And so that's sort of one direction at the beginning of my grad school career, where I felt like I had suddenly a much greater appreciation for this theorem. And on the other end, where I am now, it's also connecting very heavily to the research direction that I'm currently pursuing, which is in statistical physics, which is for me an entirely unexpected application of this sort of thing. But if you think about it, this sort of characterization of the extremal structure saying, okay, we can achieve the maximum with a complete bipartite graph, you can view this as sort of a ground state, if you will, if you want to think of the vertices as like particles in some sort of distribution, and you can take a probabilistic point of view on these sorts of counting problems. For example, it turns out that the triangle-free graphs and the bipartite graphs, if you think of these two collections, triangle-free graphs and bipartite graphs on N vertices, they're very closely related to one another. In fact, almost all triangle-free graphs are bipartite. This is a theorem by Erdős, Kleitman, and Rothschild. So you can sort of ask how far does that behavior persist if you add more constraints to your problem? And you can think about it as in a probabilistic sense of thinking, well, what if I have a probability distribution on my triangle-free graphs? And I have a probability distribution on my bipartite graphs? How are those distributions related to one another? And what is the counting statement, say, in terms of the probability distributions when we when we consider a randomness point of view on these things. And the sort of magical thing is that when you go to a probabilistic point of view, there are very natural ways that you can put it into a statistical physics context, where in statistical physics, you are thinking inherently about probability distributions on certain particles, on particles in space, or different configurations of particles in space, where there's maybe a physics motivation underlying the distribution you define. But ultimately, you can distill it down into something that is, that is simply triangle-free graphs, or different discrete structures. So one thing that I'm really interested in right now is just exploring more of this somewhat mysterious, but somewhat really amazing connection between questions that arise in graph theory and combinatorics that, you know, for a long time, we have just thought of in that context, in the graph theoretic context, and how, looking at them from a more statistical physics perspective, can help us gain new insight into how to tackle these problems.

KK: Yeah, and hopefully, it'll go in the other direction. I mean, I think we have this idea that because we learn calculus, and we think about physics being based on calculus, but inherently, right, the universe has to be kind of discrete, so you can't divide stuff forever. So I mean, it sort of makes sense that the underlying business, when you get down to it, might have to involve some kind of graph theory questions.

EL: Yeah, that is remarkable that there's this connection. So this is maybe a naive question about what you're talking about doing. Like probability distributions on graphs, are you saying things like, the likelihood that that two vertices have an edge between them? Or are we talking about some other kind of probability distribution?

CY: Yeah, so there, I purposely didn't include too many details, just because there are a lot of a lot of actually interesting and all valid ways that you can think about imposing probability, you know, into this world into these problems, these extremal combinatorics problems. So one flavor is what you said, we can think of what's called the random graph model. The most common one is the Erdős–Rényi random graph model, where you simply have your N vertices, and for each pair of vertices, you flip a coin, and it can be a P-biased coin, independently, to decide whether you put an edge there. And you can analyze what happens in that graph. What are the likely properties that this graph might have, if you, for example, change P. And what's really cool about studying this model is that there are, for a lot of graph properties, you can find these thresholds with respect to P. And this is like a huge, very active area of research right now, there have been a lot of really cool things proven just this year, in the past few years with regards to a lot of open questions here. But you can sort of if you let your P, your probability that you're adding an edge, be a function of N, the number of vertices, and you imagine N going to infinity, then you can actually sort of chart what happens if you're trying to count, let's say, the number of triangles in your graph, the expected number of triangles in your graph, or other properties. And you can see how changing P changes the value of the thing that you're trying to count. And for a lot of things, they exhibit these thresholds where the probability of finding a particular structure is close to zero. And then past a certain threshold, it jumps up to something close to one, and it happens with high probability. And this is also mimicking something in the statistical physics world where we have things like phase transitions.

KK: Right.

CY: If you think of in physics, just like water’s liquid-gas sort of phase transitions. Where we're also interested in studying what happens to certain properties of your statistical physics distribution when you change the temperature of your different parameters of your model. Can you find the sort of phase transition where the behavior changes quite drastically?

KK: Yeah.

CY: And then so GNP is, is one of these ways you can sort of input probability into — you know, take a sort of probabilistic perspective on these problems. Another is simply something a little bit more physics motivated, is by just imposing a uniform, or nonuniform, or a weighted distribution on the things that you're trying to count. For example, if you want to study triangle-free graphs, you could consider the uniform distribution on all triangle-free graphs on N vertices. And then think about the uniform distribution on bipartite graphs and ask, like, are these distributions close in total variation distance? And you can conclude things about that based on what you know about how close are triangle-free graphs and bipartite graphs to one another? Well, what does that say then about the distance between the nniform distributions that you impose on each set? And that sort of thing characterizes the different sort of distributions that come from the perspective of statistical physics. There are things called, like, the Ising model, and the Potts model and the hardcore model that were defined by physicists. And it turns out that they are simply weighted distributions on things like graph colorings, and independent sets of graphs. And so you can study them in these two different contexts, in the context of the hardcore model from the physics world, or in the context of a distribution on independent sets from the graph theory world.

KK: The hardcore model, I love that name. That’s good.

CY: Yeah.

KK: Well, we’ve gotten pretty far away from triangle free graphs can have at most N squared over four edges. So you mentioned that there were like three or four proofs of this. Do you have a favorite?

CY: I have to say my favorite is the very straightforward induction proof.

KK: Okay.

CY: And the reason I like this is because it's a proof that I've done with high school students at a summer math program I teach at called MathILy-Er. And I do it as an hour-long inquiry-based activity, where I simply pose to them this question. I let N be something like six or five, something that they could start drawing examples for them, then say, how many edges can you have before you start having to find triangles? And they often come up with the extremal construction, the complete bipartite construction first. And then I asked them how can we prove that this is actually true that this is the maximum. And they've learned induction at this point, when I do this activity. And so it's a nice lesson in induction, because it requires strong induction. And everybody wants to do weak induction, first of all, and they always want to what I call induct up instead of induct down. They always want to start with an extremal example with N vertices and try and build something with N+1 vertices. And it doesn't work.

KK: Right.

CY: And I always have to remind them, you have to start with something that has N+1 vertices, and remove something and see what happens. Yeah, yeah.

KK: And the base case of one vertex is super easy, right?

CY: And then there's also some argument about how many base cases we need and whether we need one or two or three, or where do we start? And so I think it's just a really nice exercise and practice. And it's simple, but I get to give a little, tiny spiel at the end, not nearly as much as I have said here in this podcast so far. But a tiny hint as to like, you know, what's cool about this theorem, and what more could you do? And some of the students have been interested enough to try and generalize to complete graphs or higher orders, you know, a complete graph on four vertices and try and mimic the same proof. And yeah, I think it's a really nice activity.

KK: Cool.

EL: Yeah. So a complete graph on four vertices includes a complete graph on three vertices so therefore you're trying to avoid something more, so like some of these ones that have triangles could still not have the four. Sorry I'm thinking out loud here because I have very little graph theory intuition. So okay, just like which direction are we going, and how many of these are we avoiding?

KK: You and I are probably the same, Evelyn. Like, we probably took one undergrad graph theory course and then yeah, and then then became topologists.

EL: Right. It’s kind of like it came up in, my introduction to proof class, but never a specific class dealing with graph theory things. Although the times that I've taught in high school programs or stuff, it is the kind of thing that can be quite accessible because the idea of drawing a graph, it's not hard to explain to anybody.

CY: Yeah, to answer your question. So there are actually lots of triangles in the extremal example for the complete graph on four vertices.

EL: Okay.

CY: Just to give you a sense of how it generalizes, the extremal example is the complete tripartite graph where you take three parts now sides and over three, and you have all the edges between the parts, so it looks like a giant triangle.

EL: Yeah. This kind of makes me want to go think about graphs a little bit. Yeah.

KK: Well, so the other part of this podcast is we ask our guests to pair their theorem with something. So what do you think pairs well, with with this theorem?

CY: So yeah, this this question was actually harder for me.

KK: It’s harder for everybody!

CY: I thought of something right away. And then I thought, no, I can't say that. I have to say something cool. And my pairing has to be something neat that makes me seem like a cool person. But I just couldn't think of anything better. So bear with me.

KK: Okay.

CY: My pairing is tofu. Okay. And here's why.

EL: Oh, tofu is great!

CY: Yeah. Okay, great. Great. So I thought of this because I think tofu is also somewhat of an underrated ingredient. But it is also so versatile, and you can use it in so many ways. So I grew up eating a lot of tofu because I grew up in a Filipino-Chinese household. And it was just sort of a staple of the things we were eating. But then I realized that not everybody knows or appreciates tofu. The first time I met someone who had never heard of tofu before, it just sort of shocked me, but then I realized it's not a common thing everywhere. But it's used in so many ways. And so I have been vegan since 2015. And also, every year I gain more and more appreciation of tofu as an ingredient. Like, you can use it in stir fries. There's now cheese that's made of tofu, you can make eggs using tofu. You can make a pie using tofu. There are so many ways you can use tofu. And there are so many more vegan options at restaurants and grocery stores and everywhere. So I feel like you know, for anyone who hasn't had tofu before, I would recommend at least giving it a shot.

EL: Yeah, yeah. And yeah, I mean, I grew up in a household that did not eat tofu much, my parents don't eat too much. But yeah, I'm not vegetarian or vegan, but like eat a lot — we have recently been enjoying this vegan Korean cookbook, I mean, it's called Vegan Korean. You might have seen it. [Editor’s note: It’s actually called The Korean Vegan.]

CY: Yeah. I have that!

EL: And just checked out this vegan Chinese cookbook that of course, it's like I think multiple sections are tofu because it's like the tofu tofu part and the tofu skin part, and all of this stuff.

KK: And, you know, do you use silken or what.

EL: But yeah, we’re a high-tofu household now.

CY: Nice. Yeah, there are so many different levels of tofu that you can have.

EL: Yeah, so many different textures, like, the Korean soft tofu is different from like the soft tofu in the cardboard package. Yeah, and so I finally found a Korean grocery store in Salt Lake that I could get to and got, like, the real stuff and oh man, great. That soft tofu soup, so good. And I can actually eat the kind I make because when I get it at a Korean restaurant, it's way too spicy. So I cut — in that cookbook, I think I cut at minimum, sorry, maximum spiciness is, like, a third of what the recipes start with, sometimes a sixth and see if I can work up.

KK: The correct answer level and there was you know, N squared over for, the floor.

EL: Yeah. I am impressed by the spice tolerance of Koreans.

KK: Asian cuisine in general, we once years ago, I was director of the University Honors Program, there was this place in town. It was an Asian place, and they have various stir fries. And you could ask for your spice level from zero up to no refunds, right? And so we had a student worker who was from Bangladesh, we went to lunch there one day, and he got the “no refunds.” And we were like, how is it? And he just went, eh. Like, it's just not very hot. And it's just an interesting cultural thing. Because, you know, I grew up in the Midwest, my mother's family was German, you know, we ate a lot of fried potatoes and sausage, like no flavor, you know? And it's just all what you get used to. Right?

CY: Yeah.

KK: All right. Well, this is we like to give our guests a chance to plug anything. Where can people find you online? Or you've talked about your plays, so that's good.

CY: Yeah. I mean, you can find my website. I recently updated it. And now it's got an all-purple background, which I'm very happy with. And it's corrineyap.com. That's Corrine with two R’s and one N, in case you forget. And, yeah, I continue to perform my plays. So if you're ever interested in bringing me out somewhere to perform, I am always happy to consider doing that. And I've done it at a lot of math departments. I did it at some conferences, but I don't have any conference performances coming up. So mainly like seminars and colloquia slots, things like that. So Evelyn, if you have any universities around you in Salt Lake City who might be interested in hosting a performance, you can let me know.

EL: Yeah. I'll make you vegan Korean food.

CY: Oh, amazing. But yeah, I mean, I just do this for fun. So it's not something that I'm trying to, I'm not trying to schedule, you know, 100 performances on my show this year. I just do it whenever someone is interested in having me there, but I'm always open to new inquiries. So yeah, that's, I guess, the one thing that I'll plug.

EL: Okay, great. Well, it was lovely to have you I'm so glad we finally got to at least meet online.

CY: Yeah, you as well. Thank you. Thank you so much for inviting me. This is a lot of fun.

KK: This was great.

[outro]

On this episode, we enjoyed talking with mathematician and playwright-performer Corrine Yap about Mantel's theorem in graph theory. Below are some related links you may find interesting.
Yap's website
MathILy-Er, a summer math program for high schoolers
Wikipedia on Turán's theorem, the generalization of Mantel's theorem
The Korean Vegan

Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida, and I'm joined as always by your other and let's be honest, better, host.

Evelyn Lamb: I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City. And tomorrow is my 40th birthday. So everything I do today is the last time I do it in my 30s. So, like, having my last mug of tea in my 30s, taking out the compost for the last time in my 30s, going for a bike ride for the last time in my 30s. So I'm, I'm kind of enjoying that.

KK: Well, congratulations. Let's not talk about how long ago I passed that landmark. I will say there's a switch that goes off when you turn 40. So riding your bike will be more difficult tomorrow, I assure you.

EL: Well I’d better get one in then.

KK: Any big plans?

EL: I’m actually going to the Janelle Monae concert. She's in town on my birthday. I'm sure that's a causal relationship there.

KK: It must be.

EL: So yeah, I'm excited about that.

KK: Okay, so fun fact, my Janelle Monae number is, is two. So I have a half brother. Very long story. I have a half brother, who also has a brother by — his mother has two children with — my dad was one of them. And then another man was the other one. So this other one, his name is Rico. He was a backup dancer for Janelle Monae.

EL: Wow. So yeah, brush with celebrity there.

KK: I mean, of course I've never met Janelle Monae, but you can — actually if you look him up, so there's a style of dance, sort of Memphis Jook, it’s called. Dr. Rico. He's something else. Amazing dancer.

EL: Wow. Interesting life.

KK: That's right. That's right. So anyway, enough about us. We have guests on this show. So today we're pleased to welcome Allison Henrich. Allison, introduce yourself, please.

Allison Henrich: Hi. Yes. I'm Allison Henrich. Happy birthday. I'm so excited for you.

EL: Yes, you get to be on my last My Favorite Theorem of my thirties!

AH: Yes, awesome! I feel so special. So I'm Allison Henrich. I'm a professor at Seattle University, and I'm also currently the editor of MAA Focus, which is the news magazine of the Mathematical Association of America.

KK: I have one on my desk.

AH: Woo-hoo! Is it one of mine?

EL: Yeah, and when we were chatting before we started recording, you made the mistake of mentioning that you've done some improv comedy. Is that something you do regularly?

AH: So I wasn't an improv artist. This is such a cool event. This science grad student at the University of Washington started this type of improv comedy where they have two scientists give short five-minute talks. And then this improv comedy troupe does a performance that's loosely based on things that they heard in the science talk. And so I gave a talk about some basic knot theory ideas, and it was so funny. I wish everyone could have the experience of an improv comedy troupe doing a whole set about your like research or your job. Yeah, it was so amazing.

EL: Cool. But also, it sounds a little stressful. A little bit. Yeah.

AH: Yeah. You want to not be too boring. And you gotta, like — it's really interesting. The other speaker tried to work in things for them to make jokes about, and they totally didn't take the bait. And they found like more interesting things to make jokes about, but I definitely tried to work in some things that would help them riff off of my talk, and it worked pretty well. Like just referring to knots with quirky names and making jokes about knot theorists and whatnot.

KK: Sure. What-knot. Hahaha.

AH: There are a lot of good knotty puns.

KK: Sure. Okay, so this podcast does have a theme. And the question is, what's your favorite theorem?

AH: Yes! This is a hard question.

KK: Of course.

AH: I’ve decided to tell you about my second favorite theorem. Should I admit that?

KK: Sure.

EL: I’m sorry, that’s a different podcast, My Second Favorite Theorem. It has two slightly worse hosts.

AH: It’s the cheap knockoff.

KK: No, it's the sequel, once we get rid of this one, we're gonna move on.

AH: Just, we're all out of mathematicians, we’ve got to go through them again. So my, let's call it my favorite theorem.

KK: Sure.

AH: My favorite theorem is the region crossing change theorem. So I have to tell you a bunch of stuff before I can explain what this theorem is.

KK: Sure. But it must be about knots.

AH: It is about knots. So, you know, knots we represent, typically, with two-dimensional pictures called knot diagrams, where you have ways of representing when a strand is going over and when a strand is going under at a crossing. And so every type of knot that there is has infinitely many diagrams you can draw of it. But no matter how you draw a diagram of whatever your favorite knot is, it can always be unknotted if you're allowed to do a special kind of move called a crossing change. So if you have your favorite knot diagram, and you're allowed to switch the over and under strands on whichever crossings you want, you can always turn that knot diagram into the diagram of an unknot, which is like a trivial knot that'll fall apart if you unravel it a little bit.

EL: Basically just a circle, right?

AH: Yeah, a circle. I mean, all knots are circles, so I have trouble. Like, a geometric circle.

EL: A boring circle.

AH: Yeah, a boring circle.

EL: And so this theorem, does it come with like, a number of how many of these crossing changes?

AH: Ah, so this is not my favorite theorem. This is a theorem that's going to help us understand my favorite theorem.

EL: Okay.

AH: So this theorem has a really interesting proof that Colin Adams calls “proof by roller coaster.” So the the theorem that says you can unknot any not diagram by changing crossings, you can accomplish unknotting using a certain algorithm where you choose a starting point to travel around a knot, and you decide that every time you encounter a crossing for the first time, you're going to go over it. So the fact is that you're kind of like always traveling downwards. And then when you get to the very end, you take a little elevator back up to where you started. So this will always create an unknot. So it's not that surprising that this is true, that if you're allowed to change whatever crossings you want, you can unknot things. What is surprising is my favorite theorem, which is that region crossing changes can unknot any knot diagram. So let me tell you what a region crossing change is. So you have your knot diagram in the plane. A lot of us kind of imagine that this plane is on a big sphere. So can we picture not diagram on a ball? Is that okay?

KK: Sure. Make it a big enough ball, and it looks like a knot diagram.

AH: Exactly. Yup. So we've got a knot diagram on a ball, and the knot diagram basically separates the surface of the ball into different regions, right? So this amazing theorem uses this operation called a region crossing change. And what a region crossing change is, is you choose a region in the diagram, and you change every crossing along the boundary of that region. So in my head, I'm picturing kind of like a triangular region in the diagram. And if I do a region crossing change on that region, I'm going to change all three crossings that are kind of around that region. So this is the amazing result: every not diagram can be unknotted by region crossing changes. So you no longer, seemingly, have control over individual crossings, you can only change groups of crossings at a time.

KK: Okay.

EL: But you can still do it.

AH: Yes, you can still do it.

KK: Right. That seems less likely. The other one, you told us and I thought, Well, yeah, I can kind of see, before we even saw the proof, I could sort of imagine, well, yeah, you just lift them up basically.

AH: Exactly. You lift it up, and then if it gets stuck, you know, change that crossing. But you can only change groups of crossings with the region crossing change. But amazingly, it's still an unknotting operation. So that just blew my mind when I heard that.

KK: Okay, so now I have questions. So, more than one, right? You can't expect to be able to just do one of these, right?

AH: Right. I mean, so if you have a region that just has one crossing on it, it's like a super boring region, because it's just a little loop.

KK: Yep.

AH: And that's actually called a reducible crossing.

KK: Sure.

AH: If you just have a little loop, it doesn't matter which way, which is going over and which is under.

KK: No, but I guess I meant, so you know, you've got one region, right?

AH: Yeah.

KK: So there might be multiple regions, you might have to change many of these, right?

AH: Yes, yes.

KK: What if two are adjacent, then you do one flip on one and one flip on the other, then you're undoing some of the flips from the other.

