My Favorite Theorem

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Episode 25 - Holly Krieger Aug 23, 2018

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Episode 25 - Holly Krieger Kevin Knudson & Evelyn Lamb Download EL: Hello and welcome to My Favorite Theorem, a podcast where we ask mathematicians what their favorite theorem is. I’m your cohost Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah. This is your other cohost. KK: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. How are you doing, Evelyn? EL: I’m all right. How about you? KK: Okay. So one of our former guests, who I won’t name, was giving a big lecture here at the colloquium series this week, so I got to meet that person in person. EL: Oh, excellent. KK: So I might even have a better picture for the webpage, for the post to say, hey, our hosts and guests can actually be in the same place at the same time. EL: Yeah, that would be exciting. And one of these days, maybe you and I will meet in person, which I’m pretty sure we have not yet. KK: Maybe. I know we haven’t. I keep threatening to come to Salt Lake City, but I don’t think Salt Lake can handle me. I have actually been there once. Wonderful town. It’s a great city. EL: I like it. So today we are very glad to have Holly Krieger on the show. So Holly, would you like to tell us a little bit about yourself? HK: Sure, I’d be happy to. Thanks for having me, first of all. So I am a lecturer at the University of Cambridge. I’m also a fellow at one of the constituent colleges of Cambridge, Murray Edwards College, and the kind of math I’m most interested in is complex dynamics and number theory. So I do a lot of studying of the Mandelbrot set and the arithmetic properties of these kinds of things and related questions. EL: And I see you and I have the same poster of the Mandelbrot set. Mine is not actually hanging up yet. You have been better at getting the full experience by hanging it up, but I see that poster behind you. HK: That’s right, the Mandelmap. It’s amazing, this poster. I just found it on Kickstarter, and then I sent it to a bunch of mathematician friends, and so occasionally I will go to someone to visit someone mathematically, and they have the same poster in their office. It’s very satisfying. EL: Well, we have invited you here to ask you what your favorite theorem is. So what’s your favorite theorem? HK: So here’s the thing: I shouldn’t be on this podcast because I don’t have a favorite theorem. KK: No, no, no. HK: I don’t have a favorite theorem, it’s true. Somehow I’m too much of a commitment-phobe, like I have a new favorite theorem every week or something like that. I can tell you this week’s favorite theorem. EL: That’s good enough. KK: That’s fine. Ours have probably changed too. Evelyn and I in Episode 0 stated our favorite theorems, and I’m pretty sure Evelyn might have changed her mind by now. EL: Yeah, well, one of our other guests, Jeanne Clelland, made a pretty good case for the Gauss-Bonnet theorem. KK: She really did. EL: I think my allegiance has shifted. HK: Maybe you can do a podcast retrospective, every 20 episodes or something, what are the hosts’ favorite theorems today? KK: That’s a good idea, actually. Good. HK: So, my favorite theorem for this week. I love this theorem because it is both mathematically sort of really heavy-hitting and also because it has this sort of delicious anti-establishment backstory to it. My favorite theorem this week is Brouwer’s fixed-point theorem. KK: Nice. HK: Maybe I should talk about it mathematically first, maybe the statement? EL: Yeah. HK: Okay. So I think the easiest way to state this is the way Brouwer would have thought about it, which is if you take a closed ball in Euclidean space, so you can think about an interval in the real line, that’s a closed ball in the one-dimensional Euclidean space, or you can think about a disc in two-dimensional space, or what we normally think of as a ball in three-dimensional space, and higher you don’t think about it because our brains don’t work that way. So if you take a closed ball in Euclidean space, and you take a continuous function from that closed ball to itself, that continuous function has to have a fixed point. In other words, a point that’s taken to itself by the function. So that’s the statement of the theorem. Even just avoiding the word continuous, you can still state this theorem, which is that if you take a closed ball and morph it around and stretch it out and do crazy things to it, as long as you’re not tearing it apart, you’ll have a fixed point of your function. KK: Or if you stir a cup of coffee, right? HK: That’s right, so there’s this anecdote that what Brouwer was thinking about—I have no idea if this is accurate. KK: Apocryphal stories are the best. HK: Reading about him biographically, I almost feel like coffee would be too exciting for Brouwer. So I’m not actually sure about the accuracy of this story. So the story goes that he was stirring his coffee, and he noticed that there seemed to be a point at every point in time, a point where the coffee wasn’t moving despite the fact that he was stirring this thing. So that actually leads to one of the reasons I like this in terms of real-world applications. It’s a good—well, depending on who you hang out with, it’s a good—cocktail party theorem because if you’re making yourself a cocktail and you throw all the ingredients into your shaker and you start stirring them up, well, when you’re done stirring it, as long as you haven’t done anything crazy like disconnected the liquid inside of the shaker, then you’ve got to have some point in the liquid that’s returned to its original spot. And I think that’s a fun version of the coffee anecdote. EL: But the cocktail would definitely be too exciting for Brouwer. VN: I would be really surprised. He was a vegetarian, not that you can’t be a fun vegetarian. He was a vegetarian, and he was sort of a health nut in general, and that was back in a time—he proved this theorem in the early 1900s—back in a time when I don’t think that behavior was quite so common. KK: It was more, like, on a commune. You’d go to some weird, well I shouldn’t say weird, you’d go to some rural place and hang out with other like-minded people. HK: That’s right. KK: And live this healthful lifestyle. You would eschew meat and sugar and all that stuff. HK: Right, exactly. So the other way I like to describe this in terms of the real world, and I think this is a common way Brouwer himself actually described this, is that if you take a map, so take a map of somewhere that’s rectangularly shaped. You can either think the map itself is a rectangle, so whatever it pictures is a rectangle, or you can think of Colorado or something like that. If you take a map, and you’re in the place that’s indicated by the map, then there’s somewhere on the map that is precisely in the same point on the map as it is in the place. Namely, where you are. But you can get more specific than that. So those are two sort of nice ways to visualize this theorem. One of the reasons I like it is that it basically touches every subfield of mathematics. It has implications for differential equations and almost any sort of applied mathematics that you might be interested in. Things like existence of equilibrium states and that kind of thing over to its generalizations, which touch on number theory and dynamical systems and these kinds of things through Lefschetz fixed-point theorem and trace formula and that kind of thing. So mathematically speaking, it’s sort of the precursor to the entire study of fixed-point theorems, which is maybe an underappreciated spine running through all of mathematics. KK: Since you’re interested in dynamics, I can see why you might really be interested in this theorem. HK: Yeah, that’s right. It comes up particularly in almost any kind of study of dynamical systems, where you’re interested in iteration, this comes up. EL: I like to ask our guests if this was a love at first sight theorem or if it’s grown on you over time. HK: That’s a good question. It’s definitely grown. I think when you first meet this thing, I mean let’s think about it a little bit. In one dimension, how do you think about this theorem? You think, well, I’ve got a map from, say, the unit interval to itself, right, which is a continuous map. I can draw its graph. And this is the statement essentially that that graph has to intersect the line y=x between 0 and 1. KK: So it’s a consequence of the Intermediate Value Theorem. HK: That’s right. This is one of those deals where we always tell the calc students, “Tilt your head,” and they always look at us like we’re crazy, but then they all do it and it works. I find this appealing because it’s sort of an intersection theoretic way to think about it, which is sort of the generalizations that I’m interested in. But I think that you don’t realize the scope of this kind of perspective viewing this as intersection, and how that sort of leads you into algebraic geometry versions of this theorem. You don’t realize that at first. Same with, you don’t realize the applications to Banach spaces at first, and equilibrium states at first, so understanding the breadth of this theorem is not something that happens right away. The other thing is that really why I like this theorem is the backstory. Can I tell you about the backstory? KK: Absolutely. HK: So Brouwer, you can already tell I kind of don’t like him, right? So Brouwer was a Dutch mathematician, and he was essentially the founder of a school of mathematical philosophy known as intuitionism. What these people think, or perhaps thought—I don’t know who among us is one of them at this point—what these people think is that essentially mathematics is a result of the creator of mathematics, that there is no mathematics independent of the person who is creating the mathematics. So weird consequences of this are things like not believing in the law of the excluded middle. So they think a thing is only true if you can prove it and only not true if you can provide a counterexample. So something that is an open problem, for example, they consider to be a counterexample, or whatever you want to say, to the law of the excluded middle. So it’s in some sense a time-dependent mathematical philosophy. It’s not that everything is either true or not in the system, but true or not or not yet. EL: That’s interesting. I don’t know very much about this part of math history. I’ve sort of heard of the fact that you don’t have to necessarily accept the law of the excluded middle, but I hadn’t heard people talk about this time-dependent aspect of it. I guess this is before we get into Cantor and Gödel, or more Gödel and Cohen’s, incompleteness theorems, which kind of seem like that would be a whole other wrench into things. HK: That’s right. So this does predate Gödel, but it’s after Cantor. This was basically a knee-jerk reaction to Cantor. So the reason why I’m sort of anti-this philosophy is that I view Cantor as a true revolutionary in mathematics. KK: Absolutely. HK: Maybe I’ll have a chance to say a little bit about the connection between the Brouwer fixed-point theorem and some of what he did, but Cantor sat back, or took a step back and said, “Here’s what the size of a set is, and I’m going to convince you that the real line and the real plane, this two-dimensional space, have the same size.” And everyone was so deeply unhappy with this that they founded schools of thought like intuitionism, essentially, which sort of forced you to exclude an argument like Cantor’s from being logically valid. And so anyone who was opposed to Cantor, I have a knee-jerk reaction to, and the reason I find this theorem so delicious, sort of appealing, is because it’s not constructivist. Brouwer’s fixed-point theorem doesn’t hand you the fixed point, which is what Brouwer says you should have to do if you’re actually proving something. He really believed, I mean, he worked on it from his thesis to his death, essentially, while he was active, he really believed in this philosophy of mathematics that you cannot say there exists a thing but I can’t ever tell you what it is. He thought you really had to hand over the mathematical object in order to convince somebody. And yet one of his most famous results fails to do exactly that. And the reason why is that his thesis advisor was like, “Hey, no one is going to listen to you unless you do some actual mathematics. So he put aside the philosophy for a few years, proved some nice theorems in topology, in sort of the formalist approach, and went back to mathematical philosophy. KK: I did not know any of this. This whole time-dependent mathematics, now I can’t stop thinking about Slaughterhouse-5, right, you’ve read Slaughterhouse-5? The Tralfamadorians would tell us, you know, that it’s already all there. It’s encased in amber. They can see it all, so they know what theorems we’re going to discover later. HK: That’s right. KK: So what’s your favorite proof of this theorem? HK: So I think my favorite proof of this theorem is probably not Brouwer’s. It’s probably an algebraic topology proof, essentially. KK: I thought you’d go with the iteration proof, but okay. HK: No, I don’t think so because what it’s really about to me, it really is a topological statement about the nonexistence of retractions. So let’s just talk about the disc, let’s do the two-dimensional version. So if you had, so first of all, it’s a proof by contradiction, which already Brouwer is not on board with, but let’s do it anyways. So if you had a function which was a continuous map of the closed unit disc to itself which had no fixed point, then you could define a new function which maps the closed disc to its boundary, the circle, in the following way. If you have a point inside the disc, you look at where its image is. It’s somewhere else, right, because there are no fixed points. So you can draw the ray from its image through that point in the plane. That ray will hit the unit circle exactly once. That’s the value you assign the point in this new function. This will give you a new map, which maps the closed unit disc to its boundary, so this map is a retraction, which means it acts as the identity on the unit circle, and it maps the entire disc continuously onto the boundary circle. And such a thing can never exist. KK: You’ve torn a hole in the disc. HK: You’ve torn a hole in the disc. It’s really believable, I mean, rather than a rigorous proof, think about the interval. Take every point in the interval and assign it a value of either 0 or 1. You obviously have to tear it to do it. It’s totally clear in your head. with the disc, maybe it’s not quite so obvious. Usually the cleanest proof of the non-existence of a retraction like this goes through algebraic topology and understanding what the fundamental groups of these two objects are. KK: That’s the proof I was thinking of, being a topologist. HK: You thought maybe I’d be dynamical about it? KK: Well, you could just pick a point and iterate, and since it’s a complete metric space, it converges to some point, and that thing has to be fixed. But that’s also not constructive, right? HK: It’s also not constructive. But there are approximate construction versions. KK: Right. HK: One more thing I like about this theorem, in terms of its implications, is it’s one more tool Brouwer used in his theorem proving the topological invariance of dimension, that dimension is a well-defined notion under homeomorphisms. In particular, you don’t have a homeomorphism, just stretching, continuous in both directions, from, like, R^n to R, n-dimensional Euclidean real space to the real line. This doesn’t sound earth-shattering to us now. I think we kind of take it for granted. But at the time, this was not so long after Cantor was like, “Oh, but there is actually an injection, right, from n-dimensional Euclidean space to the real line.” So it’s not that it was surprising, but it was sort of reassuring, I think, that if you impose co…

