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

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

EL: Very nice.

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

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

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

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

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

EL: Yeah.

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

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

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

EL: Yeah, very cool.

KK: Yeah.

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

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

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

LK: What do you think, Kevin?

KK: I, uh, Taxman?

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

EL: Oh, yeah.

LK: You know, shout out to that.

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

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

KK: It is.

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

KK: Yeah, Amie Wilkinson.

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

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

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

LK: Hmm.

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

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

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

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

LK: Yeah, I was thinking that.

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

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

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

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

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

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

KK: Sure.

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

KK: Sure.

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

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

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

LK: That’s right.

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

KK: Sure.

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

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

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

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

KK: It’s been verified.

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

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

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

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

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

EL: I think it’s just not that hard.

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

KK: Needing a supercomputer in 1976.

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

KK: Okay.

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

EL: Yeah.

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

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

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

EL: Yeah.

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

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

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

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

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

KK: Or fundamental theorem of algebra, right?

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

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

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

it's a nice one to record.

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

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

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

EL: Yeah.

LK: I don’t know.

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

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

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

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

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

KK: We really do.

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

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

KK: The Borsuk-Ulam theorem.

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

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

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

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

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

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

KK: Oh, the snake lemma.

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

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

EL: I think t's called It’s My Turn.

KK: That’s it.

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

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

KK: Has anybody?

LK: As mathematicians, maybe we should.

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

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

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

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

KK: Sure.

EL: Yeah.

KK: So what's the statement?

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

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

EL: Yeah.

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

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

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

EL: A can of worms.

LK: Yeah, so is it proven or not?

KK: It depends on who you ask.

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

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

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

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

KK: Yeah, yeah. Okay, so.

EL: I like this.

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

EL: I think it is.

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

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

KK: All right. Yeah.

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

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

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

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

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

KK: Sure.

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

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

LK: Oh, I that sounds fun to me.

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

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

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

LK: There’s always a first time.

EL: You’re on the hot seat!

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

LK Thank you for your time.

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

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

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