AH: Exactly.

KK: Is that why it works, maybe?

AH: That is why it works. So it’s a really cool proof. It's actually a proof by induction, which is so cool, that you can have like a proof on knot diagrams that's a proof by induction. But it's by induction on the number of reducible crossings. So the number of these crossings that you could sort of flip out of the diagram. They're not really necessary for the knotedness of the knot. But the base case is the most interesting part of the proof, where you have a knot diagram that has no reducible crossings. So no extraneous little loops or flips on it. But it's very constructive, and it uses things like checkerboard colorings, and it uses splices, or smoothings, which is where you take a crossing and you turn it into — like, you basically get rid of the crossing by cutting it and reattaching ends so that it's just — I’ve got this picture in my head, how do I say it? What's the best way to say that? So you have a crossing, and you want to get rid of it by cutting it and reattaching ends so that there's no crossing anymore. Does that make sense?

KK: Well, it’s sort of like a braid, right? I mean, so you imagine sort of a braid cross, you just clip the string above and below and then you just reattach, then you don't have it, right? Is that what you’re doing?

AH: Okay, yeah, what you just said totally makes sense because I could see your fingers.

KK: This would be a better video podcast, I suppose.

AH: I know. Yeah, at least for topology, or geometry. But the proof basically creates a checkerboard coloring that tells you how to find a collection of regions where you can basically control which crossing you're going to change. So I can change just one crossing, by carefully selecting a group of regions where exactly one, or exactly three of the regions involved in that crossing are going to get changed, but every other crossing in the diagram is next to either zero, two, or four regions that are being changed. So if it gets changed, it'll get changed back and look like it like it started.

KK: Right. Okay. All right.

EL: So I have not thought about knot theory, probably since we talked with, like, Laura Taalman on this podcast years ago. It's not something I think about a whole lot. And so I was not expecting this induction to be on the number of reducible crossings because they're so silly, you can just undo it, and then your diagram doesn't even have it anymore. So, yeah, why not the number of crossings or the number of regions or something?

KK: Yeah.

AH: So the reason reducible crossings are annoying for region crossing changes is because at a reducible crossing — you know, at any crossing, if you zero in on it, it looks like there are four different regions involved in the crossing, but with a reducible crossing, two of those four regions are actually the same region.

KK: Right.

AH: So it can look locally like you're changing two regions, so that you know, the crossing shouldn't flip. But you're really changing one, so the crossing does flip. So that's why reducible crossings are the annoying thing that you need to carefully control.

KK: Okay. All right.

AH: Yeah. And so once you get into the inductive step, you basically want to take a reducible crossing, change it so that you have two pieces, one has one fewer reducible crossings, and you know how to deal with that. And then one is a totally reduced diagram of a knot.

EL: Yeah. But the base case is the hard part, it sounds like.

AH: Yes, yes, yes. Totally.

EL: Interesting.

AH: Yeah. So yeah, so one of the reasons I love this is because I love unknotting. In general, I find unknotting questions really interesting. And I highly recommend everyone go listen to Laura Taalman's My Favorite Theorem podcast because she talks a lot about unknotting problems. But also the woman who proved this result is named Ayaka Shimizu. She’s a Japanese mathematician, probably my age, maybe a little bit younger, maybe she's about to have her 40th birthday or something, I don't know. But she is one of the coolest mathematicians I've ever met. She's definitely the cutest mathematician, and her talks are so cute that you're like, oh my gosh, I'm watching such a cute talk! And then you realize, oh my God, this result that she just proved is really amazing! So she's just super, super cool. I love her so much, and I think it's amazing that she proved this result that, you know, the Japanese math community wondered about for a long time, but no one came up with a proof before her. And she must have, maybe she was even a grad student at the time, or she was definitely a very young mathematician when she proved this result. So I love it.

KK: So here's a question: why would you want to allow such operations? I mean, because physically, changing the crossing, I mean, that would be great when your shoes are knotted, right? Like, you could just go Oh, snap, that's unknotted. Right. Is there a practical reason? And by practical, it could be including things like, it doesn't change the knot invariants or something? Or I don't know, it must if you get to the uknot, but I mean, is it… or is it just fun?

AH: Well, so the other thing — yeah, it's just fun. The other thing you need to know about me is that I study games that you can play on knot diagrams.

KK: Okay.

AH: And this result enabled this Lights Out-type game, they actually have a website. You can search for this game called Region Select. It's a really fun solitaire game that's a lot like Lights Out if you've heard of that game. And basically, the fact that the region crossing change is an unknotting operation basically means that any lights out game that you can think of or any region select game you can think of is playable, so you can have a knot diagram. Basically, at each crossing, instead of a crossing, you have a light. So it looks a lot like a graph, actually. You have a light and the lights are, some of them are on and some of them are off, and you need to select regions to try and turn all of them on or turn all of them off. And it's a really fun solitaire game that comes from this.

KK: Okay.

AH: But I’ve actually use the region crossing change to invent one of the many games that I've studied. It's called the region unknotting game. And basically, I'm super interested in these types of two-player games, where you start with a knot diagram, or maybe the shadow of a knot diagram. And you have two players doing something to the diagram, and one player wants to create the unknot and the other player wants to create something knotted. And so we have many games of this variety we've invented. One is the knotting-unknotting game. There's the region unknotting game. I’m about to publish a paper with some students called the arc unknotting game. And there are more. I could go on and on listing games.

EL: Kind of like you know that you can always unknot these things, but it's like, can you unknot it faster than someone can knot it? Is that sort of what's hard about playing this game?

AH: It doesn't have to be faster, necessarily. So the game, these games always are of the form, each player is going to move and they're going to go back and forth until everything is completely determined. And then at the very end, you see whether you have a knot or an unknot. And so you could be playing the long game, like, Oh, I'm just gonna wait it out playing on these little crossings over here to force the other player to play in this region of the knot diagram first, so that I can, you know, have the last move and turn it into a knot at the very end. So yeah, they're combinatorial games, topological combinatorial games, which is cool, because then you, then there is a player who has a winning strategy. And so your goal is to figure out which player is it? And what is a strategy that will always allow them to win?

EL: You said that the proof is constructive. So does that mean that given a knot diagram, you — someone who knew the proof — could actually say, okay, I can, you know, look at this knot diagram and do some sort of wizardry on it and say, Okay, the second player is definitely going to have a way to win this game. Or first.

AH: Yes, it can help. But of course, when you're playing two-player games, the other player can always thwart it. Like, let's say, I have to change these three regions in order to make this unknotted. Well, the other player knows that too. And so they're going to make it so that I can't change one of those regions. But actually, the the constructive way that the proof goes for region unknotting, the region unknotting operation, like basically, there's a complementary set of regions that will have the same effect. You can either do all the moves on this set of regions, or you can do all the moves on this other set of regions, and it will have the same effect on the diagram. So that does help inform game strategy, although we haven't looked at the types of diagrams that are terribly difficult to see how to unknot, because those are already hard enough to figure out strategies for.

KK: Right, right, right. Maybe it's like NIM, right? Like, if when you're playing, and if you have a huge numbers or piles of toothpicks, or whatever, you just kind of play randomly for a while, right?

AH: Yeah.

KK: And then when it gets small enough to where you can kind of analyze it, then you start to do it.

AH: Yes. Actually, I was giving a talk on not games at the Canada-USA math camp. And John Conway was in the audience. And he and all the students who were obsessively playing with him got really excited about calculating numbers for these topological combinatorial games. It's kind of a funny story. He said, he doesn't usually come to talks that speakers give at the math camp, or he didn't. And he said, normally he would leave before the speaker started speaking, because he was afraid to make speakers nervous. Like he didn't want them to be too nervous with him in the audience. But he was so intent on thinking about some problem that he was thinking about with a student there, that he just accidentally ended up in the room until it was like too late to leave. And so he told me afterwards about this dilemma he had, like, would it be worse for him to stay? Or worse for him to get up in the middle of my talk?

EL: Oh yeah, I’m glad he stayed. It would feel like a snub.

AH: Yes, he did stay. And we had a nice conversation about it afterwards, which was amazing. Because if you don't know about John Conway, he was, like, the king of knots and games and all of these things that I care about. So it was very cool.

KK: Yeah, that is cool. All right. So part two.

EH: Yes.

KK: What does this theorem pair with?

AH: This is it was such an obvious answer to me. With this paired with, because are you familiar with Nancy Scherich and her math and dance work?

EL: No, I don't think so.

AH: Nancy Scherich is a knot theorist. Actually, she works with braids. And she's also an amazing dancer, aerial acrobatics person. Acrobaticist? Acrobat. Aerial acrobat. And when she was a grad student, she won the Dance Your Ph. D competition, representing cool things about braids with dance. And since then she has recorded a number of other videos demonstrating mathematical ideas. And my husband is a musician, and he makes the music for her videos.

KK: Okay.

EL: Ao she just had a video that came out within the last month that is showing the proof of Alexander's theorem, which is a theorem about braiding. And the music that my husband composed for her dance piece, my husband's name is James Whetzel, was just beautiful, and just beautifully went with this performance that shows how the theorem works. And so I would recommend James Whetzel’s music.

KK: Unbiased, of course.

AH: I’m totally biased, and he has a new, actually, so I always forget if it's under Whetzel or James Whetzel because he has two different music personas. Right, so he has a new EP under Whetzel, W H E T Z E L, and the title track is “I want to go about my day,” and I think “I want to go about my day” would pair very well. Oddly enough, he also has songs called “Reidemeister Moves” and “This Is what Topology Sounds Like” and “Mama Proves a Theorem.” So he has some various songs with mathematical titles.

EL: So interesting that he came up with those and you also have done with things with these. What a weird coincidence.

AH: I know. How strange, isn't it? Yeah. So, but anyway, I think that everyone should go check out Nancy Scherich. I mean, you could probably just go to YouTube. Scherich is S C H, E R I C H. And check out Alexander's theorem. It's so beautiful. She does pole dancing to it.

KK: Okay, cool.

AH: Because it's about how you can turn any projection of a knot into a projection that always revolves in the same direction around a pole. So it works really well with that medium.

EL: That is so neat. So we have some things to watch and listen to after we're finished with this episode.

AH: Yeah.

KK: So we would like to give our guests a chance to plug anything that they're working on, or where we can find you on the intertubes.

AH: Yes, this is very timely because I'm trying to get out the word about an interesting event that I'm cohosting at the Joint Math Meetings.

KK: Okay.

AH: So, last Joint Math Meetings, a bunch of folks associated with Center Minorities in the Mathematical Sciences put on a storytelling event at the Joint Meetings. And it was so amazing and lovely. And they're doing it again, this Joint Meetings. But my friend Aaron Wootton and I were so inspired by this that we decided to also host a storytelling event at the upcoming Joint Meetings. And the theme is, people will be telling stories about some professional rejection that they experienced that was pretty crushing that ended up turning into something even better. So Aaron and I realized we both have stories like this where we didn't get something, we were totally feeling awful about it. And then it ended up being like a way bigger, more awesome thing. So we have a number of speakers lined up, but we need more. And so we have a web form that I created a bit.ly URL for. If you're interested in in telling a story no more than five minutes in length of the Joint Meetings, go to bit.ly/JMM2024STORY, all uppercase. Well, the bit that l y is lowercase, uppercase, JMM2024STORY, and we'd love to have people submit requests to speak, and I really hope that we have a good turnout for the event itself. It's going to be on Friday afternoon at the Joint Meetings. So mark your calendars. And let's see, the session is called Inspiring Stories: How an Academic Rejection Led to Something Amazing.

KK: Okay. In San Francisco, here we go.

AH: Yes. Can I add one more thing?

EL: Of course, you also, you also host a podcast, right?

AH: Yeah.

EL: I don't know if that's the thing you wanted to plug, but you should plug it too, and whatever you were about to say.

KK: You can plug as many things as you want.

AH: Okay, I have a lot. Okay. So we're just finishing up a book that's a handbook for math majors, called Navigating the Math Major: Charting Your Course. And it's going to be published through MAA Press by the AMS, and that should be coming out by MathFest of next year. So be on the lookout for that, especially if you're at a university that has one of these one- or two-credit freshmen seminars for math majors, or, like, an intro to the math major course. But also, it'll be good for just advisors and mentors to recommend to students and for students who might just be starting out in their college career, and they need some advice about, you know, what communities they should try and be a part of, how to apply for an REU, what kind of weird jobs are available for people. And this is what Evelyn was talking about earlier, because she did a wonderful interview for us about science writing, and that career path for math majors. So I definitely want to plug that. And regarding the podcast, it's a collaboration that I do with my friend who's an artist, Esther Loopstra. The podcast is called Flow into Authenticity. But what it's about is, if you're stuck in your life, it could be professionally or it could be personally, how can you use creativity and intuition to get unstuck? And we're actually writing a book on this called Think Like an Artist, Create Like a Mathematician, that's going to be published by 619 Wreath, which is Candice Price and Miloš Savić’s new publishing company. So that might be coming out in 2024, as well.

EL: Yeah. Well, and that sounds like something all of us can probably use it at some point.

KK: Sure.

EL: We always feel a little stuck.

AH: Yes. It's designed — we’re kind of aiming it professional stuckness in the book, but it's really broadly applicable. So very excited about that. And Esther Loopstra is amazing. She's a fine artist. She used to be an illustrator. You know, she used to work for, like, American Greetings and Target and all these places doing illustration. But now she's a fine artist and creative coach, and just super insightful about how we can use creativity to get unstuck.

EL: Cool.

AH: So, yeah, so the podcast is flow into authenticity. And the book is Think Like an Artist, Create Like a Mathematician.

KK: Cool. All right. Well, we'll try to link to everything that we can find links to.

AH: Okay, thank you.

KK: All right. Well, Allison, this has been terrific. Thanks so much for joining us.

AH: Thanks. Thanks for having me. This is such a fabulous opportunity, and I really appreciate getting to talk to you about today.

KK: Sure.

EL: Yeah. It was a lot of fun.

[outro]

On this episode, we talked with our delightful guest Allison Henrich, a mathematician at Seattle University, about the region crossing change theorem in knot theory. Here are some links to things we mentioned that might be interesting for you.
Henrich's website
MAA Focus magazine
Ayaka Shimizu's paper about the region crossing change theorem
Region Select, a game you can play where you try to unknot a knot using region crossings
An article Henrich coauthored about the region unknotting game
Nancy Scherich's YouTube channel, where she shares videos of her dances about math
James Whetzel's song I Want to Go About My Day on Bandcamp
The signup form for the mathematics storytelling event Henrich is cohosting at the Joint Mathematics Meetings in January 2024
Flow into Authenticity, the podcast she cohosts with artist Esther Loopstra

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm your host Evelyn Lamb, 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. It's Friday. Hooray!

EL: Yeah, yeah.

KK: Long Weekend. Yeah.

EL: It’s the start of a new month. Everything — anything is possible.

KK: Right.

EL: Including a great conversation with our guest.

KK: Yeah. I think it will be good. It's been an okay day so far.

EL: Great.

KK: The hurricane notwithstanding.

EL: Yeah.

KK: But yeah, that went by. But yeah, Hurricane Idalia really did some serious damage. And it’s, yeah, it's rough.

EL: Yeah, and there was recently the tropical storm on the other side of the country that actually kind of affected our weather, and today, I am hoping that the gale of wind outside my window isn't too much, too hear-able on the audio.

KK: I don't hear it, so it must be okay. Yeah.

EL: Great. Well, anyway, we are here today to talk with Tom Edgar about his favorite theorem. So Tom, would you like to introduce yourself?

Tom Edgar: Yeah, sure. Hi. Thanks for having me. It's fun to be here. I love your podcast, as you both know, but now everybody knows I love your podcast. I'm Tom Edgar. I'm a professor of mathematics at a small, comprehensive university in Tacoma, Washington called Pacific Lutheran University, just south of Seattle, about 35 minutes, maybe. Depending on traffic, like an hour and a half. I'm also currently the editor of Math Horizons, which is the undergraduate-level periodical from the Mathematics Association of America. And spend a lot of my time on those two things right there and just getting ready to go back to teaching here starting next week.

KK: Oh, you guys start after Labor Day. Okay, good for you.

EL: Oh, yeah. That is nice. Yes. And I think we've worked together a little bit on various Math Horizons things.

TE: Yeah, both of you have. So I mean, Kevin's on my editorial board, and he's written a couple of things. And then, Evelyn, I met you I think it in person at ICERM back forever ago. And I remember you were nice enough to do a piece about your awesome calendar, which I still have. I actually have a second copy now because I just have two now.

EL: Excellent. Yeah. Well, I would recommend getting one for every room.

TE: It doesn't hurt: one for the office, one at home.

EL: I’m not biased at all.

TE: No, one for your for your classrooms for your students. It's a great idea.

KK: Right. And it's universal. It's not year-specific. So reminder to all of our listeners, go to the AMS bookstore where they seem to be having a sale all the time, right?

EL: Yeah. Can’t afford not to! That's right. Anyway, Tom, now that you've so kindly plugged my calendar for me, what is your favorite theorem?

TE: And just that wasn't planned either. Right? That was just, you know, it's a nice thing that you've done. It's really cool. Yeah, so my favorite theorem is a hard thing. Because I've been listening your podcast for a number of years, and I was like, hey, if I ever get a chance, I wonder what I would talk about. And I had one that I was going to talk about, but I I've changed recently. There have been some projects that I've done in the past few years that kind of have changed my viewpoint. And so the theorem that I want to talk about is a pretty elementary theorem, in some sense. Most mathematicians will have seen it, a lot of, any math-adjacent people will have seen it. And it's the formula for the sum of the first N positive integers. So if you were to add up, say one plus two plus three plus four plus five, right, you can do this addition problem. My son, who's eight, can do this addition problem. But is there a quick way to get to the answer? And so the result is that if you add up one plus two plus three plus four plus five, you can actually get that in sort of fewer computations by multiplying five by six and dividing by two. And so the general formula is, if you were to add up the first N positive integers, pick your favorite number to stop at, N, then the theorem says that that sum should be N times N plus one divided by two. So the number that you stop at, multiplied by the next number, and then take half of that. So I really love this theorem for a variety of reasons.

KK: So there’s the apocryphal, probably apocryphal, story about Gauss, right?

TE: Yeah, for sure. So I definitely enjoy this aspect of it because most people think, oh, there is this story. So the story is, I'm not even going to tell the story because I've read — Brian Hayes has an article where he tries to get to the bottom of this actual story and where it came from, but the general idea is that, you know, some teacher of Gauss gave this as an exercise, to find this sum and expecting it to take a long time and Gauss produces the answer almost instantaneously. I like talking about this because a number of people have changed that story over the years. And so it gets more dramatic, or things like that, or a lot of people think that this is Gauss’s sum formula, that Gauss was the very first person to come up with this, like in the 1800s, like, nobody knew that, you know, this was it. But this has certainly been known — you know, one of my favorite proofs is the picture proof where you imagine the sum of the first N integers is sort of almost like a staircase diagram, one box at the top, two boxes below that, three boxes below that, and so on. And you take two copies of this staircase diagram, rotate one 180 degrees, and stick them together, and you have an N by N +1 rectangle. And Martin Gardner attributes this to the ancient Greeks, right? So presumably, people been drawing this in sands, and all sorts of things, for as long as people been thinking about counting, right?

EL: I must admit, I do — like, that story always bugs me because people, I don't know, people will use it as evidence of like this amazing genius. And I'm sorry, if this is, I don't know if I sound like I’m bragging or something. But like, I figured this out when I was in school, and I'm not a Gauss, by any stretch.

KK: Don’t sell yourself short.

EL: And it's like, you sit around playing with numbers a little bit, then, you know, you can figure this out, it's figure-out-able, which I think is good for people to know, rather than think, Oh, you have to be, you know, some native genius to be able to figure something like that out.