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Episode 24 - Vidit Nanda Aug 09, 2018

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Episode 24 - Vidit Nanda Kevin Knudson & Evelyn Lamb Download Kevin Knudson: Welcome to My Favorite Theorem! I’m your host Kevin Knudson, professor of mathematics at the University of Florida. And this is your other host. Evelyn Lamb: Hi. I’m Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City. So how’s it going, Kevin? KK: It’s okay. Classes are almost over. I’ve got grades for 600 students I still need to upload. But, you know, it’s an Excel nightmare, but once we get done with that, it’s okay. Then my son comes home for Christmas on Saturday. EL: Oh great. I don’t miss grading. I miss some things about teaching, but I don’t miss grading. KK: No. EL: I don’t envy this time of the semester. KK: Certainly not for a 600-student calculus class. But you know, I had a good time. It’s still fun. Anyway, today we are pleased to welcome Vidit Nanda. Vidit, why don’t you introduce yourself and tell everyone about you? Vidit Nanda: Hello. My name is Vidit Nanda. I’m a research fellow at the University of Oxford and the amazing new Alan Turing Institute in London. This year I’m a member at the School of Mathematics at the Institute for Advanced Study in Princeton. I’m very happy to be here. Thank you both for doing this. This is a wonderful project, and I’m very happy to be a part of it today. KK: Yeah, we’re having a good time. EL: Can you tell us a little more about the Alan Turing Institute? I think I’ve heard a little bit about it, but I guess I didn’t even know it was that new. I thought I had just never heard of it before. VN: Right. So about three years ago, and maybe longer because it takes time to set these things up, the UK decided they needed a national data science center, and what they did was they collected proposals from universities, and the ones who are now, well, the original five universities that got together and contributed funds and professors and students to the Turing Institute were Oxford, Cambridge, Warwick, UCL, and Edinburgh. Now we have a space on what they call the first floor of the British Library, and we would call the second floor of the British Library. Half of that floor is called the Alan Turing Institute, and it’s kind of crazy. You enter the British Library, and there’s this stack of books that kind of looks like wallpaper. It’s too beautiful, you know, but it is real. It’s behind glass. And then you turn to the right, and it’s Las Vegas, you know. There’s a startup-looking data science center with people dressed exactly the way you think they are with the hoodies, you know. It’s sort of nuts. But there are two things I should tell everyone about the Alan Turing Institute who’s listening. The first one is that if you walk down a flight of steps, there’s a room called Treasures of the British Library. Turn left, and the first thing you see is a table with Da Vinci’s sketches right next to Michaelangelo’s letters with the first printing of Shakespeare. Those are the first things you see. So if you’re ever thinking about cutting a corner in a paper you’re writing, you go down to that room, you feel bad about yourself for ten minutes, and you rush back up the stairs, inspired and ready to work hard. KK: Yeah. This sounds very cool. EL: Wow, that’s amazing. VN: That’s the first table. There’s other stuff there. KK: Yeah, I’m still waiting on my invitation to visit you, by the way. VN: It’s coming. It would help if I’m there. KK: Sure, once you’re back. So, Vidit, what’s your favorite theorem? VN: Well, this will not be a surprise to the two of you since you cheated and you made me tell you this in advance. And this took some time. My favorite theorem is Banach’s fixed point theorem, also called the contraction mapping principle. And the reason it’s my favorite theorem is it’s about functions that take a space to itself, so for example, a polynomial in a single variable takes real numbers to real numbers. You can have functions in two dimensions taking values in two dimensions, and so on. And it gives you a criterion for when this function has a fixed point, which is a point that’s sent to itself by the function. One of the reasons it’s my favorite theorem—well, there are several—but it’s the first theorem I ever discovered. For the kids in the audience, if there are any, we used to have calculators. I promise. They looked like your iPhone, but they were much stupider. And one of the most fun things you could do with them was mash the square root button like you were in a video game. This is what we had for entertainment. KK: I used to do this too. VN: Take a large number, and you mash the square root button, and you get 1. And it worked every time. KK: Right. VN: And this is Banach’s fixed-point theorem. That’s my proof of Banach’s fixed-point theorem. KK: That’s great. What’s the actual statement, though? Let’s be less loose. VN: Right. The actual statement requires a little bit more work than having an old, beat-up calculator. The setup is kind of simple. You have a complete metric space, and by metric space you mean a space where points have a well-defined distance subject to natural axioms for what a distance is, and complete means if you have a sequence of points that are getting close to each other, they actually have a limit. They stop somewhere. If you have a function from such a complete metric space to itself so that when you apply the function to a pair of points, the images are closer together strictly than the original points were, so f(x) and f(y), the distance between them should be strictly less, some constant less than 1 times the distance between x and y. If this is true, then the function has a unique fixed point, and the amazing part about this theorem that I cannot stress highly enough is that the way to find this fixed point is you start anywhere you want, pick any initial point and keep hitting f, this is mashing the square root button, and very quickly, you converge to the actual fixed point. And when you hit the square root button, nothing changes, you just stay at 1. KK: And it’s a unique fixed point? VN: It’s a unique fixed point because wherever else you start, you reach that same place. So I’m an algebraic topologist by trade, and this is very much not an algebraic topology fixed-point theorem. The algebraic topology fixed-point theorem makes no assumptions on the function, like it should be bringing points closer together. It makes assumptions on the space where the function is taking its values. It says if the space is nice, maybe convex, maybe contractible, then there is a fixed point, no uniqueness and no recipe for converging to the fixed point. KK: In fact, we recently had a guest who chose the Brouwer fixed-point theorem. EL: Yeah. VN: Yes, the Brouwer fixed-point theorem is one of my favorites, it’s one of the tools I use in my work a lot, but I always have this sort of analyst envy where their fixed-point theorem comes with a recipe for finding the actual fixed point. KK: Right. VN: Instead of an existence result. KK: Yeah, we just wave our hands and say, “Yeah, yeah, yeah, if you didn’t have a fixed point there’d be some map on homology that couldn’t exist and blah blah blah. VN: Right. And that’s sort of neat but sort of unsatisfying if what you actually care about are the fixed points. EL: Yeah, so in some ways I kind of ended up more of an analyst because of this. I was really attracted to algebra and that kind of thing, and I felt like at some point I just couldn’t do anything. I felt like in analysis, at least I could get a bound on something, even if it was a really ugly bound, I could at least come in with my hands and play around in the dirt and eventually come up with something. This is probably showing that somehow my brain is more likely to succeed at analysis or something because I know there are people who get to algebra and they can do things, but I just felt like at some point it was this beautiful but untouchable thing, and analysis wasn’t so pretty, and I didn’t mind going and mucking it up. KK: I had the opposite point of view. I never liked analysis. All those epsilons and deltas, and maybe it was a function of that first advanced calculus course, where you have to get at the end the thing you’re looking for is less than epsilon, not 14epsilon+3epsilon^2. It had to be less than epsilon. I was like, man, come on, this thing is small! Who cares? So I liked the squishiness of topology. I think that’s why I went there. VN: I think with those epsilon arguments, I don’t know about you guys, but I always ended up doing it twice. You do it the first time and get some hideous function of epsilon, and then you feed back whatever you got to the beginning of the argument, dividing by whatever is necessary, and then it looks like, when you submit your solution, it looks like you were a genius the whole time, and you knew to choose this very awkward thing initially, and you change the argument. KK: That’s mathematics, right, when you read a paper, it’s lovely. You don’t see all the ugly, horrifying ream of paper you used for the calculations to get it right, you know. I think that’s part of our problem as mathematicians from a PR point of view. We make it look so slick at the end, and people think, wait a minute, how did you do that? Like it’s magic. VN: We’re very much writing for people next door in our buildings as opposed to people on the street. It helps sometimes, and it also bites us. KK: This is where Evelyn’s so great, because she is writing for people on the street, and doing it very well. EL: Well thank you. I didn’t intend this to come back around here, but I’ll take it. Anyway, getting back to our guest, so when did you first encounter this theorem, and was it something you were immediately really into, or did it take some more time? VN: Actually, the first time I encountered this theorem in a semiformal setting, it just blazed by. I think this is where most people see it for the first time, is in a differential equations course. One of the things that’s so neat about this theorem is that it’s one of the things that guarantees you take f’(x), which equals some hideous expression of x, why should this have a solution, how long should it have a solution for, when is a solution unique? And this requires the hideous thing on the right side to satisfy the contraction mapping property. The existence and uniqueness of ordinary differential equations is the slickest, most famous application of the Banach fixed-point theorem. KK: I’d never thought about it. VN: And the analyst nods while Kevin stares off into space, wondering why this should be the case. KK: No, no, you had a better differential equations course than I did. In our first diffeq’s course, we wouldn’t bring this up. This is too high-powered, right? VN: It was sort of mentioned, this was at Georgia Tech. It was mentioned that this property holds, there was no proof, even though the proof is not difficult. It’s not so bad if you understand the Cauchy sequence, which not everyone in differential equations does. So we were not shown the proof, but there’s a contraction mapping principle. And then Wikipedia was in its infancy, so now I’m dating myself badly, but I did look it up then and then forgot about it. And then of course I saw it in graduate school all over the place. KK: Hey, when I was in college, the internet didn’t exist. VN: How did you get anything done? KK: You went to the library. EL: Did you use a card catalog? KK: I’m a master of the card catalog. EL: We had one at my elementary school library. KK: Geez. So growing up in high school, we used to go to the main public library downtown where they had bound periodicals and so if you needed to do your report about, say, the assassination of John Kennedy, for example, you had to go and pull the old Newsweeks off the shelf from 1963. I don’t know, there’s something to that. There’s something to having to actually dig instead of just having it on your phone. But I don’t want to sound like an old curmudgeon either. The internet is great. Well, although wait a minute, the net neutrality vote is happening right now. VN: It’s great while we speak. We don’t know what’s going to happen in 20 minutes. KK: Maybe in the middle of this conversation we’re going to get throttled. So Vidit, part of the fun here is that we ask our guest to pair their theorem with something. So what have you chosen to pair the contraction theorem with? VN: I’m certainly not going to suggest Plato like one of the recent guests. I have something very simple in mind. The reason I have something simple in mind is there’s an inevitability to this theorem, right? You will find the fixed point. So I wanted something inevitable and irresistible in some sense, so I want to pair it with pizza. EL: Pizza is the best food. Hands down. VN: Right. It is the best food, hands down. I’m imagining the sort of heathens’ way of eating pizza, right, you eat the edges and move in. I’ve seen people do this, and it’s sort of very disturbing to me. The edge is how you hold the damn thing in the first place. But if you imagine a pizza being eaten from the outside, that’s how I think of the contraction mapping, converging to the middle, the most delicious part of the pizza. I refuse to tell you what fraction of the last two weeks it took me to come up with this pairing. It’s disturbingly difficult. KK: So you argue that the middle of the pizza is the most delicious part? EL: Oh yeah. KK: See, my dog would argue with you. She is obsessed with the crust. If we ever get a pizza, she’s just sitting there: “Wait, can I have the crust?” EL: But the reason she gets the crust is because humans don’t find it the most delicious. VN: If I want to eat bread, I’ll eat bread. KK: I make my own pizza dough, so I make really good pizza crust. It’s worth eating. It’s not this vehicle. But you’re right. Yeah, sure. EL: We’re going to press you now. What pizza toppings are we talking here? We really need specifics. It’s 9 am where I am, so I can’t have pizza now unless I made my own. KK: You could. You can have it any time of the day. EL: But I don’t think there’s a store open. I guess I could get a frozen pizza at the grocery store. VN: Kevin would suggest having a quick-rise dough set up that, if you pour your yeast in it, it’ll be done in 20 minutes. I think, I’m not big into toppings, but it’s important to have good toppings. Maybe bufala mozzarella and a bit of basil, keep it simple. There’s going to be tomatoes in it, of course, some pizza sauce. But I don’t want to overload it with olives and peppers and sausage and all that. EL: Okay. So you’re going simple. That’s what we do. We make our own pizza a lot, and a couple years ago we decided to just for fun buy the fancy canned tomatoes from Italy, the San Marzanos. VN: The San Marzanos, yeah. EL: Buy the good mozzarella. And since then, that’s all we do. We used to put a bunch of toppings on it all the time, and now it’s just, we don’t even make a sauce, we just squish the tomatoes onto the pizza. Then put the cheese on it, and then the basil, and it’s so good. KK: I like to make, I assume you’ve both been to the Cheese Board in Berkeley? EL: No, I haven’t. I hear about it all the time. KK: It’s on Shattuck Ave in Berkeley, and they have the bakery. It’s a co-op. The bakery is scones—delicious scones, amazing scones—and bread and coffee and all that. And right next door is a pizza place, and they make one kind of pizza for the day, and that’s what you’re going to have. You’re going to have it because it’s delicious. Even the ones where you’re like “eh,” it’s amazing. The line goes down the block, and everybody’s in a good mood, there’s a jazz trio. Anyway, I got the cookbook, and that’s how I make my crust. There’s a sourdough crust, and then our favorite one is the zucchini-corn pizza. EL: Really. KK: It’s zucchinis, onions, and cheese, and then corn, and a little feta on top. And then you sprinkle s…