TE: Yeah, for sure. And, and I think, like, I don't know if you've read Brian Hayes’s article on it or not.

EL: I think so.

TE: Yeah. He brings up the point that maybe the reason people like it is because it's sort of, like, the student having this victory over the the mean classroom teacher. And somehow we just love this idea, not necessarily the genius myth, but this idea that like, oh, the the student won, or something like this. But yeah, but it's fun to talk about too. And just that always opens up the conversation with people about all the misattribution that we have in mathematics, right? Theorems named for people that maybe don't even have anything to do with that theorem, for one reason or another.

KK: So let's talk proofs. So you mentioned the one that Martin Gardner did with the picture. Okay. What's your favorite proof? Do you have one?

TE: Yeah. I mean, that one's pretty amazing, if you ask me. You know, I mean, another reason I like this is that this is sort of, if not the, it's probably the standard first induction proof that any undergraduate sees, right? So you learn about induction, and then you prove this formula by induction. I dislike that proof in one sense, and I love that proof in the other, right? So it's nice from learning induction. On the other hand, it's like, man, it's induction. I didn't get anything out of that. Whereas that picture proof from the ancient Greeks, right, just tells you exactly what what to do, right?

EL: Yeah. And I'm trying to remember is there a book or something called, like Proofs without Words or something like that? And it's a great proof without words, because it doesn't take a whole lot of scaffolding to show this picture and the numbers and to see exactly what's going on.

TE: For sure. Yeah, yeah. So Roger Nelson has three compendia now, like Proofs without Words, right? So this is three books, maybe almost a total of 600 pages of diagram proofs. And that one is in the first edition. And it's definitely — I mean, there's a couple iconic proofs without words, and I would put it as one of the top four iconic proofs without words. There's the Pythagorean theorem with a couple, and a couple of other ones that go along with it. But that's that. But my favorite proof actually — well, so, back in, like 2019, right at the end of 2019. Right, the beginning 2020 Before the before, sort of all the craziness, a mathematician named Enrique Treviño, who's a professor at Lake Forest College in Chicago, he was posting some things on Twitter about different proofs of this theorem and I knew a couple and I sent it to him, he's like, Hey, we should write these all up. So we got together and wrote these all up. And so we have a compendium that's online of 35 proofs so far, of the of the fact. And we finished that just before — I think it was end of January 2020, we sort of finished it. We've been working on it here and there ever since. But one that came out of there that's my favorite — and it's hard to describe, so I'll see what I can do — but it's also a picture proof. But instead of taking two triangular diagrams, so two staircase diagrams, you take eight staircase diagrams. The same kind of picture, instead of two and you just glue them together and you get a rectangle, you take eight. So again, the visual here should be sort of a right triangular stack of squares, N squares on the bottom, one square on the top, and then it's right oriented. And when you put eight of these together, you get a perfect square, except there's this one missing cell in the middle. And so it tells you that eight times this, this number, which these are called the triangular numbers, because they fit into these triangular arrays. So eight times the Nth triangular number is basically the Nth odd square. So (2N+1) squared, except missing one, missing one cell, so minus one. And this proof to me, it's much more complicated, in some sense. Like, why don't you just use the real picture proof, the easy one with two? But this one indicates that there are a lot of other things going on. So you can use this proof essentially, to prove that odd squares are congruent to one mod eight and these kinds of things right here. I mean, it sort of falls right out of that. And then this was key to Gauss’s — what's it called? — three triangle theorem, which says that every positive integer can be written as the sum of three triangular numbers. And so this fact plays a role. This visual proof plays a role there.

KK: Okay.

EL: Oh, nice.

KK: Very cool.

EL: Yeah, I'll have to draw that out later. I'm not quite sure I believe you, but I'll take your word for it for now.

TE: You’re going to have to draw it out, for sure. I was like, Oh, should I? Kevin asked my favorite. I wasn't going to necessarily going to talk about that one, but for some reason, I liked that one because it opened my eyes to a lot of other things going on in math as well. So it just has a connection, you know, thinking about what are called figurate numbers. So these are numbers that can be arranged in certain geometric patterns. So the triangular numbers, the squares, these are familiar ones to us, but there are just so many cool mathematical ideas that somehow I never picked up as an undergraduate or a graduate student about these, like Euler’s pentagonal number theorem, or Fermat’s polygonal number theorem, just amazing facts out there that I just never would have come across.

EL: Yeah, well, I guess that one is kind of an overpowered proof for that particular formula. But like you said, yeah, it kind of opens the door to a few different things, a sledge hammer for a mosquito.

KK: I like that.

TE: Those are some of my favorites of the ones that that Enrique and I compiled. One of the ones that sort of blew my mind that we came across was this idea that you can use Euler’s polyhedral formula for planar graphs, right? So the the planar graph version, you can use this and it proves the sum of the integers formula if you just find the right graph, and that's like a sledgehammer!

KK: Oh, nice.

TE: But it’s a beautiful, really powerful theorem for topologists. I think both of you somehow are topologists or topology-adjacent. Am I wrong about Evelyn? Not you?

EL: Yeah. Oh, yeah. Why not?

KK: No, it's true, Evelyn.

TE: The fact that you can use you know, this Euler’s polyhedral theorem, which I know has been featured on your podcast before, and maybe even recently, you know, to me was really powerful, like, oh, you're using something really strong. But it's also a way that you can introduce people to a cool idea with this relatively simple fact, elementary fact that they might be encountering as early undergraduate-level mathematicians, or even earlier than that.

KK: Very cool. All right, so I know visual proofs are kind of your thing. So have you animated this one? I know you like to animate these things. I see them on Twitter occasionally.

TE: Yeah, so I spend my time animating. For the past year and a half, two years this, this arose out of the pandemic, right, we all went online, and some of us were teaching online and kind of upset with maybe some of the digital content that we could produce. And so I spent some time trying to figure out how to how to do some animations. But yeah, so this one I animated, I animated 12 of them, so a dozen of the proofs from Enrique and I, that we compiled I animated a dozen of them last year. This was part of, I submitted as part of Three Blue One Brown, Grant Sanderson, runs this summer of math exposition stuff. So I submitted that last year as my video, the idea being that you should think deeply about simple things because you can encounter a lot of things along the way. And this is not my quote, this is a quote from Ken — the person who started the Ross program, and I'm forgetting the Ross program, I'm forgetting the founder. His last name is Ross but I can't necessarily remember the first name. Okay. So yeah, so I have animated some of them. And I believe I've animated, I think I've animated Euler’s polyhedral theorem, Pick’s theorem, the classic visual proof, there's combinatorial proofs. So there's like, a double counting proof. And then there's one that uses bijective proof. So just some really cool ones out there to see and explore.

KK: On YouTube? They’re on YouTube, right?

TE: Yeah, that’s on YouTube. Yeah. Mathematics Visual Proofs is the name of the YouTube channel at this point. Who knows? It changes if you have to change it, right?

KK: Well, we'll link to it. We'll find it.

TE: Okay. I appreciate that. Thank you. All right. Cool.

EL: Yeah. And so you said maybe this isn't the theorem you would have picked, if we had asked you, you know, three years ago or something. So how, how did this theorem get — Was it this project with Enrique that got you interested in it?

TE: Yeah, I mean, I've always loved the theorem, but sort of seeing all of all of the available proofs and the ways that it could open me up to things. It’s given me well, a couple of things. So when you teach a discrete math course, you can essentially teach the entire discrete math course using this theorem. You can talk about so many different discrete mathematical ideas using this and so it can be fun that way. So I've done that in a discrete math class and really enjoyed that experience with students as they see the connections being made. It's maybe a little more fun to talk about than some of the the others, I mean, the other theorem that I probably would have talked about is called Kummer’s theorem. And that one is fun to talk about, but it requires a little bit more knowledge, or a little bit more technical detail sometimes. So I like the accessibility in this one. I like that I get to speak with people — whenever I get to talk about this, I speak with people about the fact that mathematicians are looking for other proofs sometimes, right? I think mathematicians know this, we know this, that you're not always just looking for one proof. Some people say you're looking for the best proof, the so called proof “from the book.” I don't know if I agree with that. I just like the idea that we're looking for other proofs, other ways to try to understand these things to give that broad picture. And somewhere along the way, before or after, I came across this quote, It's my absolute favorite quote from a, from a mathematician, maybe ever, it's from Bill Thurston, who was a Fields medalist in the late 20th century and passed away only about a roughly a decade ago, maybe. He says, what did he say, “we're not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.” And I just, I wish I had said that. If I could have said that, I think I could die happy, like that was my quote. But I like the idea that we're not just — mathematicians aren't just sitting in the room trying to pump through more results, that we are actually interested in understanding and communicating and trying to get those ideas out.

EL: Yeah. And that, you know, what insight can we get by looking at this problem in a different way even if we already know the answer?

TE: Exactly. I think a lot of people just don't think that way about mathematics. People who are not, who haven't been around mathematics long enough, think that it's just one and done, right? You do this problem, and you move on to the next.

KK: Right, right. Or that we're just sitting around, like, doing arithmetic with really big numbers, right?

TE: Yeah, that's kind of what — that’s actually what that's what this is. This is arithmetic with really big numbers!

KK: That’s right. But clever arithmetic! They think we would just sit there and add it all up. It's like, why would I do that? I don't want to work that hard.

TE: Yeah. Yeah. I'm kidding. That's good.

KK: All right. The other thing we like to do on this podcast is ask our guests what it pairs with. What pairs well, with this formula?

TE: Yeah, so this is the greatest part about your podcast, not that there not other good things about your podcast, right? I think you two are great together. And it's fun, you know, but I think the idea of this and I was — this is the challenging part with with the other theorem I was thinking about. I was like, wow, what would I pair it with? I don't know. Presumably, I would come up with something. But this one was fairly easy for me. When I was younger, a movie came out, and over time, I guess it's become somehow I read online, that it's one of the greatest comedies of all time. I'm not sure if I agree with that. But I watched this movie a lot. And this movie is called Groundhog Day.

EL: Oh yeah!

TE: Have you seen Groundhog Day?

KK: Many times!

TE: Exactly.

EL: My thing about Groundhog Day is like watching it once is like watching it several times. Right. And then if you watch it more than once you've just like really increased your your volume of Groundhog Day.

TE: Right. So you you have no idea, exactly, you have no idea how many times you've seen this movie, you're sure you've seen this movie 30 times, but maybe you've only seen it twice. Right? But for people who haven't seen the movie, the premise is Bill Murray is a weatherman from Pittsburgh, Pennsylvania, and he's tasked with covering Groundhog Day and Punxsutawney Phil and he doesn't want to go there and essentially ends up in sort of a time loop where every morning he wakes up and it's exactly the same day and he's the only person who thinks he's reliving the day and everyone else is treating the day as the same. And so he does various things to try to, I guess the idea was to sort of “get it right,” sort of be the best possible person. But from my perspective, this is exactly — what would a mathematician do if they ended up in the Groundhog Day situation? Well, which is every single day I would just find a new proof of the sum of the integers formula and I would maybe never be bored. Maybe I'd never get it right and get out of the time loop. But I liked this idea because essentially in the movie, he learns a lot about himself, he learns a lot about the people around him. And this is sort of what happened with me working with Enrique and learning a lot of the things that come along with this theorem. You learn a lot of stuff and like, oh, this is stuff I didn't know, and it's led me to a lot of other things that I didn't know and connected me with other people. And so it's kind of like that movie, I guess. So, you know, sit down and watch that movie and figure out a couple of new proofs of the sum of the integers formula.

KK: And remind yourself of the genius of Sonny and Cher.

EL: Yes.

TE: A song that you probably probably can't listen to ever again, without automatically thinking about the movie.

KK: No, probably not.

EL: Yeah.

KK: No, that's a great pairing. I like that.

EL: Yeah, that's a nice one. I think. So I think in the movie, one of the things he does is he becomes this great piano player, right? Because he has so many times through the day. And you know, he goes, at some point, I think goes to his lesson and is like, oh, yeah, I've never played piano before and just busts out something. I always thought, like, oh, that would be — what would I have the dedication to do something like that if I got this time?

KK: What else do you have to do?

TE: Well, it's a great, that's what's so cool about the movie is, like, really, if you put yourself in that situation, you could do whatever you want. Right. I think that was what was so good about it in the end, he learned to play the piano, he learned to be a good person, I guess as well. But you know, like, you just learn a lot of things. He

KK: He learned to do ice sculpture!

EL: Yeah, that’s right.

TE: Yeah. The end scene, like, the last day when he does everything right, it’s just it really puts it, it brings it together so nicely. Like, oh, he saves that person's life and builds his ice sculpture and he's really filled himself out, right? I mean, there are some dark parts of the movie as well, but it ends nice. I can see why people might say it's the greatest comedy of all time.

EL: It’s up there, for sure, I think.

TE: And from the mathematics — there's this one scene, like from mathematics point of view, mathematicians, they famously love their coffee. And there's this one scene when he's kind of at one of his low points, and he's just eating all of the foods at the diner and he grabs this thing at coffee, and he just drinks it straight like that. I'm like, oh, okay, I could see a mathematician doing this in Groundhog Day.

EL: Yeah.

KK: All right. Well, this has been great. We always like to give our guests a chance to plug anything they want. So you've plugged the YouTube you've, you've plugged a little well, we plugged it for you.

TE: Yeah. Thank you. Oh, yeah. Plug the YouTube I appreciate.

EL: And Math Horizons, which I'm still involved with for one more year. And then there'll be someone taking over there. Yes. Yeah. It's been a long time. I don't know if I have anything else to plug otherwise, I appreciate you all having me on. It's fun to come and talk about these things. I guess I could plug — No, I don't know, for mathematicians interested about this favorite proof that I mentioned of the sum of the integers formula, this somehow told me that there's a connection between, there's sort of three famous proofs that you see as an undergraduate math major, would be the sum of the integers formula for induction, the fact that the square root of two is irrational. And then maybe the arithmetic mean, geometric mean inequality, you might learn as a first inequality type proof in a in a real analysis course or something. But somehow, there's a visual proof for all of these and the visual proof is somehow the same. So I think that possibly those theorems are somehow the same, in some realm. And so I spent a little time trying to prove one of those theorems using different techniques. So I recently had an article if people want to check in Math Magazine about the arithmetic mean, geometric mean inequality, where you prove it using moments of mass and centers of mass. And I was inspired to do this because David Treeby proved the sum of integers formula using moments of mass and centers of mass.

KK: This one? [Kevin holds up Math Magazine.] It happens to be sitting on my desk.

TE: That’s a different one.

KK: That’s not you?

TE: I didn't — I didn't know that — No, that is me, and I wasn't going to plug them both. But that's where I use the centers of mass to prove that the square root of two is irrational.

KK: Okay, that's what it is.

TE: So somehow this proof allowed me to connect those things together. And so it's been fun to play around with ideas that I that I don't know. So if you're interested in how balance plays a role in pure mathematical ideas, I would check those out. So that's one thing I can plug.

EL: Yeah, we’ll link to those. Those sounds really interesting.

TE: Thank you.

KK: All right. Well, Tom, thanks so much. It's been terrific.

TE: Yeah, thank you both. I know it's hard work, the work that you all do, but I think the community needs it and we appreciate it and it's great for my drives to work.

KK: Okay, thanks.

EL: Well thank you.

[outro]

On this episode of the podcast, we chatted with Tom Edgar of Pacific Lutheran University about the formula for the sum of integers between 1 and n. Here are some links you may enjoy:
His website and Twitter profile
Math Horizons
His collection, with Enrique Treviño, of proofs of the sum formula
His YouTube channel, Mathematical Visual Proofs, including his video on the 8-triangle proof of the sum formula
His article about proving the square root of two is irrational using centers of mass
His article about using centers of mass to prove the arithmetic-geometric mean inequality

Also, Brian Hayes’s article about Gauss: https://www.americanscientist.org/article/gausss-day-of-reckoning

Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I am one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida, and your other host is…

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, where we sadly are past our beautiful, not too hot spring and fully into summer. So we enjoyed it while it lasted. I didn't have to turn on any air conditioning until after the start of July.

KK: I think we started air conditioning in March.

EL: Slightly different.

KK: Little different vibe down here in Florida, but that's where we are. So anyway, it's summertime here, which means that there are tumbleweeds rolling through my department and I'm answering a few emails a day and trying to work, trying to do math. And boy, sometimes it's hard, you know, but sometimes it isn't. So. Anyway, so today, though, we are — this is great — we are very pleased to welcome Tatiana Toto, who will introduce herself and let us know what she's all about.

Tatiana Toro: Thank you very much for the invitation. I'm very glad to be here. And in fact, I'm very glad to see Evelyn's cloud that I had heard about in other podcasts. So I'm Tatiana Toro. I'm a mathematician at the University of Washington, where I have been a faculty member since 1996. And currently I am the director of the Simon's Lab for Mathematical Sciences Institute, formerly known as MSRI. And I'm in Berkeley, California, and summer hasn't arrived yet.

KK: It never will.

EL: Yeah, that’ll be November, right?

KK: I had actually forgotten that the name of MSRI had changed to the Simon's business. That’ll take some getting used to. I think I mentioned before we started talking, I spent a semester there, way back in 2006, and my son came with me, and my wife did too, and he was seven at the time. And now he's an adult living in Vancouver. It's weird how things change. I love that building, though. And the panoramic view you have the bay, and you can watch the fog roll in through the gate at tea time. It’s just a really wonderful place. So congratulations. How long have you been director? Has it been a year yet?

TT: It’s almost a year, a year August first.

KK: Yeah. That's fantastic. What a terrific position. And I'm glad that you're willing to take it on. Do you split your time between Berkeley and Seattle? Or are you mostly in Berkeley these days?

TT: I am mostly in Berkeley. My students are still in Seattle, so I see them mostly on Zoom. But once in a while on a Friday, in Seattle.

KK: Oh, so you go there. You don't fly them down?

TT: Some of them have come, actually one of them this year to the summer school.

KK: All right. So what is this podcast about? Favorite theorems. And you told us yours ahead of time, but we'll let you share. What is your favorite theorem?

TT: Okay, so my favorite theorem is the Pythagorean theorem, and I know that everybody's gonna say what on earth are you talking about?

EL: No, I really, really love this choice. And, you know, I've said this on many other iterations of this podcast, but I love that, you know, we'll get things that span the gamut from Pythagoras theorem, or the infinitude of primes, or something like that, all the way up to something that you, you know, you need to have been researching for 20 years in some very ultra-specific field to even understand, and so, you know, it just like shows how math connects with us in different ways at different times in our lives, and how we can appreciate some maybe things that seem very simple about math, even when we have had math careers for for many years. So yeah, tell us about how did you end up settling on the Pythagoras theorem?

TT: So, actually, it has played a very important role in my career. Like, when I describe it to my students, when I'm teaching a graduate class and I talk about the some of the theorems I'll describe in a minute, I tell them, you know, one of the key ideas in my thesis was the Pythagorean theorem. So let me explain. It appears in many other results in this area of geometric analysis. So for example — let me give you two examples. So what was my thesis about? You have a surface, a blob in space, and you're trying to — two dimensions in R3 — and you're trying to understand if you can find a parameterization, which means a good way to describe it in terms of the plane. So can you deform the plane in a nice way so that it covers the surface? And a nice way means that distances are not changed too much. So I had some specific conditions for this surface, and the answer, the key, is in the situation I was looking at, yes, you could do it. And when you go and deeply look at what makes this possible, it is the Pythagorean theorem because the basic point is that if you can control how distances are distorted, you can control how the whole shape is mapped from the plane. And at the time, it looked like a curiosity. You know, I graduated many years ago. At the time, a few years earlier, Peter Jones had solved the analyst’s traveling salesman problem, which I'm gonna — just in general terms, let's imagine you have a lot of points in a square, and you're trying to understand whether you can pass a curve of finite length to all of these points. You're going to tell me, “If they’re finite, of course you can.” But you want to do it in an efficient way, in a way that doesn't depend on the number of points. And so he had found the condition that told you if this condition is satisfied, then yes. And there's not an algorithm, that doesn't exist yet, that tells you what's the best curve, but there's a curve, and he tells you that the length is no more than something. And what's behind that is the fact that if you have a straight triangle that has sides, A and B, and the other one is B, A squared plus B squared equals C squared. And it really is understanding that. And there's another important thing, the fact that the square root also plays an important role in these, but really, really, if you ask me, “What are the tools you need in this area?” I'll tell you how the square root behaves in the Pythagorean theorem, and then a couple of good ideas and you're able to reconstruct the whole thing.