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Episode 23 - Ingrid Daubechies Jul 26, 2018

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Episode 23 - Ingrid Daubechies Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Hello and welcome to My Favorite Theorem. This is a podcast about math where we invite a mathematician in each episode to tell us about their favorite theorem. I’m one of your hosts, Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah. And this is your other host. Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. I’m excited about this one. EL: Yes, I’m very excited. I’m too excited to do any banter. We’re coming up on our one-year anniversary, and we are very honored today to have a special guest. She is a professor at Duke. She has gotten a MacArthur Fellowship, won many prizes. I was just reading her Wikipedia page, and there are too many to list. So we are very happy to have Ingrid Daubechies on the show. Hi, Ingrid. Can you tell us a little bit about yourself? Ingrid Daubechies: Hi, Evelyn and Kevin. Sure. I have just come back from spending several months in Belgium, in Brussels. I had arranged to have a sabbatical there to be close to and help set up arrangements for my very elderly parents. But I also was involved in a number of fun things, like the annual contest for high school students, to encourage them to major in mathematics once they get to college. And this is the year I turn 64, and because 64 is a much more special number than 60 for mathematicians, my students had arranged to organize some festivities, which we held in a conference center in my native village. KK: That’s fantastic. ID: A lot of fun. We had family and friends, we opted to have a family and friends activity instead of a conference where we tried to get the highest possible collection of big name marquee. I enjoyed it hugely. We had a big party in Belgium where I invited via Facebook everybody who ever crossed my timeline. There were people I went to high school with, there was a professor who taught me linear algebra. KK: Oh, wow. ID: So it was really a lot of fun. KK: That’s fantastic. EL: Yeah, and you have also been president of the International Mathematical Union. I meant to say that at the beginning and forgot. So that is also very exciting. I think while you were president, you probably don’t remember this, but I think we met at a conference, and I was trying to talk to you about something and was very anxious because my grandfather had just gone to the hospital, and I really couldn’t think about anything else. I remember how kind you were to me during that, and just, I think you were talking about your parents as well. And I was just thinking, wow, I’m talking to the president of the International Mathematical Union, and all I can think about is my grandpa, and she is being so nice to me. ID: Well, of course. This is so important. We are people. We are connected to other people around us, and that is a big part of our life, even if we are mathematicians. EL: But we have you on the show today to talk about theorems, so what is your favorite theorem? ID: Well, I of course can’t say that I have one particular favorite theorem. There are so many beautiful theorems. Right now I have learned very recently, and I am ashamed to confess how recently, because it’s a theorem that many people learn in kindergarten, it’s a theorem called Tutte’s embedding theorem about graphs, meshes, in my case it’s a triangular mesh, and the fact that you can embed it, meaning defining a map to a polygon in the plane without having any of the vertices cross, so really an embedding of the whole graph, so every triangle on the complicated mesh that you have, it’s a disk-type mesh, meaning it has no holes, a boundary, lots of triangles, but you can think of it as a complicated thing, and you can embed it under certain conditions in a convex polygon in the plane, and I really, really, really love that. I visualize it by thinking of it as a complicated shape and applying a hair dryer to it to kind of, like Saran wrap, and a hair dryer will flatten it nicely, will try to flatten it, and I think the fact that you can always do it is great. And we’re using it for an interesting, actually we are extending it to mappings, the theorem is originally formulated for a convex polygon in the plane, you can always map to a convex polygon in the plane, and we are extending it to the case where you have a non-convex polygon because that’s what we need, and then we have certain conditions. KK: Sure. Well, there have to be some conditions, right, because certainly not every graph, every mesh you would draw is planar. ID: Yeah. KK: What are those conditions? ID: It has to be 3-connected, and you define a set of weights on it, on the edges, that ensure planarity. You define weights on the edges that are all positive. What happens is that once you have it in the polygon, you can write each one of the vertices as a convex combination of its neighbors. KK: Yeah. ID: And those define your weights. You have to have a set of weights on the edges on your original graph that will make that possible. KK: Okay. ID: So you define weights on the original graph that help you in the embedding. What happens is that the positive weights are then used for that convexity. So you have these positive weights, and you use them to make this embedding, and so it’s a theorem that doesn’t tell you only that it is planar, but it gives you a mechanism for building that map to the plane. That’s really the power of the theorem. So you start already with something that you know is planar and you build that map. KK: Okay. ID: It’s really powerful. It’s used a lot by people in computer graphics. They then can reason on that Tutte embedding in the plane to build other things and apply them back to the original mesh they had in 3-space for the complicated object they had. And that’s also what we’re trying to use it for. But we like the idea of going to non-convex polygons because that, for certain of the applications that we have, will give us much less deformation. EL: So, is this related to, I know that you’ve done some work with art reconstruction, and actually in the back of the video here, I think I see some pictures of art that you have helped reconstruct. So is it related to that work? ID: Actually, it isn’t, although if at some point we go to 3-D objects rather than the paintings we are doing now, it might become useful. But right now this collaboration is with biologists where we’re trying to, well we have been working for several years and we’re getting good results, we are quantifying similarity of morphological surfaces. So the people we work with are working on bones and teeth. They’re paleontologists. Well, they’re interested in evolutionary anthropology, but they work a lot with teeth and bones. And there’s a lot of domain knowledge they have because they’ve seen so many, and they remember things. But of course in order to do science with it, they need to quantify how many similar or dissimilar things are. And they have many methods to do that. And we are trying to work with them to try to automate some of these methods in ways that they find useful and ways that they seek. We’ve gotten very good results in this over the many years that we’ve worked with them. We’re very excited about recent progress we’ve made. In doing that, these surfaces already for their studies get scanned and triangulated. So they have these 3-d triangulations in space. When you work with these organs and muscles and all these things in biology, usually you have 3-d shapes, and in many instances you have them voxelized, meaning you have the 3-d thing. But because they work with fossils, which often they cannot borrow from the place where the fossil is, they work with casts of those in very high-quality resin. And as a result of that, when they bring the cast back, they have the surface very accurately, but they don’t have the 3-d structure. So we work with the surfaces, and that’s why we work with these 3-d meshes of surfaces. And we then have to quantify how close and similar or dissimilar things are. And not just the whole thing, but pieces of it. We have to find ways in which to segment these in biologically meaningful ways. The embedding theorem comes in useful. But it’s been very interesting to try to build mathematically a structure that will embody a lot of how biologists work. Traditionally what they do is, because they know so much about the collection of things they study, is they find landmarks, so they have this whole collection. They see all these things have this particular thing in common. It looks different and so on. But this landmark point that we mark digitally on these scanned surfaces is the same point in all of them. And the other point is the same. So they mark landmarks, maybe 20 landmarks. And then you can use that to define a mapping. But they asked us, “could we possibly do this landmark-free at some point?” And many biologists scoffed at the idea. How could you do this? At the beginning, of course, we couldn’t. We could find distances that were not so different from theirs, but the landmarks were not in the right places. But we then started realizing, look, why do they have this immense knowledge? Because they have seen so many more than just 20 that they’re now studying. So we realized this was something where we should look at many collections, and there we have found, with a student of mine who made a breakthrough, if you start from a mapping between, so you have many surfaces, and you have a first way of mapping one to the other and then defining a similarity or not, depending on how faithful the mapping is, all these mappings are kind of wrong, not quite right. But because you have a large collection, there are so many little mistakes that are made that if you have a way of looking at it all, you can view those mistakes as the errors in a data set, and you can try to cancel them out. You can try to separate the grains from the chaff to get the essence of what is in there. A little bit like students will learn when they have a mentor who tells them, no, that point is not really what you think, and so on. So that’s what we do now. We have large collections. We have initial mappings that are not perfect. And we use the fact that we have the large collection to define, then, from that large collection, using machine learning tools, a much better mapping. The biologists have been really impressed by how much better the mappings are once we do that. The wonderful thing is that we use this framework, of course we use machine learning tools, we use all these computer graphics and dealing with surfaces to be efficient. We frame it as a fiber bundle, and we learn. If you think of it, every single one, if you look at a large collection, every one differs by little bits. We want to learn the structure of this set of teeth. Every tooth is a 2-d surface, and similar teeth can map to each other, so they’re all fibers, and we have a connection. And we learn that connection. We have a very noisy version of the connection. But because we know it’s a connection, and because it’s a connection that should be flat because things can be brought back to their common ancestor, and so going from A to B and B to C, it should not matter in what order you go because all these mappings can go to the common ancestor, and so it should kind of commute, we can really get things out. We have been able to use that in order to build correspondences that biologists are now using for their statistical analysis. KK: So differential geometry for biology. ID: Yes. Discrete differential geometry, which if there is an oxymoron, that’s one. KK: Wow. ID: So we have a team that has a biologist, it has people who are differential geometers, we have a computational geometer, and he was telling me, “you know, for this particular piece of it, it would be really useful if we had a generalization of Tutte’s theorem to non-convex polygons,” and I said, “well, what’s Tutte’s theorem?” And so I learned it last week, and that’s why it’s today my favorite theorem. EL: Oh wow, that’s really neat. KK: So we’ll follow up with you next year and see what your favorite theorem is then. EL: Yeah, it sounds like a really neat collaborative environment there where everybody has their own special knowledge that they’re bringing to the table. ID: Yes, and actually I have found that to be very, very stimulating in my whole career. I like working with other people. I like when they give you challenges. I like feeling my brain at work, with working together with their different expertise. And, well, once you’ve seen a couple of these collaborations at work, you get a feel for how you jump-start that, how you manage to get people talking about the problems they have and kind of brainstorm until a few problems get isolated on which we really can start to get our teeth dug in and work on it. And that itself is a dynamic you have to learn. I’m sure there are social scientists who know much more about this. In my limited setting, I now have some experience in starting these things up, and so my students and postdocs participate. And some of them have become good at propagating. I’m very motivated by the fact that you can do applications of mathematics that are really nontrivial, and you can distill nontrivial problems out of what people think are mundane applications. But it takes some investing to get there. Because usually the people who have the applications—the biologists, in my case—they didn’t say, “we had this very particular fiber bundle problem.” EL: Right. ID: In fact, it’s my student who then realized we really had a fiber bundle, and that helped define a machine learning problem differently than it had been before. That then led to interesting results. So you need all the background, you need the sense of adventure of trying to build tools in that background that might be useful. And I’m convinced that for some of these tools that we build, when more pure mathematicians learn about them, they might distill things in their world from what we need. And this can lead to more pure mathematics ultimately. KK: Sure, a big feedback loop. ID: Yes, absolutely. That’s what I believe in very, very strongly. But part of my life is being open to when I hear about things, is there a meaningful mathematical way to frame this? Not just for the fun of it, but will it help? EL: Yeah, well, as I mentioned, I was amazed by the way you’ve used math for this art reconstruction. I think I saw a talk or an article you wrote about, and it was just fascinating. Things that I never would have thought would be applicable to that sphere. ID: Yeah, and again it’s the case that there’s a whole lot of knowledge we have that could be applicable, and in that particular case, I have found that it’s a wonderful way to get undergraduates involved because they at the same time learn these tools of image processing and small machine learning tools working on these wonderful images. I mean, how much cooler is it to work on the Ghent altar piece, or even less famous artwork, than to work on standards of image analysis. So that has been a lot of fun. And actually, as I was in Belgium, the first event of the week of celebration we had was an IP4AI, which is Image Processing for Art Investigation, workshop. It’s really over the last 10-15 years, as a community is taking off. We’re trying to have this series of workshops where we have people who are interested in image processing and the mathematics and the engineering of that talk to people who have concrete problems in art conservation or understand art history. We try to have these workshops in museums, and we had it at a museum in Ghent, and it again was very, very stimulating exhilarating. KK: So another thing we like to do on this podcast is ask our guest to pair their favorite theorem with something. So I’m curious. What do you think pairs well with Tutte’s theore…