KK: I’m now curious about this traveling salesman problem. So there's no algorithm though?

TT: No, there's no algorithm. I used the word analyst’s traveling salesman problem because the analyst wants to know whether you can pass a curve of finite length. Maybe you can say you're not ambitious enough. You don't want the shortest possible curve. To build the shortest curve, there’s no algorithm. And the construction of Peter Jones builds a curve, but it's not necessarily the best one.

KK: Sure. Yeah.

TT: It doesn't tell you it tells you the length is no more than D. But it's not. Yeah, no.

EL: Yeah. I'm trying to remember if, like, I think there probably are some algorithms or some, like results that say like, you can get within a certain percentage of something. But yeah, the algorithm for the actual fastest path doesn't exist yet. Which is, you know, it's one of those things, it's like, huh, that's kind of surprising that we don't have a way to do that yet. Just means that there's still work to be done. Still jobs out there for mathematicians.

KK: Well, because the combinatorial on the graph theory one is, is NP complete, right? I mean, yeah. So that that are NP-hard, or whatever. NP-something. I've never been clear about the differences. But is this one known to be that too?

TT: I believe.

KK: Okay. All right.

TT: But you can construct — you know, so this was what was interesting about the problem, the result of Peter Jones, is that — the result of Peter Jones, and I have to say, I was very ignorant of that result, which had just happened a few years prior to my thesis. I have to remind the young audience that at the time, there was no internet the same way, and there was no arXiv, and you know, there was no Zoom. And then Peter Jones had a couple of postdocs at Yale, Stephen Semmes and Guy David, who started working on this. And the truth is, may I tell story about my thesis?

EL: Yeah.

KK: Please do.

TT: So my thesis came out of misunderstanding. I went to my advisor, and I showed that these surfaces that I was looking at, which were some that he had looked at, that there was this property about distances over the surfaces, like if an ant traveled on the surface between two points, you know, taking the shortest path, it was comparable to the Euclidean distance. And so I went to my advisor, Leon Simon, and I told him, you know, I've been able to do this about these surfaces. And then he told me, oh, then I guess they have about they admitted bilipschitz parameterization, which is this good description. So okay, so I went to the library, and I looked through every possible book that I could find, and I couldn't find that. So I went back two weeks later and asked if he’d mind giving me a reference for this results, and he said, oh, I don't have a reference. That must be true.

KK: It must be true.

TT: And that became my thesis problem. And then, oh, there were many iterations of attempts. And I could do specific cases, but I could not do the general case. And on May of my fourth year, finally, somebody gives a colloquium where he talks about good parameterizations. And he talks about things like what I was thinking. I was thrilled. I mean, I thought, oh, I'm going go read everything this guy has written and my answer will be there. And then I told my advisor afterwards, I think I'm going to go read this guy's work. And this guy was Stephen Semmes, and he comes from harmonic analysis. And my advisor says, no, stop reading, I don't want you reading anymore. You just prove that theorem and that’s it. I don't want you reading. But one good thing, you know, harmonic analysts use squares, rather than balls. That's the most useful comment my advisor had.

EL: Huh!

TT: And what's interesting is that Stephen Semmes was talking about a broader class of surfaces than mine. And for those, he was asking, “Do bilipschitz parameterizations exist?” And for those the answer still is not known. And if I had gone and read everything that he had written, I mean, he was the big shot, I was the student, I might not have gotten my result. And I remember when I told Stephen at some point in the fall, oh, you know, I proved this, his first question, his first reaction, was, “I don't believe you.” And he said, “How did you do this?” And I said, “Using the Pythagorean theorem.” And so that's why the Pythagorean Theorem really is very dear to my heart.

EL: Yeah. So I imagine that you saw the Pythagorean Theorem many years before you were in grad school. Do you remember, did it make a big impact on you when you saw it in school for the first time? I don't know what what year that would have been, elementary or middle school or whatever it was?

TT: So I remember, I think I remember when I saw it because I remember the book. I had a beautiful — I went through the French system. I'm Colombian, but I went through the French system, and in the French system at the time, they tracked us very early on. And so we had these beautiful math book that, you know, I still remember how it smelled, and it was in there. But I remember the book, not especially the theorem. I never thought much about it until I got to graduate school. I used it other times.

EL: Right. I mean, I think maybe the beauty of that kind of thing isn't necessarily what you're looking at, when you're a kid and first seeing math. You’re more like, okay, how can I use this to do the problems on the homework or something like that? So you were tracked into math pretty early on? You knew very early on that you were interested in math?

TT: Yeah.

KK: It’s nice they let you just do math. I think in the US what happens, I think, is students who are good at math are told they should be engineers. As if they're kind of the same thing, and they're not.

TT: But that, you see, now, you feel free to remove this if you want. That's what the boys were told. The girls — since math was roughly like philosophy, and I come from a South American country, it was okay.

KK: That’s fascinating. Okay, interesting.

EL: Yeah. Well, I mean, there's a lot of different, you know, philosophies about whether tracking that early, you know, kind of deciding on what direction you want to go that early, is good or not. You know, it works for some people and not others, definitely.

TT: Absolutely. I think it worked for me very well. And it didn't work on any of my classmates who were in the same class. I mean, I thought everybody loved it the same way I did and had as much fun. And then, it's interesting. Later on, I've learned that that wasn't the case. And then some of them suffered through it, you know. But to me, it was great.

KK: So this is a French system in Colombia? Okay, this is a bit — okay, let’s get there. How did that actually happen? Why were there French schools in Colombia?

TT: Well, I'll explain why there were French schools in Colombia and how I got into a French school. So there's something that's called a cooperation agreement between France and developing countries, where they have schools. The primary reason to have them is so the kids of their diplomats can continue their studies, but then they also offer them to the general population at a very reasonable price. They are private schools, but they are not as expensive. They're a fraction, or they used to be a fraction, of what the other private schools were. And at the time, so Colombia for a long time was what was called a Sacred Heart country. And so the ties with the Catholic Church were very strong. And so in terms of education for the girls, it was most girls went to nun school. But I am not Catholic, and therefore I couldn't, that was not an option for me. And so we needed a coed school. I mean, my parents wanted a coed school. The girls schools were all nuns. They wanted a coed school, and we needed an affordable coed school, and public schools were not good, and still unfortunately are not good. That's how I landed in the French school.

KK: Fascinating.

EL: Wow. Okay. Yeah.

KK: Our listeners are learning all kinds of stuff, right?

EL: Yeah, yeah, we've wandered a little away. But luckily, we know, thanks to the Pythagorean theorem, that we can walk back in a certain amount of time. So yeah, the other things that we like to do on this podcast is have you pair your theorem with, you know, some food, beverage, sport, you know, whatever, delight in life you would like.

TT: So I actually will pair it with walking. So I'm going to give myself the title of urban hiker. I do walk long distances around town and in cities on a regular basis. I mean, I walk about two hours a day, at least. And so I pair it with that, because most often when I walk, I'm actually doing exactly the opposite of the Pythagorean theorem. I want to go the longest possible way, not the shortest possible way. But once in a while, I take the diagonal. And now that I'm living here in Berkeley, there's a beautiful diagonal that I take. And so I think about that here often.

EL: Yeah. Do you like the hills?

KK: Yeah, I was about to say, do you actually hike all the way up to the building there? Because that is quite a hike.

TT: Not when I'm coming to work. But sometimes on weekends I do. You know, I want to crease and it depends. It depends what I'm doing while I walk. I use walking as a way — if I am listening to a book, then I can go up the hill. But if I want to talk on the phone, I need to go down the hill, because the reception here is terrible! I know exactly at what point on the hill, you lose AT&T.

KK: That’s true. Yeah, like I said, I was there some time ago and cell phones weren't quite as good as they are now. But yeah, my reception was terrible at the institute.

TT: Well, your cell phone might have improved, but the reception hasn’t.

EL: Yeah. I love this pairing I love walking and biking as like, ways to, you know, see the city on a human scale instead of when you're in a car or something and you just almost teleport from point A to point B, you don't like see the — you kind of don't get the same environment around you, that kind of effect. So I like that. Even though walking is also a great time to sort of, like, let your mind wander and not think about what's around you, listen to your book, or talk on the phone with someone, or think about proving that next theorem, or anything like that. So it's kind of that, it has both of those things.

KK: You live in two great cities for walking.

TT: Ye.s. With respect to, you know, seeing things differently, one thing I find amazing is that depending on what side of the street you walk, you see things differently.

KK: Absolutely. All right, this has been terrific.

EL: Gotta be some metaphor in here.

KK: I’m sure, I’m sure. Yeah. So it's always nice to get another perspective on the Pythagorean theorem. So we didn't even — it's one of those things that everyone knows so much, we didn't even tell them what it was. I think it was embedded in there somewhere.

EL: Yeah.

KK: But the idea that it is still vital, like still important in modern research mathematics, you know, is a really interesting thing to know about. We all just sort of take it for granted. Right?

EL: Yeah, this theorem that has been known by humans for millennia. And, you know, still is important.

TT: One of the things that these ways of building parameterizations, so they developed into a whole field, and then they moved to other areas. So there are some recent results by Naber and Valtorta trying to look at a singular set of minimizing surfaces, varifolds, you know, that minimize some sort of energy. And they have been able to give a very good description of the singular set by using these type of parameterizations. And they're all basically, the basis is always the Pythagorean theorem. It's really, that's how distances change.

KK: That’s right. It’s completely fundamental.

EL: Thank you so much. This was really fun.

TT: Thanks for the invitation.

[outro]

In this episode, we were happy to talk with Tatiana Toro, mathematician at the University of Washington and director of the Simons Laufer Math Foundation (formerly known as MSRI), about the Pythagorean theorem. Here are some links that you may find interesting.
Toro's website and the SLMath website
Our episodes with Henry Fowler and Fawn Nguyen, who also love the Pythagorean theorem
The analyst's traveling salesman problem on Wikipedia
Naber and Valtorta's work on singular sets of minimizing varifolds

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. Or perhaps today we should say the maths podcast with no quiz at the end. My name is 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. It's Juneteenth.

EL: It is, yeah.

KK: And I'm all alone this week. My wife's out of town. And yesterday was Father's Day and I installed cabinets in the laundry room. This is how I spend my Father's Day, something we've been talking about doing since we bought the house.

EL: That’s a dad thing to do.

KK: 14 years later, I finally installed some cabinets in the laundry room. So it looks like you had a good time in France, judging from your Instagram feed.

EL: Yes, yeah. And I'm freshly back, so I'm in that phase of jetlag where, like, you get up really early. And so it's 9am and I already went for a bike ride and did some baking and had a relaxing breakfast. At this point, I'm always like, “Why don't I do this all the time?” But eventually my natural circadian night owl rhythms will catch up with me. I'm enjoying enjoying my brief, brief morning person phase.

KK: Yeah. Never been one, won’t ever be one as far as I can.

EL: Yeah. Just keep moving west, and then you’ll be a morning person for as long as you can keep jetlag going.

KK: That’s right. That's right. Yeah.

EL: So yeah. Today we are very happy to have Sarah Hart on the show. Sarah, would you like to introduce yourself? And tell us a little bit about, you know, what you're all about?

Sarah Hart: Ah, yes. So my name is Sarah Hart. I'm a mathematician based in in London in the United Kingdom. I'm a professor of mathematics, but my true passion is finding the links and seeing them between mathematics and other subjects, whether that's music or art or literature. And so I think there's fascinating observations to be made there, you know, the symmetries and patterns that we love as mathematicians are in all other creative subjects. And it's fun to spot them and spot the mathematics that's hiding in all of our favorite things.

EL: Yeah. And of course, just a couple of months ago, you published a book about this. So will you tell us about it?

KK: Yeah,

SH: So this book, it's called Once Upon a Prime: The Wondrous Connections between Mathematics and Literature. And in the book, I explore everything from the hidden structures that are underneath various forms of poetry, to the ways that authors have used mathematical ideas in their writing to structure novels and other pieces of fiction and the ways that authors have used mathematical imagery and metaphor to enrich their writing, authors as diverse as you know, George Eliot, Leo Tolstoy, Marcel Proust, Kurt Vonnegut, you name it. And then I also look in the third section of the book at how mathematics itself and mathematicians are portrayed in fiction, because I think that's very, very interesting and shows us the ways in which those things at the time the books are written, how is the mathematics perceived? How has it made its way into popular culture? And how mathematicians are perceived as well, that tells us something fascinating, I think, about the place of mathematics in our culture.

EL: Yeah, definitely.

KK: We’re always portrayed as either mentally ill. Or just, like, absurd geniuses, you know, when really, you know, we're all pretty normal — most of us are pretty normal people, right?

SH: Yeah. Well, we are, as everybody, there's a range. There's a range of ways to be human. And there's a range of ways to be a mathematician. But yeah, we're not all tragic geniuses, or kind of amoral beings of pure logic, or any of those things that you find in books. So yeah, and there are some sympathetic portrayals of mathematicians out there, and I know I talk about some of those, but yeah, it's very interesting how these these tropes, these stereotypes can creep in.

EL: I must confess I'm about three quarters of the way through, I haven't quite finished that last section. But the first few sections that I've read, I've definitely — I keep adding books to my “Want to Read list,” so it’s a little dangerous.

SH: Oh yeah, it should have a little warning, the book, saying “You will need a bigger bookcase.” Unfortunately, you know, you will want to go and read all of these books. And yeah, “Sorry, not sorry,” I think is the phrase.

EL: Yes, definitely. I downloaded — so I don't need a bigger bookshelf because I put this one on my ereader — but I downloaded The Luminaries, which sounds like a really interesting book and excited to get to that, you know, in the neverending list of books that I'd like to read.

KK: Right, we were talking about talking about our tsundoku business before [tsundoku is a Japanese word for accumulating books but not reading them]. So I actually I did, with a friend in the lit department, or in the language department, we taught a course on math and literature a few years ago.

SH: That’s fantastic.

KK: It was. It was so much fun. It's the best teaching experience I've ever had. But I was glad to read your book because we missed so much. Right? I mean, of course, we only had 15 weeks, you know, we and we talked about Woolf, like To the Lighthouse is kind of an interesting one. And yeah, I did finish the book. So sorry, Evelyn, I won. But no, it's it's actually, you know, it is spectacularly well written and, and I'm glad you're having success with it. Because it's — again, I like this idea, that you're sort of humanizing mathematicians and mathematics and showing people how it's everywhere. Isn't that part of your job? Aren’t you the Gresham professor, is that correct?

SH: Yes, I’m the Gresham professor of geometry. So Gresham College is this really unique institution, actually. It was founded in 1597 in the will of Sir Thomas Gresham, who was a financier at the Court of Queen Elizabeth I in Tudor times. And in his will, he left provision for this college to be founded that would have seven professors, and their whole job was to give free lectures, at the time to the people of London. Of course now it's all livestreamed and it goes out and is available all over the internet. And anyone could go and it was just, you know, if you wanted to learn these subjects — and he thought there were seven most important subjects at the time that he said, I still say, geometry and mathematics more broadly, very important — but it was geometry, music, astronomy, law, rhetoric, physic, which is the old word for medicine, and I perhaps I’ve forgotten one. But yeah, these subjects, and so still today, this is what Gresham College does, free public lectures to anyone who wants to come. Now, you used to have to give them once in Latin and once in English. Now, you do not have to do it, thank goodness.

KK: Yeah. Who would come?

SH: I don’t know. Yeah, if I had to suddenly give my lectures in Latin, that might be slightly more of a challenge. My role there is to communicate mathematical ideas to anyone who wants to listen, so a general audience. And some of them will have mathematical training, but many will not. And they they're just kind of interested people who find things in general interesting, and mathematics is part of that. I love that idea, that mathematics is part of what a culturally interesting person might want to know about. And that is something that perhaps used to be more so than it is today. And I really would like mathematics to somehow be rehabilitated into what the cultural conversation involves, rather than it seems to be perhaps in a little bit, sometimes it's pigeon holed or put to one side, you have to be a geek to like mathematics. You have to be unusual. And it's really not true. It's not the case.

EL: Yeah. Wow, that sounds like a dream job. I’m writing that down and putting it on my dream board? It's yeah.

KK: I seem to remember, so I read the review of your book, I think by Jordan Ellenberg, who's also been on.

SH: Yes.

KK: It mentioned that the first person who held your chair invented long division. Is that right?

SH: It’s true.

KK: That's what used to get you a university job, is you invent long division.

SH: Yeah. So that's, you know, what a lineage to be part of. I really feel honored and humbled to be in that role. And, actually, I'm the first woman to do this job in its 400 and whatever year history which, yeah, okay, you could say, yes, we might be a bit late with that one. But I feel it's a real privilege to do it.

EL: Yeah. Well, that's wonderful. So we have invited you on this show to tell us what your favorite theorem is. So have at it.

SH: Okay, so, my favorite theorem, I guess it's could be called a collection of theorems really, but the properties of the cycloid. So the cycloid is, it’s my favorite curve. And it's my favorite curve that probably unless you're a mathematician, you may not have heard of it. So people have heard of ellipses and circles and parabolas. And they've heard of shapes like triangles and things, but cycloids, people tend not to have heard of. And for me that's a surprise because they're so lovely. And the history of the study of the cycle of which, you know, we can we can talk about, is so fascinating and fun, and so many of the most famous mathematicians that people have heard of, like Isaac Newton, and Leibniz, and Mersenne, and Descartes and Galileo, and Pascal and Fermat, all of those people worked on the cycloid and were fascinated by it. And so there are these beautiful properties that it has, which we can bundle up into a theorem. And that would be my favorite thereom.

EL: That’s great. And yeah, in case anyone listening to this doesn't know about the cycloid, it’s a cool curve. And it's actually, you know, it's a curve that a lot of people haven't seen as such, but it's one that does kind of arise sort of in everyday life, kind of. So yeah, do you want to describe what a cycloid is?

SH: You can make a cycloid quite easily. It’s a fairly natural idea, I would say. Imagine a wheel rolling along the road. And now somewhere on the rim of the wheel, you paint the put a little blob of paint, or something like that, or if it's in the dark, you can put a little light. And then and then as the wheel rolls along, that blob of paint or little light will be following a particular path, as the wheel rolls.

EL: Going up and down.

SH: Kind of up and down. And eventually, sometimes it'll touch where the ground is. And then we'll go up and down again. And what you get is a series of arches, they look like arches. And that's what the cycloid is, normally you take one arch and call that the cycloid.

KK: Right.

SH: So this is quite a natural idea, what kind of shape will that be? And what is this arch shape? And the first thing you can say is, yeah, is it something I already know about? So early on in the study of this curve, which is first written down as a question, what is this shape? About 1500. Marin Mersenne, who is famous for Mersenne primes, among other things, so he thought maybe it's half an ellipse. And that's not too bad an approximation, but it isn't quite that. And so that's sort of question one. Is it something we already know? And it wasn’t. So then, people like Galileo started to ask, well, what do we what do we like to know about shapes and curves? So there are two questions really, at the time, they were called the quadrature question and the rectification. So quadrature is what's the what's the area? So if you make this arch, what's the area underneath this arch, between the arch and the road, I guess. That's question one. And the other one is the rectification: what's the length? So how long is this arch in terms of the circle that makes that makes the arch, the cycloid. And Galileo didn't know how to calculate either of those things. But he actually made, he physically made a cycloid. So he got a piece of sheet metal, and he rolled a circle along it, and he got the path. And then he cut it out and he weighed, he weighed the bit of metal that he had.