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Episode 22 - Ken Ribet Jul 12, 2018

Show notes

Episode 22 - Ken Ribet Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Welcome to My Favorite Theorem, a podcast about math. I’m Evelyn Lamb, one of your cohosts, and I’m a freelance math and science writer in Salt Lake City, Utah. Kevin Knudson: Hi, I’m Kevin Knudson, a professor of mathematics at the University of Florida. How are you doing, Evelyn? Happy New Year! EL: Thanks. Our listeners listening sometime in the summer will really appreciate the sentiment. Things are good here. I promised myself I wouldn’t talk about the weather, so instead in the obligatory weird banter section, I will say that I just finished a sewing project, only slightly late, as a holiday gift for my spouse. So that was fun. I made some napkins. Most sewing projects are non-Euclidean geometry because bodies are not Euclidean. KK: Sure. EL: But this one was actually Euclidean geometry, which is a little easier. KK: Well I’m freezing. No one ever believes this about Florida, but I’ve never been so cold in my life as I have been in Florida, with my 70-year-old, poorly insulated home, when highs are only in the 40s. It’s miserable. EL: Yeah. KK: But the beauty of Florida, of course, is that it ends. Next week it will be 75. I’m excited about this show. This is going to be a good one. EL: Yes, so we should at this point introduce our guest. Today we are very happy to have Ken Ribet on the show. Ken, would you like to tell us a little bit about yourself? Ken Ribet: Okay, I can tell you about myself professionally first. I’m a professor of mathematics at the University of California Berkeley, and I’ve been on the Berkeley campus since 1978, so we’re coming up on 40 years, although I’ve spent a lot of time in France and elsewhere in Europe and around the country. I am currently president of the American Mathematical Society, which is how a lot of people know me. I’m the husband of a mathematician. My wife is Lisa Goldberg. She does statistics and economics and mathematics, and she’s currently interested in particular in the statistics of sport. We have two daughters who are in their early twenties, and they were home for the holidays. KK: Good. My son started college this year, and this was his first time home. My wife and I were super excited for him to come home. You don’t realize how much you’re going to miss them when they’re gone. KR: Exactly. EL: Hi, Mom! I didn’t go home this year for the holidays. I went home for Thanksgiving, but not for Christmas or New Year. KK: Well, she missed you. EL: Sorry, Mom. KK: So, Ken, you gave us a list of something like five theorems that you were maybe going to call your favorite, which, it’s true, it’s like picking a favorite child. But what did you settle on? What’s your favorite theorem? KR: Well, maybe I should say first that talking about one’s favorite theorem really is like talking about one’s favorite child, and some years ago I was interviewed for an undergraduate project by a Berkeley student, who asked me to choose my favorite prime number. I said, well, you really can’t do that because we love all our prime numbers, just like we love all our children, but then I ended up reciting a couple of them offhand, and they made their way into the publication that she prepared. One of them is the six-digit prime number 144169, which I encountered early in my research. KK: That’s a good one. KR: Another is 1234567891, which was discovered in the 1980s by a senior mathematician who was being shown a factorization program. And he just typed some 10-digit number into the program to see how it would factor it, and it turned out to be prime! KK: Wow. KR: This was kind of completely amazing. So it was a good anecdote, and that reminded me of prime numbers. I think that what I should cite as my favorite theorem today, for the purposes of this encounter, is a theorem about prime numbers. The prime numbers are the ones that can’t be factored, numbers bigger than 1. So for example 6 is not a prime number because it can be factored as 2x3, but 2 and 3 are prime numbers because they can’t be factored any further. And one of the oldest theorems in mathematics is the theorem that there are infinitely many prime numbers. The set of primes keeps going on to infinity, and I told one of my daughters yesterday that I would discuss this as a theorem. She was very surprised that it’s not, so to speak, obvious. And she said, why wouldn’t there be infinitely many prime numbers? And you can imagine an alternative reality in which the largest prime number had, say, 50,000 digits, and beyond that, there was nothing. So it is a statement that we want to prove. One of the interesting things about this theorem is that there are myriad of proofs that you can cite. The best one is due to Euclid from 2500 years ago. Many people know that proof, and I could talk about it for a bit if you’d like, but there are several others, probably many others, and people say that it’s very good to have lots of proofs of this one theorem because the set of prime numbers is a set that we know a lot about, but not that much about. Primes are in some sense mysterious, and by having some alternative proofs of the fact that there are infinitely many primes, we could perhaps say we are gaining more and more insight into the set of prime numbers. EL: Yeah, and if I understand correctly, you’ve spent a lot of your working life trying to understand the set of prime numbers better. KR: Well, so that’s interesting. I call myself a number theorist, and number theory began with very, very simple problems, really enunciated by the ancient Greeks. Diophantus is a name that comes up frequently. And you could say that number theorists are engaged in trying to solve problems from antiquity, many of which remain as open problems. KK: Right. KR: Like most people in professional life, number theorists have become specialists, and all sorts of quote-on-quote technical tools have been developed to try to probe number theory. If you ask a number theorist on the ground, as CNN likes to say, what she’s working on, it’ll be some problem that sounds very technical, is probably hard to explain to a general listener, and has only a remote connection to the original problems that motivated the study. For me personally, one of the wonderful events that occurred in my professional life was the proof of Fermat’s last theorem in the mid-1990s because the proof uses highly technical tools that were developed with the idea that they might someday shed light on classical problems, and lo and behold, some problem that was then around 350 years old was solved using the techniques that had been developed principally in the last part of the 20th century. KK: And if I remember right — I’m not a number theorist — were you the person who proved that the Taniyama-Weil conjecture implied Fermat’s Last Theorem? KR: That’s right. The proof consists of several components, and I proved that something implies Fermat’s Last Theorem. KK: Right. KR: And then Andrew Wiles partially, with the help of Richard Taylor, proved that something. That something is the statement that elliptic curves (whatever they are) have a certain property called modularity, whatever that is. EL: It’s not fair for you to try to sneak an extra theorem into this podcast. I know Kevin baited you into it, so you’ll get off here, but we need to circle back around. You mentioned Euclid’s proof of the infinitude of primes, and that’s probably the one most people are the most familiar with of these proofs. Do you want to outline that a little bit? Actually not too long ago, I was talking to the next door neighbors’ 11-year-old kid, he was interested in prime numbers, and the mom knows we’re mathematicians, so we were talking about it, and he was asking about what the biggest prime number was, and we talked about how one might figure out whether there was a biggest prime number. KR: Yeah, well, in fact when people talk about the proof, often they talk about it in a very circular way. They start with the statement “suppose there were only finitely many primes,” and then this and this and this and this, but in fact, Euclid’s proof is perfectly direct and constructive. What Euclid’s proof does is, you could start with no primes at all, but let’s say we start with the prime 2. We add 1 to it, and we see what we get, and we get the number 3, which happens to be prime. So we have another prime. And then what we do is take 2 and multiply it by 3. 