EL: Oh wow!

SH: To find an estimate for the area. Okay? So this is a real hands on thing.

EL: Yeah, that’s commitment.

SH: Because he did not know. So he physically made it and weighed it. And he got an answer that was around about three times the area of the of the circle that makes it, roughly speaking, and he said, Okay, if we all think, what’s a number that's roughly three, that's to do with circles, right? And so he wondered, could it be pi times the area of the circle? It isn't. It isn't pi times the area of the circle! Galileo never managed to work out exactly what it was. But this guy Roberval, Gilles de Roberval, did manage to work out what the area is. He didn't tell anyone how he'd done it because at this time in history, there were all these priority disputes, who sorted this thing first, who has done what first? People would sometimes go to the length of writing their solutions in code. So Thomas Hooke, who was another Gresham professor, when he worked out what we call Hooke’s law now, he wrote Hooke’s law down as an anagram in Latin, before he told anyone else. And then if anyone else came up with it, he could say, look, here's my anagram that I did earlier to prove that I thought of it first. So there were all these weird and wonderful things that people did at that time to establish priority. But Roberval, he had this incentive for not telling that he knew the area under a cycloid. And the incentive was this — it was not a good idea for them to do this — the job he had at the time, Roberval, was renewed every three years. And to get the job every three years, there were some questions that were set. And if you could answer those questions the best out of all the people who tried to do it, you could get that job for the next three years. But the person setting the question was the incumbent professor. So if you're the incumbent professor, you need to set questions that only you know the answer to, and then you get to keep your job. So for a few years, Roberval could say, you know, what's the area under this cycloid, and no one else knew. So he worked it out. And his proof was quite nice, but it wasn't published until 30 or 40 years after his death. But it actually — and this is the first lovely thing about the cycloid — the area, if you have a circle that's making this cycloid by rolling along road, the area underneath one of these arches is exactly not pi times, exactly three times the area of the generating circle. So a lovely whole number, simple relationship between the arts.

EL: What are the odds? It’s almost miraculous.

SH: Fantastic. So here's another equally miraculous thing that kind of adds to the first one. Then people try to work out what's the length of this cycloid? And the person who managed to solve that was, in fact, Christopher Wren. So he's well known as an architect, and he designed St. Paul’s, the wonderful dome of St. Paul's in London, and many other churches in London. But he was also a mathematician among many other things. So he solved the rectification problem, what's the length, and if the circle that makes this, the cycloid has diameter d. So we know that the circumference of that circle, the length around the circle would be pi times d. Well, another beautiful whole number relationship, the length of the cycloid arch is exactly four times the diameter. A beautiful whole number relationship. It's fantastic. So you've got these two lovely properties of the cycloid. And people were fascinated by it. So it had this nickname, the Helen of geometry, as in Helen, you know, face that launched a thousand ships.

KK: Right.

SH: It was a very beautiful curve with beautiful properties. But there's another reason why it was called the Helen of geometry. And it was because, like Helen of myth, it started lots of squabbling. So I mentioned Roberval, who had proved the area formula for the cycloid. Someone else came along a few years later, and found out this this result, and Roberval immediately accused him of plagiarism. And this guy was like, No, I didn't do that. But they argued about it. I think it was Torricelli. And and When Torricelli died a few years later, team Roberval said he's died of shame because of being a plagiarist. He may have died of shame. But he also happened to have typhoid at the same moment. So you know.

KK: Sure.

EL: Shame-induced typhoid?

SH: But you know, so that was one squabble, but then Fermat and Descartes had an argument because they both proved something about the tangents to the cycloid. And they hated the way each other done this. So I think it was Fermat did have a particular method. Descartes said that this method was ridiculous gibberish. So you know, he's not mincing his words, he’s not saying “I prefer my method” but “Fermat is speaking gibberish nonsense.” So they argued. But, you know, this beautiful curve has other exciting properties. And this is where it goes for me from, “Okay, nice whole number relationships, cute.” But then one of the things that we all love in mathematics is where something you've studied over here, reappears in a completely different context. And this is what happens with the cycloid. So it comes up to in connection with trying to make a better clock. So there's this mathematician, Christiaan Huygens, who is trying to make a better clock. And he comes up with a pendulum clock. And so pendulum clocks improved timekeeping dramatically. Before the pendulum clock came along, basically, it was a sundial or nothing, really. There were no good mechanical clocks. And the ones that existed would lose about 15 minutes a day or something of time. The pendulum clock comes along. And so you can do kind of the mathematics of a swinging pendulum, and if you make a little approximation, so the approximation that you make is that for a small angle, theta, the sine of theta is approximately theta. So you can make that approximation. And it's pretty good for small angles. And if you do that, then when you work out what the forces are acting on the pendulum, you find that, roughly speaking, it'll take the same time to do its swing wherever you release it from. So it has this kind of constant period, basically. And that's why pendulum clocks are useful for telling for time. But they're not perfect, because we had to use an approximation to get to that point. So Christiaan Huygens is wondering, is there actually a curve that I can make, that will really genuinely have this constant period property, that wherever I release a particle from on this curve, it will reach the bottom in the same time?

KK: Right.

SH: Because that's what the pendulum almost does, but doesn't quite do. And so he said — and this problem is known as the tautochrone problem, because it's “the same time” in Greek. And it turns out, guess what, the cycloid solves the tautochone problem. It's precisely — so we have an arch, you've got to turn the arch upside down. So now you can roll, your particle can roll down. And wherever you release a particle from on the cycloid, it will reach the bottom in exactly the same time.

KK: Remarkable.

SH: I mean, assuming you know, it's smooth, no friction or whatever. It's just rolling down under gravity. And I mean, it's not even clear that such a curve could exist, right? It's quite a thing to ask. And yet, the cycloid has this property, and it's fantastic. So that's an amazing thing. And few years later — so Huygens worked this out. A few years later, a different problem was posed. It's kind of a related question, or it's something to do with particles anyway. And the question here is called the brachistochrone problem. And it was proposed by Johann Bernoulli, one of the Bernoulli brothers. And he posed this kind of publicly in a journal saying, Okay, if you now have two points A and B, A is above B, and you want to have a curve such that when a particle rolls down that curve from A to B, it will reach point B in the quickest time, so what might that be? Is it sort of a parabola, maybe a straight line, what's it going to be like? And this problem was posed to the mathematicians of Europe as a challenge, and quite a few big names enter this competition to see if they could do this. So Leibniz was one, Gottfried Leibniz, Bernoulli himself solved it, his older brother solved it, and then they got this anonymous entry. And it was so beautifully done, and elegantly produced, the solution to this, that, even though it was anonymous, when Bernoulli he saw it, he said this famous phrase, “I recognize the lion by his claw.”

KK: Right.

SH: And it was Isaac Newton, who had solved this problem. And guess what? It's the cycloid again. The cycloid solves this problem as well. So you've got this amazing curve, which is a natural idea. It's got these lovely whole number relationships about its length and its area, and then it suddenly also can solve these totally different questions about particles rolling down in the quickest time or constant time. And so that is why I love the cycloid so much. Everybody’s worked on it. It's got this amazing history, it's really beautiful.

KK: This sounds like a good public lecture.

EL: Yeah.

SH: I just get really.

EL: The cornucopia of the cycloid.

KK: Yeah, so question, the original area calculation that Roberval did, did he use calculus? Or was this a geometric argument?

SH: So he used something that isn't quite calculus yet, Cavalieri’s principle. If you're comparing areas, if you have got two shapes where if you slice through, the length of those slices is the same at every point, then the areas are the same. So he used that principle, which you can extend to volumes as well. And he kind of did a particular, so he managed to do this. And he had the curve that you make for the cycloid, he made it up from three different pieces. And he did this sort of slicing argument to compare it to with things he already knew, one of which was the sine curve, although I don't think he noticed it was a sine curve at the time, but we can now see that. So now, you would make that argument with calculus. But it's the same basic idea. You're slicing something very finely.

KK: Right. You could almost imagine Archimedes figuring this out.

SH: Yeah. Yeah, exactly.

EL: Yeah. So I mean, you've made a very compelling case that this is a very cool curve that has all these properties, So like, why is this your favorite? Or I know it's hard to pick a true favorite. But yeah, can you talk a little bit about, like, how you encountered it and what makes it so appealing to you?

SH: Well, there's at least two things. There might be three. One is, I love the simplicity of the results about the area and the length, that they are just lovely, simple relationships there comparing to the circle that makes this this curve, which itself is easy to think about what it is. So it's not contrived at all. It arises fairly naturally from just thinking about wheels rolling along roads. You get this curve, and then these relationships are very simple. The second reason I love it so much is because of this unexpected appearance of the cycloid in this totally different context from from how you imagined it. When it's generated by just, you know, a wheel, but then a curve that has these other properties, that’s very surprising. There are other things we could talk about to do with it. involutes, and other kinds of things where it crops up, but that for me, it encapsulates why it's such an exciting thing. And it's like when you first encounter pi or something, or you see the e to the i pi plus one equals zero, it gives you that same kind of feeling, that thing's from over here, and this other constants from over there, you know, that they're linked together seems really surprising. But the final thing, I suppose this kind of links in again with what we were saying about mathematics and literature, is how the cycloid has caught people's imagination over time. And it's both of mathematicians, but outside. And there are several books that mention cycloids. So Moby Dick is one. That's got a lovely little passage about cycloids. But also, Gulliver’s Travels mentions cycloids, Tristram Shandy by Laurence Sterne, this amazing, crazy 18th century book talks about cycloids. And those are just three that are really classic books. It was in the air at the time, and perhaps we don't necessarily — like, a modern and modern person may not have heard of cycloids. But certainly if you were educated in the 18th, 19th century, you may well have heard about cycloids. And that, to me, is very interesting too.

EL: Yeah, do write a little bit about this in your book that Moby Dick part, I have gotten to that part. And apparently, did you say that Melville apparently had some amazing math teacher in high school. And so, you know, kind of was able to really capture his imagination about math and then bring that into literature later, which is just kind of a cool thing to think about as math teachers, people who teach math. It's like, yeah, even if your your students don't end up in math or something, they might, you know, hopefully bring some of what you teach them that direction.

SH: Yeah, absolutely. I mean, it’s the value of having a great inspirational teacher. Just look at with Melville. So he had a teacher. He went to a school called the Albany Academy, and he was good at school in some areas, mathematics was something he was particularly good at. And he actually won a prize for being the first best at ciphering, was what it was for. Cipher, the old word for calculation.

KK: Right.

SH: His prize was a book of poetry, which I liked, because for me, that's absolutely a natural prize, but it wouldn't necessarily be thought so. But his teacher was a man called Joseph Henry. And Joseph Henry was no ordinary schoolteacher. He was a very good scientist in his own right, he went on to become the first secretary of the Smithsonian. So you know, pretty impressive. But physicists will know the name Henry, because the Henry is the scientific unit of inductance. And that's for Joseph, that is Herman Melville's maths teacher at school. So he was by all accounts an exceptionally good teacher, to the extent that some of his classes were actually, members of the public were allowed to come in and attend as public lectures. So there's a record that says, a request of his that he wants to have additional books for the more advanced students to entertain them beyond the normal curriculum. And so I don't know, and we can't know for sure, how Herman Melville learned about cycloids. But I could very easily imagine that a lesson on Friday afternoon, let's just talk about this fascinating curve because it's really interesting. And Melville did have a love, then, of mathematics, which just comes out in his writing. You can just see it, the way he chooses metaphors and imagery, they're often mathematical. And you can just see it's, it's not thinking “I must include some mathematics.” It's just the sheer pleasure of it. The delights, the joy of mathematics just comes out in his writing, which is wonderful to see.

EL: Well that was such a cool story that I read in there. And I have loved to revisit this. I don't think I've actually thought about cycloids since I taught calculus, right, which, it's been quite a while since I taught calculus. It is a fun, it’s a very common example in calculus books now. You'll kind of go through and solve some of these, these things. And I think, when you do parametric curves, maybe?

SH: Yes.

EL: So yeah, lots of fun, but I don't think I had really appreciated it as this whole whole thing before. So the other thing we like to do on this podcast is ask our guests to pair their theorem, or their bouquet of cycloid facts, with something else in life. So what have you chosen for your pairing?

SH: Well, so I've chosen Moby Dick.

EL: Okay.

SH: Because, I mean, he does talk about cycloids in the book. It’s not just because of that, but with the cycloids is this lovely passage where Ishmael, who is, you know, traveling as a deckhand on a whaling ship with Captain Ahab, who perhaps is not entirely sane, and we discovered that through the book, but there are many — Ishmael sort of has these wonderful meditations, he's just thinking about things. And some of them are mathematical, and some of them aren't. But there's one particular point where he is cleaning the the try pots. A try pot is something you had on a whaling ship, where there's great cauldron like pots where they kind of render the whale blubber down, and then you have to clean them. And so he says, you know, this is a place for wonderful mathematical meditation. And he and he talks about, as his soapstone is circling around the inside of the try pot, he says, I was struck by the fact that in mathematics, the cycloid is the curve where you can you can release something and it falls to the bottom in a constant time. And so he's just sort of drops in, the cycloid, just mentions it while he's daydreaming about something else. But Moby Dick, it's full of mathematical ideas. And it’s, you know, they are interested in numbers, to the extent that Ishmael keeps, he has the data or information about whales, measurements and statistics about whales, he has them tattooed onto his body, because as he says, you know, I didn't have a pen to hand, kind of thing, there was no other way to record. So he just has them tattooed on his body. Ahab is doing calculations on his ivory leg, you know, there are all these discussions about number. But there are lovely pieces of imagery around the infinite series of ripples in waves in the sea. There's a metaphor about loyalty where Ahab says to the cabin, boy, you are loyal as the circumference to the center, you know, the circumference always stays the same distance from the center. And it's just lovely little pieces of mathematical imagery throughout, and throughout all Melville's work. So I thought, yes, Moby Dick would be a very good pairing.

KK: Yeah. And so you actually have a paper about this in the Journal of Humanistic Mathematics, right?

SH: Yeah.

KK: Ahab’s arithmetic?

SH: Yes. And that itself is a little bit of a reference to a discussion that happens in Moby Dick, which is where two of them were talking about a book called Daboll’s Arithmetic, which was the kind of classic text in American schools, I think, at the time, which had all these rules about how to do calculations. And you could do mysterious things with with this book because, you know, if perhaps the mathematics hadn't been taught by a teacher like Joseph Henry, perhaps you learnt you've learned these rules off by heart, you don't quite understand them. And so they talk in the book about cabbalistic contrivances of producing these things. And at one point, someone says, “I have heard devils can be raised with Daboll’s arithmetic.” So, you know, this is the other side of mathematics, where people sort of hold it in or but also, perhaps, they have some suspicions around what do all these symbols mean? And it's very interesting, if you look at that book, Daboll’s Arithmetic, it isn't like a mathematics book would now be. So when he talks about how to find the areas of circles, for instance, pi is not mentioned at all. He says, you square the radius, and you multiply it by 22/7, or if you want a more accurate thing, you could multiply it by what's that other approximation right? 355/113? But he doesn't say “because these are approximations to pi,” it's just like, you can do this or you can do that.

EL: Here’s a number.

SH: Tust where's that come from? So that's a very interesting thing. And so there are mathematics books discussed or mentioned in Moby Dick as well. And if you know a little bit about them, so Euclid, of course, is mentioned a little bit. Yeah. So the book is full of mathematics. And I really wanted to think about in the article I wrote, why — how did Herman Melville know all this stuff? Why, you know, where does it come from? Because, you know, he's not a mathematician. And this is why, you know, nowadays, we're sort of taught to believe, or somehow we come to believe, quite often, that you're either a mathematics personal, or you're not. And if you're not, then you don't know any and you don't care. But this is absolutely not the case for one of our greatest writers, Herman Melville. And so you know, yeah, where did that come from? And, you know, it was just lovely to, to find out a little bit more about what he knew and how he knew it, and where it all came from.

KK: Very cool.

EL: Yeah, that's great. I have confessed to you already, but I will confess to our listeners that I have not read Moby Dick, but it is on my list that I hope to get to this year. It's a little daunting.

KK: You’d better get cracking.

EL: I know. I’ve only got six months.

KK: I read it at bedtime. That's when I tend to read, and so I read it, you know, maybe 10 or 12 years ago, and it took me quite a while. Yeah, yeah. It's pretty dense too.

SH: It is. I mean, I didn't read it till I was older. Because, you know, you hear this is the “great American novel,” and you should, everyone should have read this book, and then you feel bad that you haven't read it, and then you feel annoyed that you feel bad that you haven't read it. So there's all these barriers that you put up for yourself. And, you know, I'm so glad that I did eventually read it, because I loved it. It's so rich. And there is, you know, many many, many layers of interpretation and depth in the writing, but it is a great book. So yeah, I hope you will enjoy it when you read it.

EL: Yeah.

SH: You know that there are books that we all — I haven't yet read, I don't know if I will ever read, maybe one day, Finnegans Wake. I do mention it in the book because James Joyce, I talk about Ulysses a little bit and Dubliners in the book. But Finnegan’s Wake for me, I tried and I didn't quite quite get there. All I can say is in the middle of Finnegans Wake, there is a picture which could have come straight out of Euclid’s Elements. It's got equilateral triangles, two circles intersecting. But yeah, that for me, maybe one day, maybe I'll have a sabbatical one day and that will be what I do in that sabbatical.

EL: There just, there is so much. There's so many good books published now, you can't you can't read them because you’ve got to read last year's good books. But I mean, it's just you — Yeah, anything you read is great. And you’re never going to get to all of it. Enjoy what you read.

SH: Exactly. Amnesty of all our unread books. It's fine. We forgive ourselves.

EL: Yeah. Thank you so much. for joining us. This has been a lot of fun. You know, we do like to give our guests a chance to plug things but we've already talked about your book quite a bit. Is there anything else that you'd like to to mention about what you're working on or other things that you've published that you'd like us to share?

SH: Oh, no, I think I'm alright. So coming up. I mean, not for US listeners, but I've got an event coming up in a couple of weeks is going to be really fun because we're going to watch a classic B movie from the 1950s, which is this film about giant ants terrorizing the New Mexico desert. It’s called Them! with an exclamation point.

KK: Yeah, I've seen the posters.

SH: Yeah, right. Yeah. Which is super fun. But that's about, yeah, something has happened. Who knows? But there are giant ants. They have a lot of fun with it. But we're going to watch the film at the Barbican Centre in London. And then we're going to talk about, yeah, what does mathematics tell us about what life is like? Could giant ants exist, could giant spiders exist? Or giants, or tiny people like Lilliputians. And so that's a kind of fun thing that's coming up. But yeah, you've already, if you look at my book, you will already have a reading list that’s like 100 new books that are gonna be fun, fun to read and explore. So yeah, there's plenty to go on.

EL: Great.

KK: Thanks so much, Sarah, this has been great fun.

SH: Thank you for having me. Yeah. I’ve loved it.

EL: Bye.

SH: Bye.

[outro]

In this episode, we were delighted to talk with Sarah Hart, the Gresham Professor of Geometry at the University of London, about the serendipitous cycloid. Below are some links you might enjoy as you listen.
Hart's website and Twitter profile
Her book Once Upon a Prime and its review in the New York Times
Hart's article Ahab's Arithmetic about mathematics in Moby-Dick
The Wikipedia entry for the cycloid, which has links to many of the people we discussed

Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I am joined today as always by my fabulous co-host.