2 and 3 are the primes that we’ve listed, and we add 1 to that product. The product is 6, and we get 7. We look at 7 and say, what is the smallest prime number dividing 7? Well, 7 is already prime, so we take it, and there’s a very simple argument that when you do this repeatedly, you get primes that you’ve never seen before. So you start with 2, then you get 3, then you get 7. If you multiply 2x3x7, you get 6x7, which is 42. You add 1, and you get 43, which again happens to be prime. If you multiply 2x3x7x43 and add 1, you get a big number that I don’t recall offhand. You look for the prime factorization of it, and you find the smallest prime, and you get 13. You add 13 to the list. You have 2, 3, 7, 43, 13, and you keep on going. The sequence you get has its own Wikipedia page. It’s the Euclid-Mullin sequence, and it’s kind of remarkable that after you repeat this process around 50 times, you get to a number that is so large that you can’t figure out how to factor it. You can do a primality test and discover that it is not prime, but it’s a number analogous to the numbers that occur in cryptography, where you know the number is not prime, but you are unable to factor it using current technology and hardware. So the sequence is an infinite sequence by construction. But it ends, as far as Wikipedia is concerned, around the 51st term, I think it is, and then the page says that subsequent terms are not known explicitly. EL: Interesting! It’s kind of surprising that it explodes that quickly and it doesn’t somehow give you all of the small prime numbers quickly. KR: It doesn’t explode in the sense that it gets bigger and bigger. You have 43, and it drops back to 13, and if you look at the elements of the sequence on the page, which I haven’t done lately, you’ll see that the numbers go up and then down. There’s a conjecture, which was maybe made without too much evidence, that as you go to the sequence, you’ll get all prime numbers. EL: Okay. I was about to ask that, if we knew if you would eventually get all of them, or end up with some subsequence of them. KR: Well, the expectation, which as I say is not based on really hard evidence, is that you should be able to get everything. KK: Sure. But is it clear that this sequence is actually infinite? How do we know we don’t get a bunch of repeats after a while? KR: Well, because the principle of the proof is that if you have a prime that’s appeared on the list, it will not divide the product plus 1. It divides the product, but it doesn’t divide 1, so it can’t divide the new number. So when you take the product and you factor it, whatever you get will be a quote-on-quote new prime. KK: So this is a more direct version of what I immediately thought of, the typical contradiction proof, where if you only had a finite number of primes, you take your product, add 1, and ask what divides it? Well, none of those primes divides it. Therefore, contradiction. KR: Yes, it’s a direct proof. Completely algorithmic, recursive, and you generate an infinite set of primes. KK: Okay. Now I buy it. EL: I’m glad we did it the direct way. Because setting it up as a proof by contradiction when it doesn’t really need the contradiction, it’s a good way, when I’ve taught things like this in the past, this is a good way to get the proof, but you can kind of polish it up and make it a little prettier by taking out the contradiction step since it’s not really required. KR: Right. KK: And for your 11-year-old friend, contradiction isn’t what you want to do, right? You want a direct proof. KR: Exactly. You want that friend to start computing. KK: Are there other direct proofs? There must be. KR: Well, another direct proof is to consider the numbers known as Fermat numbers. I’ll tell you what the Fermat numbers are. You take the powers of 2, so the powers of 2 are 1, 2, 4, 8, 16, 32, and so on. And you consider those as exponents. So you take 2 to those powers of 2. 2^1, 2^2, 2^4, and so on. To these numbers, you add the number 1. So you start with 2^0, which is 1, 2^1 is 2, and you add 1 and get 3. Then the next power of 2 is 2. You add 1 and you get 5. The next power of 2 is 4. 2^4 is 16. You add 1, and you get 17. The next power of 2 is 8. 2^8 is 256, and you add 1 and get 257. So you have this sequence, which is 3, 5, 17, 257, and the first elements of the sequence are prime numbers. 257 is a prime number. And it’s rather a famous gaffe of Fermat that he apparently claimed that all the numbers in the sequence were prime numbers, that you could just generate primes that way. But in fact, if you take the next one, it will not be prime, and I think all subsequent numbers that have been computed have been verified to be non-prime. So you get these Fermat numbers, a whole sequence of them, an infinite sequence of them, and it turns out that a very simple argument shows you that any two different numbers in the sequence have no common factor at all. And so, for example, if you take 257 and, say, the 19th Fermat number, that pair of numbers will have no common factor. So since 257 happens to be prime, you could say 257 doesn’t divide the 19th Fermat number. But the 19th Fermat number is a big number. It’s divisible by some prime. And you can take the sequence of numbers and for each element of the sequence, take the smallest prime divisor, and then you get a sequence of primes, and that’s a infinite sequence of primes. The primes are all different because none of the numbers have a common factor. KK: That’s nice. I like that proof. EL: Nice! It’s kind of like killing a mosquito with a sledgehammer. It’s a big sequence of these somewhat complicated numbers, but there’s something very fun about that. Probably not fun to try to kill mosquitoes with a sledgehammer. Don’t try that at home. KK: You might need it in Florida. We have pretty big ones. KR: I can tell you yet a third proof of the theorem if you think we have time. KK: Sure! KR: This proof I learned about, and it’s an exercise in a textbook that’s one of my all-time favorite books to read. It’s called A Classical Introduction to [Modern] Number Theory by Kenneth Irleand and Michael Rosen. When I was an undergraduate at Brown, Ireland and Rosen were two of my professors, and Ken Ireland passed away, unfortunately, about 25 years ago, but Mike Rosen is still at Brown University and is still teaching. They have as an exercise in their book a proof due to a mathematician at Kansas State, I think it was, named Eckford Cohen, and he published a paper in the American Mathematical Monthly in 1969. And the proof is very simple. I’ll tell you the gist of it. It’s a proof by contradiction. What you do is you take for the different numbers n, you take the geometric mean of the first n numbers. What that means is you take the numbers 1, 2, 3, you multiply them together, and in the case of 3, you take the cube root of that number. We could even do that for 2, you take 1 and 2 and multiply them together and take the square root, 1.42. And these numbers that you get are smaller than the averages of the numbers. For example, the square root of 2 is less than 1.5, and the cube root of 6, of 1x2x3, is less than 2, which is the average of 1, 2, and 3. But nevertheless these numbers get pretty big, and you can show using high school mathematics that these numbers approach infinity, they get bigger and bigger. You can show, using an argument by contradiction, that if there were only finitely many primes, these numbers would not get…

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Episode 21 - Jana Rodriguez Hertz Jun 28, 2018

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Episode 21 - Jana Rodriguez Hertz Kevin Knudson & Evelyn Lamb Download

Evelyn Lamb: Hello and welcome to My Favorite Theorem. I’m one of your hosts, Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah. And this is your other host.
Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. How are you doing, Evelyn?
EL: I’m all right. I’m excited because we’re trying a different recording setup today, and a few of our recent episodes, I’ve had a few connection problems, so I’m hoping that everything goes well, and I’ve probably jinxed myself by saying that.
KK: No, no, it’s going to be fine. Positive thinking.
EL: Yeah, I’m hoping that the blips that our listeners may have heard in recent episodes won’t be happening. How about you? Are you doing well?
KK: I’m fine. Spring break is next week, and we’ve had the air conditioning on this week. This is the absurdity of my life. It’s February, and the air conditioning is on. But it’s okay. It’s nice. My son is coming home for spring break, so we’re excited.
EL: Great. We’re very happy today to have Jana Rodriguez-Hertz on the show. So, Jana, would you like to tell us a little bit about yourself?
Jana Rodriguez-Hertz: Hi, thank you so much. I’m originally from Argentina, I have lived in Uruguay for 20 years, and now I live in China, in Shenzhen.
EL: Yeah, that’s quite a big change. When we were first talking, first emailing, I mean, you were in Uruguay then, you’re back in China now. What took you out there?
JRH: Well, we got a nice job offer, and we thought we’d like to try. We said, why not, and we went here. It’s nice. It’s a totally different culture, but I’m liking it so far.
KK: What part of China are you in, which university?
JRH: In Southern University of Science and Technology in Shenzhen. It’s in Shenzhen. Shenzhen is in mainland China in front of Hong Kong, right in front of Hong Kong.
KK: Okay. That’s very far south.
EL: I guess February weather isn’t too bad over there.
JRH: It’s still winter, but it’s not too bad.
EL: Of course, that will be very relevant to our listeners when they hear this in a few months. We’re glad to have you here. Can you tell us about your favorite theorem?
JRH: Well, you know, I live in China now, and every noon I see a dynamical process that looks like the theorem I want to talk to you about, which is the dynamical properties of Smale’s horseshoe. Here it goes. You know, at the canteen of my university, there is a cook that makes noodles.
EL: Oh, nice.
JRH: He takes the dough and stretches it and folds it without mixing, and stretches it and folds it again, until the strips are so thin that they’re ready to be noodles, and then he cuts the dough. Well, this procedure can be described as a chaotic dynamical system, which is the Smale’s horseshoe.
KK: Okay.
JRH: So I want to talk to you about this. But we will do it in a mathematical model so it is more precise. So suppose that the cook has a piece of dough in a square mold, say of side 1. Then the cook stretches the dough so it becomes three times longer in the vertical sense but 1/3 of its original width in the horizontal sense. Then he folds it and puts the dough again in the square mold, making a horseshoe form. So the lower third of the square is converted into a rectangle of height 1 and width 1/3 and will be placed on the left side of the mold. The middle third of the square is going to be bent and will go outside the mold and will be cut. The upper third will be converted to another rectangle of height 1 and width 1/3 and will be put upside down in the right side of the mold. Do you get it?
KK: Yeah.
JRH: Now in the mold there will be two connected components of dough, one in the left third of the square and one in the right third of the square, and the middle third will be empty. In this way, we have obtained a map from a subset of the square into another subset of the square. And each time this map is applied, that is, each time we stretch and fold the dough, and cut the bent part, it’s called a forward iteration. So in the first forward iteration of the square, we obtain two rectangles of width 1/3 and height 1. Now in the second forward iteration of the square, we obtain four rectangles of width 1/9 and height 1. Two rectangles are contained in the left third, two rectangles in the right third. These are four noodles in total.
Counting from left to right, we will see one noodle of width 1/9, one gap of width 1/9, a second noodle of width 1/9, a gap of 1/3, and two more noodles of width 1/9 separated by a gap of width 1/9. Is that okay?
KK: Yes.
JRH: So if we iterate n times, we will obtain 2n noodles of width (1/3)n. And if we let the number of iterations go to infinity, that is, if we stretch and fold infinitely many times, cutting each time the bent part, we will obtain a Cantor set of vertical noodles.
KK: Yes.
EL: Right. So as you were saying the ninths with these gaps, and this 1/3, I was thinking, huh, this sounds awfully familiar.
KK: Yeah, yeah.
EL: We’ll include a picture of the Cantor set in the show notes for people to look at.
JRH: When we iterate forward, in the limit we will obtain a Cantor set of noodles. We can also iterate backwards. And what is that? We want to know for each point in the square, that is, for each flour particle of the dough in the mold, where it was before the cook stretched vertically and folded the dough the first time, where it came from. Now we recall that the forward iteration was to stretch in the vertical sense and fold it, so if we zoom back and put it backwards, we will obtain that the backward sense the cook has squeezed in the vertical sense and stretched in the horizontal sense and folded, okay?
EL: Yes.
JRH: Each time we iterate backwards, we stretch in the horizontal sense and fold it and put it in that sense. In this way, the left vertical rectangle is converted into the lower rectangle, the lower third rectangle. And the right side rectangle, the vertical rectangle, is converted into the upper third rectangle, and the bent part is cut. If we iterate backwards, now we will get in the first backward iteration four horizontal rectangles of width 1/9 and the gaps, and if we let the iterations go to infinity, we will obtain a Cantor set of horizontal noodles.
When we iterate forward and consider only what’s left in the mold, we start with two horizontal rectangles and finish with two vertical rectangles. When we iterate backwards we start with two vertical rectangles and finish with two horizontal rectangles. Now we want to consider the particles that stay forever in the mold, that is, the points so that all of the forward iterates and all the backwards iterates stay in the square. This will be the product of two middle-thirds Cantor set. It will look more like grated cheese than noodles.
KK: Right.
JRH: This set will be called the invariant set.
KK: Although they’re not pointwise fixed, they just stay inside the set.
JRH: That’s right. They stay inside the square. In fact, not only will they be not fixed, they will have a chaotic behavior. That is what I want to tell you about.
KK: Okay.
JRH: This is one of the simplest models of an invertible map that is chaotic. So what is chaotic dynamics anyways? There is no universally accepted definition about that. But one that is more or less accepted is one that has three properties. These properties are that periodic points are dense, it is topologically mixing, and it has sensitivity to initial conditions. And let me explain a little bit about this.
A periodic point is a particle of flour that has a trajectory that comes back exactly to the position where it started. This is a periodic point. What does it mean that they are dense? As close as you wish from any point you get one of these.
Topologically mixing you can imagine that means that the dough gets completely mixed, so if you take any two small squares and iterate one of them, it will get completely mixed with the other one forever. From one iteration on, you will get dough from the first rectangle in the second rectangle, always. That means topologically mixing.
I would like to focus on the sensitivity to initial conditions because this is the essence of chaos.
EL: Yeah, that’s kind of what you think of for the idea of chaos. So yeah, can you talk a little about that?
JRH: Yeah. This means that any two particles of flour, no matter how close they are, they will get uniformly separated by the dynamics. in fact, they will be 1/3 apart for some forward or backward iterate. Let me explain this because it is not difficult. Remember that we had the lower third rectangle? Call this lower third rectangle 0, and the upper third rectangle 1. Then we will see that for some forward or backward iterate, any two different particles will be in different horizontal rectangles. One will be in 1, and the other one will be in the 0 rectangle. How is that? If two particles are at different heights, than either they are already in different rectangles, so we are done, or else they are in the same rectangle. But if they are in the same rectangle, the cook stretches the vertical distance by 3. Every time they are in the same horizontal rectangle, their vertical distance is stretched by 3, so they cannot stay forever in the same rectangle unless they are at the same height.
KK: Sure.
JRH: If they are at different heights, they will get eventually separated. On the other hand, if they are in the same vertical rectangle but at different x-coordinates, if we iterate backwards, the cook will stretch the dough in the horizontal sense, so the horizontal distance will be tripled. Each time they are in the same vertical rectangle, they cannot be forever in the same vertical rectangle unless they are in the same, unless their horizontal distance is 0. But if they are in different positions, then either their horizontal distance is positive or the vertical distance is positive. So in some iterate, they will be 1/3 apart. Not only that, if they are in two different vertical rectangles, then in the next backwards iterate, they are in different horizontal rectangles. So we can state that any two different particles for some iterate will be in different horizontal rectangles, no matter how close they are. So that’s something I like very much because each particle is defined by its trajectory.
EL: Right, so you can tell exactly what you are by where you’ve been.
JRH: Yeah, two particles are defined by what they have done and what they will do. That allows something that is very interesting in this type of chaotic dynamics, which is symbolic dynamics. Now you know that any two points in some iterate will have distinct horizontal rectangles, so you can code any particle by its position in the horizontal rectangles. If one particle is in the beginning in the 0 rectangle, you will assign to them a sequence so that its zero position is 0, a double infinite sequence. If the first iterate is in the rectangle 1, then in the first position you will put a 1. In this way you can code any particle by a bi-infinite sequence of zeroes and ones. So in dynamics this is called conjugation. You can conjugate the horseshoe map with a sequence of bi-infinite sequences. This means that you can code the dynamics. Anything that happens in the set of bi-infinite sequences, happens in the horseshoe and vice versa. This is very interesting because you will find particles that describe any trajectory that you wish because you can write any sequence of zeroes and ones as you wish. You will have all Shakespeare coded in the horseshoe map, all of Donald Trump’s tweets will be there too.
KK: Let’s hope not. Sad!
JRH: Everything will be there.
EL: History of the world, for better and worse.
KK: What about Borges’s Library of Babel? It’s in there too, right?
JRH: If you can code it with zeroes and ones, it’s there.
EL: Yeah, that’s really cool. So where did you first run into this theorem?
JRH: When I was a graduate student, I ran into chaos, and I first ran into a baby model of this, which is the tent map. A tent map is in the interval, and that was very cool. Unlike this model, it’s coded by one-sided sequences. And later on, I went to IMPA [Instituto de Matemática Pura e Aplicada] in Rio de Janeiro, and I learned that Smale, the author of this example, had produced this example while being at IMPA in Rio.
KK: Right.
JRH: It was cool. I learned a little more about dynamics, about hyperbolic dynamics, and in fact, now I’m working in partially hyperbolic dynamics, which is very much related to this, so that is why I like it so much.
KK: Yeah, one of my colleagues spends a lot of time in Brazil, and he’s still studying the tent map. It’s remarkable, I mean, it’s such a simple model, and it’s remarkable what we still don’t know about it. And this is even more complicated, it’s a 2-d version.
EL: So part of this show is asking our guests to pair their theorem with something. I have an idea of what you might have chosen to pair with your theorem, but can you tell us what you’ve chosen?
JRH: Yeah, I like this sensitivity to initial conditions because you are defined by your trajectory. That’s pretty cool. For instance, if you consider humans as particles in a system, actually nowadays in Shenzhen, it is only me who was born in Argentina, lived in Uruguay, and lives in Shenzhen.
EL: Oh wow.
JRH: This is a city of 20 million people. But I am defined by my trajectory. And I’m sure any one of you are defined by your trajectory. If you look at a couple of things in your life, you will discover that you are the only person in the world who has done that. That is something I like. You’re defined, either by what you’ve done, or what you will do.
EL: Your path in life. It’s interesting that you go there because when I was talking to Ami Radunskaya, who also chose a theorem in dynamics, she also talked about how her theorem related to this idea of your path in life, so that’s a fun idea.
JRH: I like it.
KK: Of course, I was thinking about taffy-pulling the whole time you were describing the horseshoe map. You’ve seen these machines that pull taffy, I think they’re patented, and everything’s getting mixed up.
EL: Yeah.
JRH: All of this mixing is what makes us unique.
EL: So you can enjoy this theorem while pondering your life’s path and maybe over a bowl of noodles with some taffy for dessert.
KK: This has been fun. I’d never really thought too much about the horseshoe map. I knew it as this classical example, and I always heard it was so complicated that Smale decided to give up on dynamics, and I’m sure that’s false. I know that’s false. He’s a brilliant man.
JRH: Actually, he’s coming to a conference we’re organizing this year.
EL: Oh, neat.
KK: He’s still doing amazingly interesting stuff. I work in topological data analysis, and he’s been working in that area lately. He’s just a brilliant guy. The Fields Medal was not wasted on him, for sure.
EL: Well thanks a lot for taking the time to talk to us. I really enjoyed talking with you.
JRH: Thank you for inviting me.
[outro]

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Episode 20 - Francis Su Jun 14, 2018