Evelyn Lamb: Well, thank you. I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City. And anyone who's on this Zoom, which is only us and our guest, can see that I am bragging with my Zoom background right now. We just got back from a trip to southern Utah, and I took possibly the best picture I've ever taken in my life. And 95% of the credit goes to the clouds because they just — above these red rock hoodoos outside of Bryce Canyon, I turned around and looked at it while we were hiking, and I was like, Oh, my gosh, I have to capture this.

KK: It is quite the picture.

EL: My little iPhone managed.

KK: Yeah. Well, they're pretty good now. Yeah. Anyway, so yeah, I'm getting ready to — I have three trips in the next three weeks. So lots and lots of travel, and I'm gonna make sure I mask up and hopefully I don't come home with COVID, but we'll see.

EL: Yes.

KK: Anyway. So today, we are pleased to welcome Matthew Kahle. Matt, why don’t you introduce yourself?

Matthew Kahle: Hi, everyone. Thanks for having me, Kevin and Evelyn. I'm a mathematician here at The Ohio State University in Columbus, Ohio. I've been here for 11 or 12 years now, and before that, I spent a good part of my life in the western United States. So those clouds look familiar to me, Evelyn. I miss the Colorado sky sometimes.

EL: Yeah, just amazing here.

KK: You did your degree in Seattle, right?

MK: I did. I did my PhD at the University of Washington.

KK: Yep. Yep.

EL: Great. And what is your general research field?

MK: I work a little bit between fields. My main interests are topology, combinatorics, and also probability and statistical physics. And I think I usually feel most comfortable, or maybe I should say most excited mathematically, when there's sort of more than one thing going on, or when it's in the intersection of more than one field.

KK: Yeah, lots of randomness in your work. He’s got this very cool stuff with random topology. And I remember, some paper you had few years ago, I remember really sort of blew my mind, where you had some, you're just computing homology of these random simplicial complexes, and, like, some four- or five-complex had torsion of order, you know, I don't know, 10 to the 12th, or some crazy torsion coefficient. Yeah.

MK: Yeah. So we were really surprised by this too, and we still don't really have any way to prove it, or really understand it very deeply. Kevin was mentioning some experimental work I did with some collaborators a few years ago. But yeah, that is the gist of a lot of what I think about, is random topology, which I sometimes try to sum up as the study of random shapes. And one of the original motivations for this was as sort of a null hypothesis for topological data analysis, that if you want to do statistical methods — if you want to use topological and geometric methods, and statistics and data science, you need a probabilistic foundation. But one of the things we've discovered over the last 15 years or so is that these random shapes are interesting for their own sake as well. And sometimes they have very interesting, even bizarre, properties, where we don't even know how to construct shapes that have these properties at all, but they're they are there. And we know they exist, because of the probabilistic method. Yeah.

EL: So let me be the very naive person who asks, like, how do you, I guess, come up with — like, what do you randomize about shapes? Or you know, if I think about, I don't know, randomly drawing from from some sort of, I don't know, bucket of properties, is it that or is it… Just what is random? What quantity or quality is being randomized?

MK: Right. So a lot of the random shapes or spaces that I've studied have have been on the combinatorial or discrete side. So for example, there are lots of different types of random simplicial complex that people have studied by now. And typically, you have just some probability distribution, some way of making a random simplicial complex on n vertices. And n can be anything, but then the yoga of the subject is that typically n goes to infinity. And then we're interested in sort of the asymptotic properties as your random shape grows. So one of the early motivations, or early inspirations, for the subject of random, simplicial complexes was random graph theory. So you can create random networks various ways, and people have been studying that for for a bit longer, probably at least 60 years or so now, with new models and new interesting ideas coming along all the time. For example, there was originally the Erdős–Rényi model of random graph where the edges all have equal probability, and they're all independent. This is a beautiful model mathematically, and it's been studied extensively. We really know lots and lots about that model of random graph now, although surprisingly, people can continue to discover new things about it as well. But in today's world, some people have studied other models of random graphs that they say may have made better model real world networks, for example, social networks, or what we see in epidemiology, and so on. The Erdős–Rényi model is something that's tractable, and that we can prove deep math theorems about, but it might not be the best model for real world networks. But, you know, I think of the random simplicial complexes that I study sometimes as just higher-dimensional versions of random graphs.

EL: Okay.

MK: So as well as as well as vertices and edges, we can have higher-dimensional cells in there, and and that starts to sort of enrich the space. It's not just one-dimensional now, it could be two-dimensional, or it could be any dimension.

EL: So you might not know. You've got some large number n, and you might not know what dimension this random — you’re, like ,attaching with edges with some sort of probability between any two things. And so you might not know what dimension your simplicial complex is going to be until after you randomly assign all of these edges and faces and, you know, and n- whatever the word is for that, n-things. [Editor’s note: it’s n-simplex.]

MK: Yeah, absolutely. That's right. It could be that the dimension of the random simplicial complex is itself a random variable. And you know, that we don't ahead of time even know what the dimension of it is.

EL: Cool!

KK: So there's lots to do here. This is why Matt has lots of students and lots of lots of good projects to work on. But anyway, we invited you on not just to talk about this really interesting mathematics, but to find out what your favorite theorem is. So what is it?

MK: Okay, so I've been thinking about this. Well, I have to admit, I think I asked myself this just knowing of your podcast in case I ever got invited on. And then I've been thinking about it since you invited me. I would say my favorite math theorem, probably the one I've thought about the most, the one maybe that affects me the most, is Euler’s polyhedral formula, which is V−E+F=2. Right? So let's just start out saying, well, you know, what do we mean by this? I think my understanding of the history of it is that it was something that as far as we know, the Greeks didn't observe even though they were interested in convex polyhedra. And sometimes people consider the classification of the perfect Platonic solids is one of the peak contributions of Euclid’s Elements. But we don't know that they recognized this pattern that Euler noticed thousands of years later. If you take any convex polyhedron, a cube or an icosahedron, or a pyramid, a bi-pyramid, any kind of three dimensional polyhedral shape that you can imagine that's convex, V, the vertices is the sort of number of corners of the shape and E is the number of edges. And then F is the faces. It always is the case that V−E+F=2. So Euler noticed this. And it's not clear if he gave a rigorous proof or not. I don't even know if he felt like anything needed to be proved, maybe it was obvious to him. And nowadays, we have many, many beautiful proofs of this fact. But one of the things that strikes me about it is that, it’s sort of in hindsight, is that this is just sort of the tip of a very big iceberg. There's a much more general fact that we are just kind of getting our first glimpses of, and nowadays, we would think of this as not just a phenomenon about convex polyhedron, 3-dimensional space, that it’s just a general phenomenon in algebraic topology, or you can say =more generally, in homological algebra. It's just sort of a feature of nature somehow.

KK: Right, right.

EL: I think something that I really enjoy about this fact is you can present it at first as a theorem or as a fact. But then this fact kind of leads you to this new definition that you can observe about all sorts of different shapes, you know, this number that is the vertices minus the edges plus the faces, hopefully, I got it in the right order, yes. Then you can assign that, you know, you can say, like, what does, you know, if you've got a torus, like a polyhedral torus, or, you know, a higher-genus object or a higher-dimensional thing, you can sort of use this, and so it's like a fact becomes a definition or a new thing to observe.

MK: That’s right. Are you saying, for example, you know, we have the Euler characteristic is an invariant of a space?

EL: Right.

MK: And that might, if you're introduced to a new topological space, that might be one of the first things you might like to know about it. And so yeah, it becomes its own invariant. It’s a way of telling some different spaces apart, for example.

EL: Yeah. So do you have a favorite proof of this favorite theorem?

MK: I do. I present it and the graduate combinatorics and graph theory course when I teach this course. So already, we're looking at a little bit more general formulation than what Euler looked at. We don't just have a convex polyhedron in 3-dimensional space, what we have is a connected planar graph. So we have some kind of network with nodes and connections between them, and it's one that you can draw on the plane without any of the edges or connections crossing. And in this case, the faces now are just going to be the connected components, or the regions, in the complement of the graph that then comes with an embedding into the plane. And then V is the number of vertices of the graph, and E is the number of edges. So V−E+F=2 in this case, so for for just any connected planar graph, this might seem totally unrelated, but it's actually a more general version than what we just saw with convex polyhedra because you could take any convex polyhedron and unwrap it, or stereographically project it into the plane and get a planar graph. But planar graphs could have lots of other features. So when I present this in class, I tend to give three or four different proofs of it. There's a beautiful proof that I've heard attributed to John Conway, where he says something about, like, letting in the ocean or something. So your graph is connected, but there may be some cycles in it. And anytime you have cycles, the Jordan curve theorem tells us there's an inside and outside. So John Conway wants to let the ocean in. The ocean is the sea, is the outside of the graph, let it in until it touches. So what he's saying is if there's any cycle, delete one edge from it, and so what this does is it reduces the number of edges by one because you deleted an edge, but it also reduces the number of faces by one because that two regions that were inside and outside of that cycle are now the same region, so V−E+F has stayed the same.

KK: Right.

MK: And then eventually, you've just got a tree. There's no more cycles left, but your graph is still connected, so it must be a tree. And we know that every finite tree with at least two vertices has a leaf, has a vertex of degree one. And again, you can prune away that, and then you've reduced the number of vertices by one and the number of edges by one. And V−E+F is again not changed. So at the very, very end, we're just left with a single vertex in the plane. There's one vertex and there's one region, which is everything except that vertex. So at the very end, V−E+F=2. But through all those steps, we know that it never changed. So it must have been V−E+F, it must have been 2 at the very beginning. So I love that proof.

EL: Yeah.

MK: There's another proof that I think I like even better, which is that you consider the dual graph and a spanning tree. You pick a spanning tree on the original graph and a spanning tree and a dual graph at the same time.

EL: So the dual graph being where you replace, you swap vertices and faces.

KK: Yes. For every face there’s a vertex and you join two when the two faces share an edge.

MK: That’s right, exactly. So one thing that's tricky about that is that now the dual of even just a nice planar graph might be a multi-graph. So just imagine a triangle in the plane. The dual graph has two vertices, one inside the triangle and one outside, but there's three edges connecting those two vertices, because there's three edges in the original graph. And the edges in the dual graph have to correspond to edges in the original graph, and that's important. They cross them transversely. So then you choose a spanning tree on each one, and you and you realize that — you count the number of edges in each and you somehow — now I'm getting a little stuck remembering the proof, but the punch line is in the original graph, I guess the number of edges is V−1. And then the dual graph, the spanning tree, the number of edges is F−1. And these have to be in correspondence. So you just immediately just write down V−1=, sorry, no, I don't remember exactly how that the end of that proof goes. But there was something about it I liked. It seemed like the other proofs, you're kind of doing induction on either the number of vertices or the number of edges or the number of faces, and that you have to make some arbitrary choices. And this proof by duality doesn't use any induction and doesn't require any choices. It just kind of comes for free. And you sort of immediately see where the 2 comes from, because there's a V−1 on one side and an F−1 on the other side, so the 2 just sort of pops out immediately from the proof. There’s — I think it’s Eppstein? — some mathematician collects proofs of Euler’s polyhedra formula on his website, and he has at least 10 or 20 different proofs. And when you read them all, some of them start to remind you of each other, and who knows what counts as the same proof or different proofs.

KK: Sure.

MK: But there are some neat contributions in there. One of them he attributes to Bill Thurston in the middle of some very influential notes that Thurston had in differential geometry. And he's talking, he's giving his own proof, I think, that the Euler characteristic of a differential manifold really is an invariant of the manifold, for a smooth manifold, let’s say. You could triangulate it, and then the Euler formula, the Euler characteristic, you could just say is the alternating sum of the faces of every dimension. But why doesn't that depend on which triangulation you pick? And Thurston gave a really beautiful kind of almost physical argument with, like, moving charges around. I like to show the class this one also. At that point, we leave — I don't know how to make that proof work for planar graphs, but it works beautifully for polyhedra, for convex polyhedra, like what Euler first noticed. And apparently, it works also for higher dimensional manifolds, too, although I've never gone through that proof carefully.

KK: Yeah. Right. Well, the proof that you said might be attributed to Conway is sort of the one that I always knew, and I never heard it attributed to him, but that's good. It's sort of nice. You can explain that one to just about anybody right? You just sort of imagine plugging away an edge and a face at the same time basically, yeah.

EL: Yeah. A proof that proof that is of something that is so visual, but you can really understand over a podcast, is a special proof. Because I do think that it doesn't take a whole lot of you know, imagination, to be able to follow this audially.

KK: Audially, is that a new word?

EL: There is a real word that is embedded in that word. Aurally, that’s the real word I was trying to say.

KK: Yeah, so is this sort of a love at first sight theorem? I think I first learned this theorem in the context of graph theory.

MK: I think for me, too.

KK: And then I became a topologist kind of later. And then of course, now I think of it as, oh, it's the alternating sum of the Betti numbers, but that those two quantities are equal is an interesting theorem in its own right.

MK: Right. Yeah. So I was trying to think about this. When did I learn about this theorem? And I think I first learned it in graph theory also. But then I know now that it's much more general, and I don't even know if I ever remember anyone telling me that specifically in a class or reading it in a particular book or paper. I think this to me, maybe part of what I like about the Euler formula is that I feel like my understanding of it has just deepened over time, and that there’s kind of a series of small revelations. At some point, I started thinking of it as the alternating sum of the Betti numbers, and things like that. And since I like the combinatorial side of topology, and have simplicial complexes or cell complexes, also the alternating sum of the number of faces of each dimension. But then even in the last couple of years, my understanding has continued to develop because now I think, you know, well, you could just have a chain complex, and all you know is the dimensions of the vector spaces, but it makes sense to ask what's the homology of the chain complex, so they're the Betti numbers again, and again, the alternating sum of the Betty numbers now is the alternating sum of the dimensions of the vector spaces of your chain complex. But I think I probably first saw, you know, the graph theory version of it, maybe in an undergrad or a first graduate combinatorics course.

KK: All right. So the other thing on this podcast is we like to ask our guest to pair their theorem with something. So what have you chosen to pair Euler’s formula with?

MK: Well, you know, I've been stumped by this. But you know, thanks for the warning that I am going to get asked this question. So I had a little time to think about it, and I'm not totally stumped on the spot. But the thing that keeps coming to mind the most when I ask myself that question is some of Bach's music. Johann Sebastian Bach is really known for his four-part harmonies and for counterpoint, and it feels a little bit like this: You're listening to a beautiful piece of — it could be anything, you know: a fugue on an organ, or four-part harmonies that were written for choral music or something like that. And when you listen to it, you can listen to a recording of it two or three times and each time pick out a different voice to follow along. And there are just these independent melodies, harmonies that he's somehow weaving together. You can also just relax and just let the whole thing wash over you. And honestly, that's most often how I listen to music. But it's completely fascinating to just hone in on one particular thread. And so I think a lot of people feel like Bach's music has maybe a mathematical feeling to it, or that it's mathematically perfect or precise. So you could say that Bach, pairs with mathematics already, but the reason I want to try to connect it with the Euler formula that I like as my favorite theorem is that there are these sort of different layers. And just the same way you can kind of listen for one voice, and then tune your ear and listen to a different voice and emphasize that, I feel like this is one of these areas, of one of these kinds of mathematical phenomena, that’s just sitting there in, you know, platonic space, or wherever it lives. And you can look at it. So if you look at it from the topological point of view, it's the alternating sum of the Betti numbers, the number of holes in each dimension. But if you look at it through a combinatorial lens, then it's the alternating sum of the number of faces of each dimension. Or you can just step back and it's just its own thing. It's just an invariant of the space, the Euler characteristic, and these just happen to be different ways to compute it. But it has that feeling to me that you can look at it different ways. But you're really always looking at the same thing. Just we're putting on different glasses or looking at it through different lenses, and so it reminds me of that sort of, I don't know, counterpoint and music or something.

EL: Yeah. Oh, I love this pairing! I'm also, I play viola and I sing, and when you get to a point when you’ve learned a piece that you've learned it enough that you don't have to be just concentrating on, like, am I singing the right note at the right time, but you can actually start hearing like, oh, I didn't originally hear that the parallel that the bass and the soprano line has right here, or the way we come in and then the altos come in and something like that. I've been singing a lot of, you know, things that have these fugal sections in them, which is — I haven't actually sung much Bach recently, but similar things — and I just love that pairing and how seeing the same thing, or singing the same music over and over again, you hear something different every time. You know, just a little easter egg that you didn't pick up the first 20 times you practiced this piece, and then now you hear and you say, oh, next time I really want to make sure that I, you know, do that crescendo with the tenors just perfectly or something. I love that.

KK: Yeah, so I see the edge of a keyboard there in your Zoom, Matt. Do you play?

MK: A little bit. I mean, nothing to write home about, but it's something I enjoy. I took it back up during the pandemic as a hobby. And I've been practicing a little bit. This over here, I have a little portable keyboard, and then I have an electric piano out in the living room. But I've been practicing music with one of my friends. We get together, like, once a week and and just play some cover songs. And I like what you're saying, Evelyn, about hearing different things. Even, you know, we'll be playing some song by REM or somebody that I've known, I don't know, it seems like my whole life, it’s very familiar. But once we start to play it, once we start to sing it, then I hear all kinds of different things in it that just listening to the same recording that I've listened to before all of a sudden, I'm like, wait, Mike Mills is actually doing some really interesting harmonizing and this track, and not only is he harmonizing, like singing different notes, than what Michael Stipe is singing, he’s actually singing different words. He's saying something in that song I never even noticed he was saying. So anyway, music and mathematics, I think that's probably another big thing that they have in common, is that, you know, a little bit can go a long way, and even just entry-level, you can already start to appreciate the beauty of it, but that it's sort of almost inexhaustible how deep it goes and that you can always, there's always more to learn. There's there's always more to notice.

EL: Yeah, with Bach specifically, you know, as a viola student, I think I started playing the Bach cello suites, an octave up on the viola, I was probably 10 or 11 years old? And it's like, I will still play those same suites that I started learning when I was in fifth grade. And it's like, it always has something to teach me. It's something that I can always get something more out of.

KK: Yeah. I think we can all agree that that Mike Mills is REM’s secret weapon. I took up the guitar about 10 years ago, so I'm terrible, and I play alone. But it's still something that I enjoy to do. It's certainly, it's a good way to exercise your — what was it Leibniz said? That music is the pleasure the brain derives from counting without knowing that it's counting?

EL: Oh, yeah. That’s a good little quote, to file away for us math-musician-type people.

KK: That’s right. All right, well, so we always like to give our guests a chance to plug anything. Where can we find you on the interwebs?

MK: Yeah, I don't have anything particular to plug, but you can find, you know, all my mathematical work on my professional webpage, matthewkahle.org. There's links to all my papers and everything there. And, you know, if my friend and I get our REM cover band off the off the ground, we’ll keep you posted.

EL: All of our Columbus area listeners can find you.

KK: I can play rhythm guitar on some of the tracks if you need somebody.

MK: All right. We'll have to all get together if you come out and visit in Columbus.

KK: Well this has been great fun.

EL: Thanks so much for joining us.

MK: Thanks for having me today.

[outro]

On this episode, we were delighted to talk with Matthew Kahle of the Ohio State University about Euler's polyhedral formula, also known as V−E+F=2. Here are some links you might find useful as you listen to the episode.
Kahle's website
His paper about torsion in homology groups of random simplicial complexes
The Erdős–Rényi model of random graphs
Euclid's Elements, book 13, is devoted to the classification of Platonic solids. Also found herestarting on page 438.
The Jordan curve theorem has made a previous appearance on the podcast in our episode with Susan D'Agostino.
David Eppstein's website with 21 different proofs of Euler's formula. Thurston's proof is here.

Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Kevin Knutson, professor of mathematics at the University of Florida. And today I am flying solo while I am at Texas Christian University in Fort Worth, where I'm serving as the Green Honors Chair for the week. And I've been given some talks and meeting the fine folks here at TCU. And today, I have the pleasure to talk with some of their students. And they're going to tell us about their favorite theorems and what they pair well with. And we're just going to jump right in. So my first guest is Aaryan. Can you introduce yourself?

Aaryan Dehade: My name is Aaryan. I'm a sophomore computer science major at TCU, and I'm from India. And I've chosen to go with the fundamental theorem of calculus.

KK: Okay. The fundamental calculus.

AD: Yeah.

KK: Okay, so now there are two parts of the fundamental theorem of calculus. Do you have a favorite part?

AD: So that's what I like about it. Honestly, I can't choose a favorite part because one of the parts is very important, and the other is interesting.

KK: Yes.

AD: So the first part, basically, it tells us the relationship between the integral and derivative. The second part tells us — like, basically, you can use that second part supply and to get solutions for questions in calculus. Does that makes sense?

KK: Sure.

AD: So what I like about this theorem is that it's not like other theorems where it has two parts. So it's kind of interesting how everything in calculus is based off of these two theorems. And if you didn't know what the relationship between the derivative and the integral was, probably you wouldn't be able to do anything with mathematics with it.

KK: Sure, maybe. Yeah. So you said one part was useful, and one part was interesting. Which part do you think is interesting?

AD: I feel like the first part is interesting, because it's almost intuitive. Like, you know that should happen, like the relationship between integrals and derivatives should be, like, one is an inverse of the other. And that is intuitive. So it's almost given. And it's interesting that we have to say that.

KK: Really, you think that’s intuitive? I mean, I’m not — I don’t know, when I first learned the fundamental theorem, I thought it was kind of shocking that somehow this this thing, this integral, which is sort of defined as in terms of these Riemann sums, somehow that went, you know, if you let x be the upper limit there, and you differentiate that function, you get your original function back. That’s intuitive? That’s amazing.

AD: Yeah, basically what it is, it's just an area of a rectangle. So I just thought of it as just decreasing the width of the rectangles, and then you get smaller and smaller rectangles, so you get the area. And then if you take the function at that point…that’s what I think.

KK: You’re cleverer than I am. I was just kind of dim, I guess, and I didn't think it was so intuitive. I mean, I saw the proof and believed it. But then, yeah, then the second part is how you actually evaluate integrals.

AD: Yeah, so that's what you use to calculate the area between two points.

KK: Yeah. Although I guess the problem is, right. So the theorem says that, you know, if you want to find the, the integral, the value of this definite integral, all you have to do — and our listeners can't see me doing the scare quotes — “all” you have to do is find an antiderivative of the function. Right?

AD: Right.

KK: Yeah. And then you spend all of Calc II learning how to find antiderivatives.

AD: Yeah.

KK: And even then, if I hand you an arbitrary function, you can't even do it, right? Like that's the sort of disappointing part of that theorem, is that most functions, you can't find a closed form antiderivative for. And so what do you do? But you’re a computer science major. You know what you do, right? You do it numerically, right?

AD: Yeah.

KK: Okay. Very cool. So you've known this theorem for quite some time, I guess.

AD: Yeah, I've done it since high school.

KK: So you love the theorem.

AD: I really, yeah, I do. Because when I was in high school, I used to sit at my dining table and study because I wanted to have some snacks at the same time.

KK: Sure. As we all do.

AD: I would just spend hours just doing sums on integrals, or basically just integrals. That was difficult at that point.

KK: Sure.

AD: And yeah, it was interesting, because I got used to that at some point. And then it got easier. And I just started liking the satisfaction of being able to do this. That was fun.

KK: Cool. All right. So on this podcast, we also like to ask our guests to pair their theorem or something. So what pairs well with the fundamental theorem?

AD: So as I said, I used to sit at the dining table and have snacks. And there's this really, really popular biscuit in India called Parle-G. And I used to have that with tea while doing my sums. So that was the highlight of it, that's why I used to look forward to studying, just for those biscuits.

KK: Okay, so I assume there's an Indian market in town somewhere, right? Can you get these?

AD: Yeah, I do have them in my dorm right now. Yeah. I have them every day.

KK: All right. So what are these called again?

AD: Parle-G.

KK: Parle-G. Okay. So I'll have to go to the Indian market when I get back home and see if I can find these because I am always on the lookout for a good new biscuit.

AD: Yeah, they’re amazing. Okay, so you should you should know, our very first episode of this podcast, aur guest, who was Amie Wilkinson, who is on the faculty at University of Chicago, chose the fundamental theorem as her favorite theorem. So you're in very good company, because she's a phenomenal mathematician. And okay, thanks.

AD: Awesome. Thanks so much.

KK: All right, up next, we have Toan. So why don't you tell us about yourself and what your favorite theorem is?

Duc Toan Nguyen: Okay. My name is Duc Toan Nguyen. People usually call me Toan. I’m an international student from Vietnam, and I'm a sophomore majoring in math and computer science

KK: Okay, great. So, favorite theorem. What’ve you got?

DTN: Yeah. So as my peer Aaryan, he chose the fundamental theorem of calculus, right?

KK: Yes.

DTN: But I want to bring another fundamental theorem in analysis, which is the mean value theorem.

KK: Oh, okay. So I have a theory, okay. I call the mean value theorem, the real fundamental theorem of calculus.

DTN: Yeah, me too!

KK: So why do you like it so much?

DTN: Yeah, I think I have the same idea with you of why it is called the real fundamental theorem. I think, because to prove the fundamental theorem of calculus, you need the mean value theorem.

KK: You absolutely do.

DTN: Also for analysis, the most popular and common tool in calculus, which is a derivative test, also has the mean value theorem behind it.

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

DTN: So when I first so I first approached the mean value theorem when I was in high school. I took the Math Olympiad in Vietnam. So I had to prepare for that, and there is a section about that.

KK: Okay.

DTN: So it's called the Lagrange Theorem, it was kind of very fancy. Yeah. And it usually applies to — so you know, in the exam, we had some of the problems related to the continuous version, and f(a) minus f(b), something like that. Most of time, we used the mean value theorem. So yeah, it was kind of cool at the time, but I really enjoyed that until last semester, I took real analysis. So I could see the whole process was using the mean value theorem. That's why it can be taught in one lecture or one unit. Even today, this semester, I’m taking multivariate analysis. And it's also a very fundamental thing in proof, everything from differentiability on. It also even has a mean value theorem in it higher-dimensional space.

KK: So maybe we should remind our listeners what the mean value theorem actually says,

DTN: Oh, okay. So, let f be a function defined on an interval [a,b], so that f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b). So the theorem say that there exists a point c between a and b that is not is not inclusive so that f(b)−f(a) is equal to f’(c)(b−a). So I think the mean value theorem, the name comes from the quantity f(b)−f(a) divided by (b−a).

KK: Yeah. Right. So the average rate of change over the interval is equal somewhere to the instantaneous rate.

DTN: Yeah.

KK: Yeah, that's right. That's how you prove the fundamental theorem, too, because it's just a telescoping sum when you write it out correctly. And the mean value theorem sort of pushes everything away, and then you're done.

DTN: Yeah, that's also my favorite part. Because it can tell you the relationship between the integral of functions and their derivatives.

KK: Right, right. So what do you want to pair with your theorem, what pairs with the mean value theorem?

DTN: Yeah, I want to pair with something really weird. Which is a phone with FaceTime. Okay, so I'm here. I study. I'm far from my home. My home is in Vietnam, which is on the other side of the Earth.

KK: Almost exactly opposite, right?

DTN: Actually it takes 20 hours from this time to my country time. So it's actually like, opposite, it's more. So yeah, and the FaceTime, why? Because through FaceTime, I can see what people in my home are doing and they also [can see me]. So it's kind of a bridge or relationship that connects what I'm doing here and what my family is doing there. And, you know, my family is always wants the best for me and hopes everything is good for me here. And me too. So that's a very meaningful thing for me.

KK: Yeah. That's great. And I'm glad that technology exists. When I was in college, the internet didn't exist. So you know, I had to call people on the phone and phone calls to Vietnam, I imagine, would be — I can't imagine what that would cost. It was expensive enough to call my girlfriend who lived four hours away.

DTN: It’s kind of more romantic. And you can give them a love letter.

KK: We wrote letters too. All right. Well, Toan, thanks so much. That was great.

DTN: Yeah. Thank you.

KK: Up next, we have Maiyu Diaz. You can introduce yourself.

Maiyu Diaz: My name is Maiyu Diaz. I'm a second-year graduate student here at the department of mathematics at TCU.

KK: Cool, and you have the best shirt. You win the shirt contest today.

MD: Oh, thank you.

KK: I’m actually kind of wishing you would like give me that shirt. [Editor’s note: How dare you say that and not send us a picture?!] All right, so you’re a second-year grad student here in the math department. Okay, great. And so yeah, so what your favorite theorem?

MD: So my favorite theorem is — I don't see it really formally presented, but Stirling's formula where n factorial can be approximated by n^n e^−n times square root of 2πn.

KK: Yes.

MD: That is my favorite theorem.

KK: Yeah. So that’s a really interesting approximation for factorials. So okay, where did you come across this?

MD: I first came across this, I want to say, when I was first studying the factorial back in grade school, and it was just more of like, looking up on a Wikipedia page, which was something that I really relied on when it came to writing essays for my English classes. And when I found out that it could also be used as a resource for mathematics as well, I thought, oh, factorial, let's learn interesting things about this. And that's where I found an example about Stirling's formula, under that. It wasn’t until much later that I was able to understand how that was derived, and there were a variety of proofs for proving Stirling's formula. One of them relies on probability distributions, which are looking at how the factorial works. And then there's another way of finagling a little bit with the integral formula for n factorial that comes from the gamma function, and that's another way of deriving Stirling's formula.

KK: Do you have a favorite proof of your favorite theorem?

MD: I do, but it's a very uncommon proof.

KK: Okay.

MD: The proof relies on a contour integral involving the derivative of the Riemann zeta function.

KK: Oh! I don't know if I know this proof, but what contour do you use?

MD: So you're going to go ahead and do a contour that is on the half strip, so you're just going to fix a number like σ and since the derivative of the Riemann zeta function, you just want to pick a σ just a little bit to the right of where it converges. So real part greater than one, and that's going to be a line integral from σ − i infinity to σ + i infinity, okay? And what you want to do is that you kind of want to start pushing that back a little bit so you can start picking up the residues of the derivative of the zeta function. You’re going to go ahead and possibly the non trivial zeros you could possibly hit, though. You’re definitely going to hit the pole at s=1. That's where the n log n minus n term comes from, when you look at the log of n factorial.

KK: Right, okay, sure.

MD: You keep pushing that more, and then you're going to be picking up the rest of the terms from the Stirling’s formula as well. It's pretty interesting.

KK: Yeah. All right. I haven't seen this proof. That's very cool. Yeah, okay. Cool. All right. So what do you think pairs well with Stirling's formula?

MD: I am going to say chicken tikka masala.

KK: Chicken Tikka Masala. Okay. I do like chicken tikka masala. What in particular makes you want to link those two things together?

MD: So Stirling's formula, I would say is a little bit spicy. And I underestimated it at first.

KK: Right.

MD: Because that was just something I would not have been — I just, if you’d asked me if the formula was intuitive, it’s just like, absolutely not. Where's where does this e term come from? Where does the square root of two pi come from?

KK: Sure.

MD: It’s not until you start familiarizing with the proof more. It's like, okay, at this point, it's not that I'm used to that, I’ve just seen this too many times.

kk: Sure. Sure. Yeah.

MD: But I want to say it’s spicy because of a paper, a PhD thesis, that was published in 2014 by Matthew Lamoureux, whose advisor is Keith Conrad, who wrote a paper on Stirling's formula, just devoting a series of notes on it, you can see all the things he has compiled and I like reading through his notes. And this PhD thesis, he was looking at a generalization of the factorial, which is the factorial for number fields. So instead of looking at the derivative of the zeta function, what it was looking at was a modification of the derivative of the Dedekind zeta function because they had a function defined over number fields. And that's the same technique as well. You want to keep pushing it along the line, be able to pick up non-zeros, possible poles, and it just more or less the same outline as well, and spicy because number one, there is a ton of information coming about the factorial just from the location of zeros and poles of these zeta functions. I was like, Whoa, this is some pretty advanced stuff. I don’t want to be messing with this. But then it's also really delicious because one, I think chicken tikka masala is very delicious.

KK: It is. Agreed.

MD: But also delicious in the context of these formulas, because the approximation formula for factorials is reliant on the poles and zeros of these zeta functions. So it's like, whatever I want to know about on the left side over here, the approximation for the factorials I'm looking at, all I’ve got to know is the information about the zeros and poles of these zeta functions.

KK: Right.

MD: That’s the spicy and delicious part.

KK: Very cool. All right. That's a good pairing. I like that. Cool. Thanks, Maiyu.

MD: Thank you.

KK: Up next, we have Hope Sage. Why don’t you introduce yourself?

Hope Sage: Hey, I'm Hope. I'm a junior physics major at TCU.

KK: Cool.

HS: My favorite theorem is Bell's theorem.

KK: Okay, I don't think I know this theorem.

HS: Okay. It’s kind of Physics-y. It's from quantum mechanics. And John Bell wrote it as a response to the EPR [Einstein-Poldosky-Rosen] paper, which is kind of a famous paper in quantum physics, where it talks about how quantum physics is probably incomplete, and there's probably some hidden variable that's underlying it. And Bell uses this inequality to calculate the probabilities based on what you would expect classically. So if the quantum particles are not entangled, then you would get this expected probability. And then he shows that experimental results kind of conflict with that. And so the underlying assumption is that from that, local realism isn't a thing. So then the universe is like, super wacky.

KK: I think we knew that, right?

HS: Yeah.

KK: So okay, so Alright, so my quantum mechanics is — well, calling it rusty would be, like, an insult to rusty things. So the probability of what? The state that a particle is in?

HS: Yeah, so whenever you have two entangled particles, one might be spin up, one might be spin down, okay? And you can run an experiment. They have a beam splitter and two detectors, and you measure the different states. It's kind of a traditional example. And the probabilities depend on the angle that everything is situated at. So you have nine, and then you were to take all those probabilities, he basically proves this inequality that just shows that mathematically, it can't be possible for there to be hidden variables, which means that things are paired and they would have to be, like, communicating at faster than the speed of light, which doesn't happen. So then there are all these theories about what could theoretically be the underlying nature of the universe from that okay, in different interpretations.

KK: Okay. So which is your favorite interpretation of what might be going wrong here? Or right, whatever the right word is.

HS: There’s this interpretation kind of extended from this called the many worlds interpretation of quantum mechanics. I don't know if I’d necessarily say it is the most likely to be correct. But I think it's the most fun one. It's also fun because you can read cool science fiction books about it. Like, every action you take, there's a different universe, different paths. I think it's kind of fun.

KK: Sure, right. So so right, so like, right now, what we're doing, we could, like, split into any number of paths. And there are all these weird different outcomes that could happen, depending on whether or not some quantum state is what it is or not.

HS: Yeah, basically. One of my favorite books is Dark Matter. It's by Blake Crouch. And it's about, like, every action that you take, there’s a different universe. And then there's infinitely many possibilities based on every single decision you make, which is kind of interesting, because every decision that you make does create a different next possible decision.

KK: Sure.

HS: But yeah.

KK: Okay. Well, all right, so our minds are getting blown. All right, what pairs well with this theorem?

Well, sometimes it's called “spooky action at a distance” and so I was going to go with Halloween candy. Because spooky.

KK: Yeah, yeah. All right. Okay, so what what's your favorite Halloween candy though?

HS: Probably Reese’s.

KK: What? Okay, I have strong Reese’s opinions. So which, like the full size or the miniatures or what?

HS: Okay, a Reese's Peanut Butter Cup, but the dark chocolate version.

KK: Okay. All right. I can respect that. I am team miniature. I think that's the correct ratio of chocolate to peanut butter. But the dark, I get it, I understand. Okay. All right. That was great. Thanks, Hope. Jonah, why don’t you introduce yourself?

Jonah Morgan: All right. Well, I'm Jonah Morgan. I'm a freshman engineering major here at TCU, and my favorite theorem is Gödel’s incompleteness theorem.

KK: Gödel’s incompleteness theorems. So that's more than one theorem. All right, so let's remind our listeners what what at least one of them is.

JM: Sure. So the first one: The first of Gödel’s incompleteness theorems is effectively any — I'll call it interesting, okay — any sufficiently interesting or complex set of axioms, it fundamentally has theorems or statements that cannot be proven, but which are true.

KK: Cannot be proven inside the system, right?

JM: Yeah, cannot be proven inside the system. Right.

KK: And okay, you hedged around, but I think “sufficiently complicated” just means, like, you can do arithmetic.

JM: Yeah, you can add numbers. Because if you just can't do anything, then well, you can’t say anything.

KK: Okay. Yeah. So that's the first one. What's the second one? I think I don't even remember the second one.

JM: So the second one was, I think — I'll make some background. I just think background’s kind of fun to know.

KK: Sure.

JM: His first theorem kind of says, Okay, well, if you have a set of axioms, you can have effectively a statement that says this statement cannot be proven by the axioms. So if you have that statement, then well, if that statement is true, then it's a true statement within the system that cannot be proven by the axioms. But if it's false, then it is a statement which cannot be proven. Or it is a false statement, which cannot be proven. I'm a little rusty on that aspect of it. But effectively, the idea is, so there's this weird statement that you can have in any system, which makes it he says incomplete, where incomplete is the word for it. But then, so mathematicians are like, “Okay, well, we want to prove things. And you gave one example, it's a bit of a weird example. I don't want to prove that a statement is unprovable.” So then he has a second theorem that says, well, also there are true statements which are unprovable which we cannot prove or unprovable.

KK: Yes.

JM: And so that's the second theorem. And that's like, okay, so you can spend your life working on a theorem or working on a problem, and then you can't even know whether or not you can know the answer to this problem within within the set of axioms that you're working with.

KK: Right. So it's hopeless, in some sense. Yeah. You can't fix this issue.

JM: No. It’s — some people say math is broken. But really, it just incomplete. There are certain things —

KK: Yeah, I mean, I think it caused a crisis amongst certain elements of the mathematical community, but I'm with you, I just sort of view it as, well, okay, so there are unprovable statements. It doesn't mean the bridges that we build are going to fall down,

JM: Right. Everything we have proven still stands. You can still prove a lot of things.

KK: All right, still lots more to prove. Where did you come across this?

JM: I think some time in grade school, middle school, high school, I got really into just watching videos about mathematics. And at first they were just little conjectures and little fun things and then Gödel’s Incompleteness Theorem stuck out because it's like, I was diving into the world of math for the first time. And there are so many things that you can prove, and even these, like, crazy things that I never thought were provable, or like things that are so complex. Fermat’s last theorem took 300 years to be proven.

KK: Right.

JM: And I don't understand any part of that proof, to be honest with you.

KK: Same.

JM: But we did it. And it took a long time, but sort of my idea after seeing all of this was that, well, anything can be proven if you have a sufficiently — or maybe we're not smart enough to find the proof, but everything should be provable. Mathematics, it’s a language, you should be able to explain things in that language. And then Gödel’s incompleteness theorem says no. And also, there are things that you just — it kind of changes your perspective on that.

KK: Right.

JM: And so then you get questions like, well, you know, the Riemann zeta hypothesis, one of the most famous unsolved hypotheses. And a lot of people see this and they're like, Oh, well, does this mean that this this million dollar problem, one of the millennium problems, could just be unsolvable? And we can't even know that it's unsolvable? And I think that was the first thing I looked up when I heard about this theorem. And something that was even more interesting to me was that if the Riemann zeta hypothesis is false, it’s provably false. Because it is equivalent to saying that there exists some number on the real part one half line, yeah. So you can write an algorithm, and given infinite time, you will find sure if there is one, you will find it. So it is provably false, which also means that proving that the Riemann zeta hypothesis cannot be proven means that it must be true.