Show notes

Episode 20 - Francis Su Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Hello and welcome to My Favorite Theorem. I’m your host Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah. And this is your other host. Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. How are you doing, Evelyn? EL: I’m all right. I am hanging in there in the winter as a displaced Texan. KK: It’s not even winter yet. EL: Yeah, well, somehow I manage to make it to the end of the season without dying every year outside of Texas, but yeah, the first few cold days really throw me for a loop. KK: Well my son’s in college now, and they had snow last week. EL: Well the south got a bunch of snow. Is he in South Carolina, is that right? KK: North Carolina, and he’s never driven in snow before, and we told him not to, but of course he did. No incidents, so it was okay. EL: So we’re very glad to have our guest today, who I believe is another displaced Texan, Francis Su. Francis, would you like to tell us a little bit about yourself? Francis Su: Hi, Evelyn and Kevin. Sure. I’m a professor of mathematics at Harvey Mudd College, and that’s a small science and engineering school in southern California, and Evelyn is right. I am a displaced Texan from a small town in south Texas called Kingsville. EL: Okay. I grew up in Dallas. Is Kingsville kind of between Houston and Beaumont? FS: It’s between Houston and the valley. Closer to Corpus Christi. EL: Ah, the other side. Many of us displaced Texans end up all over the country and elsewhere in the world. FS: That’s right. I’m in California now, which means I don’t have to deal with the winter weather that you guys are wrestling with. KK: I’m in Florida. I’m okay. EL: Yeah. And you’re currently in the Bay Area at MSRI, so you’re not on fire right now. FS: That’s right. I’m at the Math Sciences Research Institute. There’s a semester program going on in geometric and topological combinatorics. KK: Cool. EL: Yeah, that must be nice. Is this your, it’s not too long after your presidency of the Mathematical Association of America, so it must be nice to be able to not have those responsibilities and be able to just focus on research at MSRI this semester. FS: That’s right. It was a way of hopping back into doing research after a couple of years doing some fun work for the MAA. EL: So, what is your favorite theorem? We would love to hear it. FS: You know, I went around and around with this because as mathematicians we have lots of favorite theorems. The one I kept coming back to was the Brouwer fixed point theorem. KK: I love this theorem. FS: Yes, so the Brouwer fixed point theorem is an amazing theorem. It’s about a hundred years old. It shows up in all sorts of unexpected places. But what it loosely says is if you have a continuous function from a ball to itself—and I’ll say what a ball means in a minute—it must have a fixed point, a point that doesn’t move. And a ball can be anything that basically has no holes. EL: So anything you can make out of clay without punching a hole in it, or snaking it around and attaching two ends of it together. I’m gesturing with my hands. That’s very helpful for our podcast listeners. KK: Right. FS: Exactly. KK: We don’t even need convexity, right? You can have some kind of dimpled blob and it still works. FS: That’s right. It could be a blob with a funny shape. As long as it can be deformed to something that’s a ball, the ball has no holes, then the theorem applies. And a continuous function would be, one way of thinking about a continuous function from a ball to itself is let’s deform this blob, and as long as we deform the blob so that the blob stays within itself, then the blob doesn’t move. A very popular way of describing this theorem is if you take a cup of coffee, let’s say I have a cup of coffee and I take a picture of it. Then slosh the coffee around in a continuous fashion and then take another picture. There is going to be a point in the coffee that is in the same spot in both pictures. It might have moved around in between, but there’s going to be a point that’s in the same spot in both pictures. And then if I move that point out of its original position, I can’t help but move some other point into its original position. EL: Yeah, almost like a reverse diagonalization. In diagonalization you show that there’s a problem because anything you thought you could get on your list, you show that something else, even if you stick it on the list, something else is not on the list still. Here, you’re saying even if you think, if I just had one fixed point, I could move it and then I wouldn’t have any, you’re saying you can’t do that without adding some other fixed point. FS: That’s right. The coffee cup sloshing example is a nice one because you can see that if I take the cup of coffee and I just empty it and pour the liquid somewhere else, clearly there’s not going to be a fixed point. So you sort of see the necessity of having the ball, the coffee, mapped to itself. KK: And if you had a donut-shaped cup of coffee, this would not be true, right? You could swirl it around longitudinally and nothing would be fixed. FS: That’s right. If you had a donut-shaped coffee mug, we could. That’s right. The continuity is kind of interesting. Another way I like to think about this theorem is if you take a map of Texas and you crumple it up somewhere in Texas, there’s a point in the map that’s exactly above the point it represents in Texas. So that’s sort of a two-dimensional version of this theorem. And you see the necessity of continuity because if I tore the map in two pieces and threw east Texas into west Texas and west Texas into east Texas, it wouldn’t be true that there would be a point exactly above the point it represents. So continuity is really important in this theorem as well. KK: Right. You know, for fun, I put the one-dimensional version of this as a bonus question on a calculus test this semester. FS: I like that version. Are you referring to graphing this one-dimensional function? KK: Right, so if you have a map from a unit interval to itself, it has a fixed point. This case is nice because it’s just a consequence of the intermediate value theorem. FS: Yes, that’s a great one. I love that. KK: But in higher dimensions you need a little more fire power. FS: Right. So yeah, this is a fun theorem because it has all sorts of maybe surprising versions. I told you one of the popular versions with coffee. It can be used, for instance, to prove the fundamental theorem of algebra, that every polynomial has a root in the complex numbers. EL: Oh, interesting! I don’t think I knew that. KK: I’m trying to think of that proof. FS: Yeah, so the idea here is that if you think about a polynomial as a function and you’re thinking of this as a function on the complex plane, basically it takes a two-dimensional region like Texas and maps it in some fashion back onto the plane. And you can show that there’s a region in this map that gets sent to itself, roughly speaking. That’s one way to think about what’s going on. And then the existence of a zero corresponds to a fixed point of a continuous function, which I haven’t named but that’s sort of the idea. EL: Interesting. That’s nice. It’s so cool how, at least if I’m remembering correctly, all the proofs I know of the fundamental theorem of algebra are topological. It’s nice, I think, for topology to get to throw an assist to algebra. Algebra has helped topology so much. FS: I love that too. I guess I’m attracted to topology because it says a lot of things that are interesting about the existence of certain things that have to happen. One of the things that’s going on at this program at MSRI, as the name implies, geometric and topological combinatorics, people are trying to think about how to use topology to solve problems in combinatorics, which seems strange because combinatorics feels like it just has to do with counting discrete objects. EL: Right. Combinatorics feels very discrete, and topology feels very continuous, and how do you get that to translate across that boundary? That’s really interesting. FS: I’ll give you another example of a surprising application. In the 1970s, actually people studied this game called Hex for a while. I guess Hex was developed in the ‘40s or ‘50s. Hex is a game that’s played on a board with hexagonal tiles, a diamond-shaped board. Two players take turns, X and O, and they’re trying to construct a chain from one side of the board to the other, to the opposite side. It turns out that the Brouwer fixed-point theorem, well you can ask the question: can that game ever end in a draw configuration where nobody wins? For large boards, it’s not so obvious that the game can’t end in a draw. But in a spectacular application of the Brouwer fixed-point theorem it can’t end in a draw using the Brouwer fixed-point theorem. EL: Oh, that’s so cool. KK: That is cool. And allegedly this game was invented by John Nash in the men’s room at Princeton, right? FS: Yes, there’s some story like that, though I think it actually dates back to somebody before. KK: Probably. But it’s a good story, right, because Nash is so famous. EL: So was it love at first sight with the Brouwer fixed-point theorem for you, or how did you come across it and grow to love it? FS: I guess I encountered it first as an undergraduate in college when a professor of mine, a topology professor of mine, showed me this theorem, and he showed me a combinatorial way to prove this theorem, using something known as Sperner’s lemma. There’s another connection between topology and combinatorics, and I really appreciated the way you could use combinatorics to prove something in topology. EL: Cool. KK: Very cool. KK: You know, part of our show is we ask our guest to pair their theorem with something. So what have you chosen to pair the Brouwer fixed-point theorem with? FS: I’d like to pair it with parlor games. Think of a game like chess, or think of a game like rock-paper-scissors. It turns out that the Brouwer fixed-point theorem is also related to how you play a game optimally, a game like chess or rock-paper-scissors optimally. KK: So how do you get the optimal strategy for chess from the Brouwer fixed-point theorem? FS: Very good question. So the Brouwer fixed-point theorem can’t tell you what the optimal strategy is. KK: Just that it exists, right, yeah. FS: It tells you that there is a pair of optimal strategies that players can play to play the game optimally. What I’m referring to is something known as the Nash equilibrium theorem. Nash makes another appearance in this segment. What Nash showed is that if you have a game, well there’s this concept called the Nash equilibrium. The question Nash asked is if you’re looking at some game, can you predict how players are going to play this game? That’s one question. Can you prescribe how players should play this game? That’s another question. And a third question is can you describe why players play a game a certain way? So there’s the prediction, descriptions, and prescription about games that mathematicians and economists have gotten interested in. And what Nash proposed is that in fact something called a Nash equilibrium is the best way to describe, prescribe, and predict how people are going to play a game. And the idea of a Nash equilibrium is very simple, it’s just players playing strategies that are mutually best responses to each other. And it turns out that if you allow what are called mixed strategies, every finite game has an equilibrium, which is kind of surprising. It suggests that you could maybe suggest to people what the best course of action is to play. There is some pair of strategies by both players, or by all players if it’s a multiplayer game, that actually are mutual best replies. People are not going to have an incentive to change their strategies by looking at the other strategies. KK: The Brouwer fixed point theorem is so strange because it’s one of those existence things. It just says yeah, there is a fixed point. We tend to prove it by contradiction usually, or something. There’s not really any good constructive proofs. I guess you could just pick a point and start iterating. Then by compactness what it converges to is a fixed point. FS: There is actually, maybe this is a little surprising as well, this theorem I mention learning as an undergrad, it’s called Sperner’s lemma, it actually has a constructive proof, in the sense that there’s an efficient way of finding the combinatorial object that corresponds to a fixed point. What’s surprising is that you can actually in many places use this constructive combinatorial proof to find, or get close to, a proposed fixed point. KK: Very cool. FS: That’s kind of led to a whole bunch of research in the last 40 years or so in various areas, to try to come up with constructive versions of things that prior to that people had thought of as non-constructive. EL: Oh, that’s so cool. I must admit I did not have proper appreciation for the Brouwer fixed-point theorem before, so I’m very glad we had you on. I guess I kind of saw it as this novelty theorem. You see it often as you crumple up the map, or do these little tricks. But why did I really care that I could crumple up the map? I didn’t see all of these connections to these other points. I am sorry to the Brouwer fixed-point theorem for not properly appreciating it before now. FS: Yes. I think it definitely belongs on a top ten list of top theorems in mathematics. I wonder how many mathematicians would agree. KK: I read this book once, and the author is escaping me and I’m kind of embarrassed because it’s on the shelf in my other office, called Five Golden Rules. Have you ever seen this book? It was maybe 10 or 15 years ago. EL: No. KK: One of the theorems, there are like five big theorems in mathematics, it was the Brouwer fixed-point theorem. And yeah, it’s actually of fundamental importance to know that you have fixed points for maps. They are really important things. But the application he pointed to was to football ranking schemes, right? Because that’s clearly important. College football ranking schemes in which in essence you’re looking for an eigenvector of something, and an eigenvector is a fixed point with eigenvalue 1, and of course the details are escaping me now. This book is really well-done. Five Golden Rules. EL: We’ll find that and put it in the show notes for sure. FS: I haven’t heard of that. I should look that one up. KK: It’s good stuff. FS: I’ll just mention with this Nash theorem, the basic idea of using the Brouwer fixed-point theorem to prove it is pretty simple to describe. It’s that if you look at the set of all collections of strategies, if they’re mixed strategies allowing randomization, then in fact that space is a ball. KK: That makes sense. FS: And then the cool thing is if players have an incentive to deviate, to change their strategies, that suggests a direction in which each point could move. If they want to deviate, it suggests a motion of the ball to itself. And the fact that the ball has a fixed point means there’s a place where nobody is incentivized to change their strategy. EL: Yeah. KK: Well I’ve learned a lot. And I even knew about the Brouwer fixed-point theorem, but it’s nice to learn about all these extra applications. I should go learn more combinatorics, that’s my takeaway. EL: Yeah, thanks so much for being on the show, Francis. If people want to find you, there are a few places online that they can find you, right? You’re on Twitter, and we’ll put a link to your Twitter in the show notes. You also have a blog, and I’m sorry I just forgot what it’s called. FS: The Mathematical Yawp. EL: That’s right. We’ll put that in the show notes. I know there are a lot of posts of yours that I’ve really appreciated, especially the ones about helping students thrive, doing m…