KK: Okay. Yeah.

JM: So I can't give you the formal explanation on that. I don't know what it is. But the general idea is that you can prove that something is true by proving that you cannot prove it.

KK: Yeah, that's a little mind-twisting. But I can see why this would appeal to you, as you’re coming into your own intellectual being sort of state and moving out of being a kid. Yeah, that's really great. Okay, so. So what do you think pairs well with the incompleteness theorems?

JM: So, this isn't sponsored, but GrubHub, or Uber Eats or whatever.

KK: Okay.

JM: Because, occasionally, I order food there. It's easy, it's convenient. Most of the time, you get what you ordered. And sometimes your driver takes a nugget and you don’t — and, you know, that's gone. And sometimes he just doesn't show up at the door, and you're left wondering, where's the stuff that I paid my money for? And I think the feeling is similar there that you can you can pay for something, like you can spend your time working on this theorem, and it just unprovable, and you'll never know where it went or where it goes. And also you can pay for your food and just have no idea where.

KK: Yeah, well, that's a good pairing right there.

JM: Thank you.

KK: Thanks a lot, Jonah.

JM: Thank you.

KK: All right, up next, we have Anna Long. Anna?

Anna Long: Yeah, so my name is Anna. I'm a senior math and French double major. Actually.

KK: Nice!

AL: I’ll stick with English for you.

KK: Je ne parle pas bien le français.

AL: Tres bien!

KK: So I can say I don't speak the language really well in several languages.

AL: Well, that’s all you really need.

KK: That’s right. It was only that and “toilette,” you know, yeah.

AL: Yeah, so my favorite theorem I picked is the invertible matrix theorem from linear algebra.

KK: Okay.

AL: And it's really a pretty big theorem. It's 24 equivalent statements for a square matrix, that'll call A. So just a few of my favorite little statements in there. So obviously, we have that the matrix is invertible, that the columns then form a linearly independent set. So there's always a solution. And then that, for it to equal zero, it's only the trivial solution, that your vector is zero. That it has n pivot positions, or that it has full rank. And then that the linear map is both one-to-one and onto, and that the determinant is nonzero and that zero is not an eigenvalue.

KK: Right. And that's only, like, six of the equivalent conditions. So yeah, if you open up a linear algebra book, there will usually be at some point, some page where they list all these, I had forgotten there were 24. I can probably — so I'm teaching our senior-level analysis course this year. And part of it, we do some stuff with operator theory, and they remind, in our text that we wrote ourselves, they use of 11 of the equivalent conditions, but not all of them. So yeah, I can't imagine. I don't even think I know what some of the other ones are. I'm sure I would if you told me.

AL: A lot of them are, like, jumbled together. So, like, n pivot positions and full rank I've seen defined as two different things, but they’re really the same. But essentially, because of the theorem, they're all the same.

KK: Yeah. So yeah. So why do you love this theorem so much?

AL: I just love it because it's so useful. I’ve had two linear algebra classes now, and it just makes life so much easier trying to prove that any various things in class. So yeah, it's just great being able to find the easiest statement and prove that one, and then you just know all of them are true.

KK: Right. Yeah, I do like those things when, like, 50 things are equivalent. that's really nice. Yeah. Okay. So I guess you learned this in your linear algebra course. You had a second linear algebra course? What’s the second one?

AL: I’m in applied linear algebra right now.

KK: Okay. All right. So you're doing, like, singular values and things like that?

AL: Yes, we are.

KK: Okay. All right. So that's super useful stuff. Linear algebra, of course, I think is one of those things that we don't teach enough of. And basically, any problem in math comes down to either making some estimate, like analysis, or some linear algebra problem, it seems to me. so yeah, the more you learn, the more you know, then the better off you'll be. So you're a senior. What’s next for you?

AL: Graduate school.

KK: In what? In French?

AL: No. In math. Yeah. So I haven't decided on a school yet, but I'm looking at PhD programs in applied or computational math.

KK: Excellent. That’s great. Well, good luck to you. So what pairs well with this theorem?

AL: Yeah, so I picked chicken tortilla soup, mostly because it's my favorite soup. But you kind of have all sorts of different things piled in there. You've got your spices, you've got your chicken, you got your, you know, pieces of tortilla, you've got maybe chives or your different little vegetables in there. So you have a whole bunch of things that may or may not look very similar or different to each other. But you get one scoop of it, and you have the whole thing. So it's kind of like a 24-for-one deal.

KK: It’s like 24 equivalent soups in one.

AL: Right!

KK: Okay. Very cool. Yeah. So are you from Texas?

AL: I’m from Oklahoma.

KK: Okay. So this region, yeah, that sort of makes sense. That’s a popular sort of soup. Yeah. Okay. All right. Excellent. Well, thanks so much, Anna, that was great.

AL: Thank you.

KK: Up next, we have Matthew Bolding. Matthew, welcome!

Matthew Bolding: Hey there. Thank you. Yeah. So my name is Matthew. I am a senior dual degree student for mathematics and computer science, and I'm from around here as well.

KK: Okay. Cool. All right. So what's your favorite theorem?

MB: My favorite theorem today, or really, I guess all time, is the four color theorem.

KK: The four color theorem. Oh boy. Okay.

MB: Are you familiar?

KK: Oh, yeah, I am. So there's a lot to be said about this theorem. So yeah, tell us what it is, and then we'll unpack it.

MB: Sure. So I suppose in the most dry mathematical language, the four color theorem states that the chromatic number for a graph, for a simple planar graph specifically, is no more than four. And I guess there are some things to unpack there.

KK: Yes. Right.

MB: So a planar graph is a graph that can be constructed, drawn, if you will, such that no two edges cross one another. And a simple graph is one that does not have any self loops. And I guess the other part of it is, what's a chromatic number, right? And so a chromatic number is essentially the smallest k for which the graph is k-colorable. And then that also takes us down the rabbit hole with what is k-colorable? So a k-coloring of a graph is an assignment of at most k colors to the vertices of the graph in such a way so that no two adjacent vertices are the same color.

KK: Right? Okay. So most people might know this in terms of maps.

MB: Yes, that is correct. That actually what got me interested in it. It seems so deceivingly simple, I guess, you know, four colors, all you need are really at most four colors. You could do it in three or two, depending on on the map or graph.

KK: Sure.

MB: But, you know, you can ask people, what do you think? How many colors you think you might need to color this graph under these conditions and constraints? Oh, six, seven? No, you only need four.

KK: Right.

MB: And although I'm no cartographer, you know, I've never really colored a map maybe since pre-K, I think it is just so interesting, and sort of out of the blue, that you only need four colors. And with that, I also think it is really interesting that the proof that the chromatic number is no more than four hasn't been proved by humans, by hand.

KK: Right.

MB: And we've had to rely on computers to facilitate that proof. And being a computer science, or within the computer science field to study, I think that's really, really interesting.

KK: I thought that was part of part of your motivation here. I mean, so although, yeah, recently there, some people announced a proof by hand of the four color theorem. And it's wrong.

MB: Really?

KK: So I mean, anybody who's tried to prove this thing just by hand has come up short. And, you know, so the question of the maps, right? So, for a map, it's like, you know, you don't want two states that share a border to be colored the same, you know, and you can draw examples where you need four, but it's interesting that you can always do it with four, but then it does turn into a graph or a question, because how do you how do you create a graph out of this? Well, you stick a vertex for each state, and you join them if they share a border, and now you've converted it to a graph theory question. So that's how they come along.

MB: Exactly. And actually, I found this, or was presented this theorem in graph theory last spring. And I mean, I really — of course, with a computer science background, I mean, I just, you know, jumped in headfirst. I thought it was the coolest class. You know, it maybe doesn't have the same rigor as, like, really analysis or something.

KK: Oh no, it’s hard.

MB: Well, don't get me wrong. There are some difficult concepts. You know, I guess, with a computer science mindset, you know, I mean, of course, not every topic, you know, had roots in computer science, but maybe with Huffman encoding, or shortest path algorithms, you know, it was just so interesting and fascinating.

KK: Yeah, graph theory stuff is vital in computer science. I mean, it's everywhere, and having good algorithms for that is really important. You know, decision trees and all these kinds of things that you need to know. Cool. All right. So, yeah, but it's true that the first proof was given in, what, 1976?

MB: Around there.

KK: And yeah, and basically, it reduces to some couple hundred special cases that you just check. And then you get a computer to check it. And yeah, so for mathematicians, that's unsatisfying, right? We would just like a nice clean, wordy proof that works instead of relying on computer code. But I mean, I'm okay with it personally.

MB: I am too, but according to the Wikipedia page for this theorem, there are still many doubters. I guess we just have to prove it by hands.

KK: I guess. So that's how it goes. All right. So what pairs well with the four-color theorem?

MB: I really struggled trying to think about something that went along with this, but I landed on something that you could actually, you know, show the four color theorem with, and that would be Skittles.

KK: Okay.

MB: You know, you could lay them out all flat, you know, make it a planar collection of Skittles, basically. you could arrange the Skittles in such a way that no adjacent Skittles share the same color. Of course, there are more than four colors in a Skittle pack.

KK: Right.

MB: I would think it's been a while since I've had Skittles.

KK: Yeah, too sweet. Although, so the five color theorem is really not so hard to prove. Apparently, I've been told. I think I may have even read the five color theorem proof. It's not so bad. But four is tricky.

MB: I looked back at my graph theory notes, and we worked from a chromatic number no greater than six to five. And then we sort of just had a blank, you know, statement. Well, you can prove that the chromatic number is no more than four. But right, the ones for five and six aren't, too, too — I mean, compared to having to do it on a computer.

KK: Right. Right. Compared to people, you know, not necessarily believing the computer proof, right? Yeah. Yeah. We’re convinced about five, so we’re good. All right. Well, Matthew, that was great. Thank you.

MB: Thank you.

KK: All right, up next we have Brandon Isensee. Brandon?

Brandon Isensee: I'm Brandon Isensee. I'm a math major at TCU. I'm a senior. I'll be graduating this semester.

KK: What’s next?

BI: Grad school at Rice University.

KK: In math?

BI: Yeah. Computational and applied mathematics.

KK: Okay, that’s That’s great. It’s a terrific university. You're going to have a great time there. Well, I mean, grad school is what it is. It's fun and work and all those things. But it will really a great experience. All right. So, what’s your favorite theorem?

BI: So my favorite theorem is called Sharkovskii's theorem.

KK: Sharkovskii's theorem?

BI: Yes.

KK: Okay.

BI: Have you heard of it?

KK: I have. But let's tell our listeners.

BI: So kind of the theoretical way of saying it is that the theorem relates to discrete equations that are in, it’s one-dimensional discrete equation, so you only have one variable, and if a certain period exists — if a certain periodic orbit exists — that implies the existence of other periodic orbits. And if there's a three-cycle in particular, that implies the existence of all the other cycles

KK: Yes.

BI: And so that's more of the theoretical way of saying it. But if we put this in more concrete terms, if you have an equation that models a population over time, and it's discrete time, so it's years 0, 1, 2, 3, etc. So this equation tells you the population values for each year, right? And say there's a population growth parameter within this equation that you can vary, so we'll call it K. And this parameter tells you how fast the population is growing. And so say your growth parameter is two. And for this growth parameter, no matter which population value you choose, your population ends up oscillating between three different values across time.

KK: Right.

BI: So that's the long term behavior. So as time goes on, your population oscillates between, say, four individuals, five individuals, six individuals, and it keeps repeating. So it's 4, 5, 6, 4, 5, 6, 4, 5, 6. So if that's the case, then that implies that there is a four-cycle. So maybe it's 8, 9, 10, 11, there's a five-cycle, there's a six-cycle, there’s a 1-million-cycle, all the other cycles exist.

KK: Right.

BI: And so it's quite interesting, because when you look at particular examples of equations, like the discrete logistic map, and you look at where the three-cycle exists, you only see that three-cycle. So you may be wondering why? Why am I only seeing that three-cycle and not the four-cycle or the five-cycle? That's because Sharkovskii's theorem tells you that these cycles exist, but doesn't tell you whether or not they're stable. And so when you see the three cycle, it's stable, because that's where your populations are oscillating between. But when it's unstable, well those unstable cycles repel the population values away from them, and it ends up settling at that three cycle. So it's almost like there's an infinite number of fixed points, if I'm understanding this correctly. It’s like there's an infinite number of fixed points where you see the three-cycle, but all of them are unstable, except for the three-cycle, because that's what you're seeing on the graph.

KK: Right. Yeah. So sometimes this theorem is stated as “period three implies chaos.” Right?

BI: Right. So there, so there was actually like a difference between those two. So Sharkovskii's theorem is the one that's stronger, because that tells you exactly which periods imply the existence of other periods.

KK: Yes, right.

BI: And so period three, that relates to the three-period, right, the three-period implies everything, but Sharkovskii's theorem tells you exactly which periods imply the existence of others.

KK: Yeah. And if I remember right, he puts some weird order on the natural numbers.

BI: Yes. So the order the order is, if you're doing this in rows, the first row is your odd numbers, so 3,5,7…. Your second row is your odd numbers times two. And then the next row is your odd numbers times two to the second power.

KK: Sure. Right.

BI: So it's odd numbers times powers of twos. It’s very interesting. It took me some time to understand the order, but now I get it.

KK: And then, like, one is at the end or something, right?

BI: Yeah. Your two-cycles and such, you know, your periods of twos are all the way at the bottom.

KK: And then this is very important for discrete dynamics, right? It's just kind of the whole story. That's very cool. It's very, very cool. All right, so what do you think pairs well, with Sharkovskii's theorem?

BI: So maybe it's very a superficial connection.

KK: That doesn’t matter.

BI: There's a story with Alan Turing. He's the mathematician — for those that don't know, he solved the Enigma code during World War Two, the Germany Enigma code, and because of that, I forget the exact estimates, but it's like at least a million lives were saved because of that in, like, two years.

KK: It was vitally important.

BI: That shortened the war by two years. And so there's a story with Alan Turing where he rode his bicycle, and after a certain number of revolutions of the bike wheels, the bike chain would fall off. But instead of him just fixing it, he would just count the number of revolutions as he's riding the bike, and right before the bike chain would fall, he’d get off the bike and he would just put the bike chain back on.

KK: I wonder if that’s true.

BI: I suppose so.

KK: It’s a good story either way. Yeah. Okay. That's that's a good pairing. Excellent. Thanks.

BI: Thank you.

KK: All right, and our last willing volunteer today — I think they were all willing — is Julia Goldman.

Julia Goldman: Hi. Yeah, I'm Julia. I'm in my first year of grad school here at TCU.

KK: In math?

JG: Yes.

KK: Okay. How do you like it so far?

JG: I like it so far.

KK: Math’s pretty cool.

JG: I think so.

KK: All right. So what's your favorite theorem?

JG: So my favorite theorem today is Brouwer’s fixed point theorem.

KK: Brouwer! All right, good. Finally a topology theorem. Good.

JG: So last week, I was trying to come up with a theorem talk about, and one of my professors suggested this one. And when I was going online, and reading it and learning about it, I was looking at all these proofs of it that seem fairly technical, and I’d probably want to take a topology class to really get into it.

KK: Sure, right.

JG: But the theorem itself is, I think, very understandable. And I came across so many of these cute little fun real world examples that make the theorem pretty explainable to anyone of any math background, and I really appreciated that aspect of it is approachable.

KK: Okay, so what's the theorem? Let’s remind everyone.

JG: Oh yeah. The theorem is for any continuous function of a convex compact set onto itself, there's going to be at least one fixed point.

KK: Right. One point that doesn't move.

JG: Exactly. I think that’s really fun. There’s a couple examples. Like if I had a map of Fort Worth right now, and I laid it on the floor, there would be at least one point on that map lying directly on top of the point it's supposed to represent.

KK: That’s right.

JG: That’s kind of fun.

KK: Yeah. That's a good example.

JG: Then my other favorite example is a cup of tea. You stir the cup of tea. When you're done stirring, there’s going to be one little bit of your tea that's in the same spot as when you started stirring.

KK: That’s right. Okay. Are you a tea drinker or a coffee drinker?

JG: A little bit of both. Maybe it's because I'm not a topologist, but I read that example, and I just thought, I feel like I could stir my tea enough.

KK: Sure.

JG: But Brouwer says I'm wrong.

KK: That’s right. That's right. Okay. So maybe you don't know enough topology to prove this yet? Did you find a favorite proof that sort of made sense to you?

JG: Not a favorite. I just glanced over it.

KK: Right. So the algebraic topology proof involves, like, something with the homotopy groups, or homology groups and things like that. So that's kind of weird. There are sort of analysis-type proofs. So we just talked about dynamical systems a little bit. So if you just pick any point, and you start iterating the function, right? Just keep pushing it around, then eventually it will converge to a fixed point. Well, some subsequence of it will, because you're in a compact set. So that sort of analysis thing. So that's another way to think about it. But yeah, this is a popular theorem among topologists. You know, we all really dig this theorem a lot. It's probably one of our favorite examples. It is a fan favorite. Absolutely. Yeah. Okay, good. I like the map example, too, because that's really illustrative. Okay. So what do you think pairs well with Brouwer’s fixed point theorem?

JG: Well, I wanted to pair my favorite theorem with my current favorite TV show, which I'm only a little bit embarrassed to say is a reality show called Love Island.

KK: Okay. I've heard of it. I have not watched it.

JG: I think they must have some topologists on set there because I think the show itself is kind of an example of a theorem if you stretch some definitions a little bit.

KK: Okay. Let’s hear it.

JG: So if you're unfamiliar with the show, it’s a dating show. The very first episode, all the participants are put into couples. And then there’s, like, 65 episodes of just, like, fighting and breaking up and getting into other couples, whatever. From the seasons I've seen, at the end of the show, there's always at least one couple that ends up back in their original pairing.

KK: Right.

JG: So thinking of the participants as our convex set, and all the show drama as a function, then there’s your example of the theorem.

KK: Yeah, it's funny how there are always two people who were like, you know, we were we were right all along.

JG: Exactly.

KK: Do they do it randomly? I’m sure the producers don’t actually do it randomly.

JG: I mean, they kind of mix it up. They let the girls choose the first time, or the boys.

KK: All right. Okay. Well, I’ll have to check this out.

JG: Great show. Highly recommended.

KK: Okay. All right. Excellent. Okay, well, thanks, Julia.

JG: Thank you so much.

[outro]

In another Very Special Epsiode of My Favorite Theorem, Kevin had the privilege of asking a group of nine TCU students about their favorite theorems. We loved the variety of theorems and pairings they picked! Below are some links to more information about their favorites.

Aaryan Dehade led off with the fundamental theorem of calculus.

Duc Toan Nguyen's favorite is the mean value theorem, which some would argue is the real fundamental theorem of calculus. It was also a hit with our past guests Amie Wilkinson and Aris Winger.

Maiyu Diaz shared Stirling's formula for approximating factorials.

Hope Sage chose Bell's theorem from physics, which was a response to a paper by Einstein, Podolsky, and Rosen.

Jonah Morgan shared his love for Gödel's incompleteness theorems, which also came up when we talked with math students from CSULA last year.

Anna Long chose the invertible matrix theorem, a behemoth of a theorem that gives scads of ways to show that a matrix is invertible.

Matthew Bolding highlighted the four-color theorem. 

Brandon Isensee chose Sharkovskii's theorem, which was also the favorite of past guest Kimberly Ayers.

Julia Goldman finished out the episode with a perennial MFT favorite, the Brouwer fixed point theorem. We have talked about it on past episodes, most recently with Priyam Patel. See if you agree with Julia that it is the mathematics underlying the TV show Love Island!