Full show notes at the publisher

Episode 19 - Emily Riehl May 24, 2018

Show notes

Episode 19 - Emily Riehl Kevin Knudson & Evelyn Lamb Download Kevin Knudson: Welcome to My Favorite Theorem, a podcast about mathematics and everyone’s favorite theorem. I’m your host Kevin Knudson, professor of mathematics at the University of Florida. This is your other host. Evelyn Lamb: Hi, I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City. So how are things going, Kevin? KK: Okay. We’re hiring a lot, and so I haven’t eaten a meal at home this week, and maybe not last week either. You think that might be fun until you’re in the middle of it. It’s been great meeting all these new people, and I’m really excited about getting some new colleagues in the department. It’s a fun time to be at the University of Florida. We’re hiring something like 500 new faculty in the next two years. EL: Wow! KK: It’s pretty ambitious. Not in the math department. EL: Right. KK: I wish. We could solve the mathematician glut just like that. EL: Yeah, that would be great. KK: How are things in Salt Lake? EL: Pretty good. It’s a warm winter here, which will be very relevant to our listeners when they listen in the summer. But it’s hiring season at the University of Utah, where my spouse works. He’s been doing all of that handshaking. KK: The handshaking, the taking to the dean and showing around, it’s fun. It’s good stuff. Anyway, enough about that. I’m excited about today’s guest. Today we are pleased to welcome Emily Riehl from Johns Hopkins. Hi, Emily. Emily Riehl: Hi. KK: Tell everyone about yourself. ER: Let’s see. I’ve known I wanted to be a mathematician since I knew that that was a thing that somebody could be, so that’s what I’m up to. I’m at Johns Hopkins now. Before that I was a postdoc at Harvard, where I was also an undergraduate. My Ph.D. is from Chicago. I was a student of Peter May, an algebraic topologist, but I work mostly in category theory, and particularly in category theory as it relates to homotopy theory. KK: So how many students does Peter have? Like 5000 or something? ER: I was his 50th, and that was seven years ago. EL: Emily and I have kind of a weird connection. We’ve never actually met, but we both lived in Chicago and I kind of replaced Emily in a chamber music group. I played with Walter and the gang I guess shortly after you graduated. I moved there in 2011. They’re like, oh, you must know Emily Riehl because you’re both mathematicians who play viola. I was like, no, that sounds like a person, though, because violists are all the best people. KK: So, Emily, you’ve told us, and I’ve had time to think about it but still haven’t thought of my favorite application of this theorem. But what’s your favorite theorem? ER: I should confess: my favorite theorem is not the theorem I want to talk about today. Maybe I’ll talk about what I don’t want to talk about briefly if you’ll indulge me. KK: Sure. ER: So I’m a category theorist, and every category theorist’s favorite theorem is the Yoneda lemma. It says that a mathematical object of some kind is uniquely determined by the relationships that it has to all other objects of the same type. In fact, it’s uniquely characterized in two different ways. You can either look at maps from the object you’re trying to understand or maps to the object you’re trying to understand, and either way suffices to determine in. This is an amazing theorem. There’s a joke in category that all proofs are the Yoneda lemma. I mean, all proofs [reduce] to the Yoneda lemma. The reason I don’t want to talk about it today is two-fold. Number one, the discussion might sound a little more philosophical than mathematical because one thing that the Yoneda lemma does is it orients the philosophy of category theory. Secondly, there’s this wonderful experience you have as a student when you see the Yoneda lemma for the first time because the statement you’ll probably see is not the one I just described but sort of a weirder one involving natural transformations from representable functors, and you see them, and you’re like, okay, I guess that’s plausible, but why on earth would anyone care about that? And then it sort of dawns on you over however many years, in my case, why it’s such a profound and useful observation. And I don’t want to ruin that experience for anybody. KK: You’re not worried about getting excommunicated, right? ER: That’s why I had to confess. I was joking with some category theorists, I was just in Sydney visiting the Center of Australian Category Theory, which is the name of the group, and it’s also the center of Australian category theory. And I want to be invited back, so yes, of course, my favorite theorem is the Yoneda lemma. But what I want to talk about today instead is a theorem I really like because it’s a relatively simple idea, and it comes up all over mathematics. Once it’s a pattern you know to look for, it’s quite likely that you’ll stumble upon it fairly frequently. The proof, it’s a general proof in category theory, specializes in each context to a really nice argument in that particular context. Anyway, the theorem is called right adjoints preserve limits. EL: All right. KK: So I’m a topologist, so to me, we put a modifier in front of our limit, so there’s direct and inverse. And limit in this context means inverse limit, right? ER: Right. That’s the historical terminology for what category theorists call limits. KK: So I always think of inverse limits as essentially products, more or less, and direct limits are unions, or direct sum kinds of things. Is that right? ER: Right. KK: I hope that’s right. I’m embarrassed if I’m wrong. ER: You’re alluding to something great in category theory, which is that when you prove a theorem, you get another theorem for free, the dual theorem. A category is a collection of objects and a collection of transformations between them that you depict graphically as arrows. Kind of like in projective geometry, you can dualize the axioms, you can turn around the direction of the arrows, and you still have a category. What that means is that if you have a theorem in category theory that says for all categories blah blah blah, then you can apply that in particular to the opposite category where things are turned around. In this case, there are secretly two categories involved, so we have three dual versions of the original theorem, the most useful being that left adjoints preserve colimits, which are the direct limits that you’re talking about. So whether they’re inverse limits or direct limits, there’s a version of this theorem that’s relevant to that. KK: Do we want to unpack what adjoint functors are? ER: Yes. EL: Yeah, let’s do that. For those of us who don’t really know category theory. ER: Like anything, it’s a language that some people have learned to speak and some people are not acquainted with yet, and that’s totally fine. Firstly, a category is a type of mathematical object, basically it’s a theory of mathematical objects. We have a category of groups, and then the transformations between groups are the group homomorphisms. We have a category of sets and the functions between them. We have a category of spaces and the continuous functions. These are the categories. A morphism between categories is something called a functor. It’s a way of converting objects of one type to objects of another type, so a group has an underlying set, for instance. A set can be regarded as a discrete space, and these are the translations. So sometimes if you have a functor from one category to another and another functor going back in the reverse direction, those functors can satisfy a special dual relationship, and this is a pair of adjoint functors. One of them gets called a left adjoint, and one of them the right adjoint. What the duality says is that if you look at maps out of the image of the left adjoint, then those correspond bijectively and naturally (which is a technical term I’m not going to get into) to maps in the other category into the image of the right adjoint. So maps in one category out of the image of the left adjoint correspond naturally to maps in the other category into the image of the right adjoint. So let me just mention one prototypical example. KK: Yeah. ER: So there’s a free and forgetful construction. So I mentioned that a group has an underlying set. The reverse process takes a set and freely makes a group out of that set, so the elements of that group will be words in the letters and formal inverses modulo some relation, blah blah blah, but the special property of these free groups is if I look at the group homomorphism that’s defined on a free group, so this is a map in the category of groups out of an object in the image of the left adjoint, to define that I just have to tell you where the generators go, and I’m allowed to make those choices freely, and I just need to find a function of sets from the generating set into the underlying set of the group I’m mapping into. KK: Right. ER: That’s this adjoint relationship. Group homomorphisms from a free group to whatever group correspond to functions from the generators of that free group to that underlying set of the group. EL: I always feel like I’m about to drown when I try to think about category theory. It’s hard for me to read category theory, but when people talk to me about it, I always think, oh, okay, I see why people like this so much. KK: Reading category theory is sort of like the whole picture being worth a thousand words thing. The diagrams are so lovely, and there’s so much information embedded in a diagram. Category theory used to get a bad rap, abstract nonsense or whatever, but it’s shown to be incredibly powerful, certainly as an organizing principle but also just in being able to help us push boundaries in various fields. Really if you think about it just right, if you think about things as functors, lots of things come out, almost for free. It feels like for free, but the category theorist would say, no, there’s a ton of work there. So what’s a good example of this particular theorem? ER: Before I go there, exactly to this point, there’s a great quote by Eilenberg and Steenrod. So Eilenberg was one of the founders of category theory. Saunders MacLand wrote a paper, the “General Theory of Natural Equivalences,” in the ‘40s that defined these categories and functors and also the notion of naturality that I was alluding to. They thought that was going to be both the first and last. Anyway, ten years later, Eilenberg and Steenrod wrote this book, Foundations of Algebraic Topology, that incorporated these diagrammatic techniques into a pre-existing mathematical area, algebraic topology. It had been around since at least the beginning of the twentieth century, I’d say. So they write, “the diagrams incorporate a large amount of information. Their use provides extensive savings in space and in mental effort. In the case of many theorems, the setting up of the correct diagram is a major part of the proof. We therefore urge that the reader stop at the end of each theorem and attempt to construct for himself (it’s a quote here) the relevant diagram before examining the one which is given in the text. Once this is done, the subsequent demonstration can be followed more readily. In fact, the reader can usually supply it himself.” KK: Right. Like proving Meier-Vietoris, for example. You just set up the right diagram, and in principle it drops out, right? ER: Right, and in general in category theory, the definitions, the concepts are the hard thing. The proofs of the theorems are generally easier. And in fact, I’d like to prove my favorite theorem for you. I’m going to do it in a particular example, and actually I’m going to do it in the dual. So I’m going to prove that left adjoints preserve colimits. EL: Okay. ER: The statement I’m going to prove, the specific statement I’m going to prove by using the proof that left adjoints preserve colimits, is that for natural numbers a, b, and c, I’m going to prove that a(b+c)=ab+ac. KK: Distributive law, yes! ER: Distributive property of multiplication over addition. So how are we going to prove this? The first thing I’m going to do is categorify my natural numbers. And what is a natural number? It’s a cardinality of a finite set. In place of the natural numbers a, b, and c, I’m going to think about sets, which I’ll also call A, B, and C. The natural numbers stand for the cardinality of these sets. EL: Cardinality being the size, basically. ER: Absolutely. A, B, and C are now sets. If we’re trying to prove this statement about natural numbers, they’re finite sets. The theorem is actually true for arbitrary sets, so it doesn’t matter. And I’ve replaced a, b, and c by sets. Now I have this operation “times” and this operation “plus,” so I need to categorify those as well. I’m going to replace them by operations on sets. So what’s something you can do to two sets so that the cardinalities add, so that the sizes add? KK: Disjoint union. EL: Yeah, you could union them. ER: So disjoint union is going to be my interpretation of the symbol plus. And we also need an interpretation of times, so what can I do for sets to multiply the cardinalities? EL: Take the product, or pairs of elements in each set. ER: That’s right. Absolutely. So we have the cartesian product of sets and the disjoint union of sets. The statement is now for any sets a, b, and c, I’m going to prove that if I take the disjoint union B+C, and then form the cartesian product with A, then that set is isomorphic to, has in particular the same number of elements as, the set that you’d get by first forming the products A times B and A times C and then taking the disjoint union. KK: Okay. ER: The disjoint union here is one of these colimits, one of these direct limits. When you stick two things next to each other — coproduct would be the categorical term — this is one of these colimits. The act of multiplying a set by a fixed set A is in fact a left adjoint, and I’ll make that a little clear as I make the argument. ER: The disjoint union here is one of these colimits, one of these direct limits. When you stick two things next to each other — coproduct would be the categorical term — this is one of these colimits. The act of multiplying a set by a fixed set A is in fact a left adjoint, and I’ll make that a little clear as I make the argument. EL: Okay. ER: Okay. So let’s just try and begin. So the way I’m going to prove that A times (B+C) is (AxB) +(AxC) is actually using a Yoneda lemma-style proof because the Yoneda lemma comes up everywhere. We know that these sets are isomorphic by arguing that functions from them to another set X correspond. So if the sets have exactly the same functions to every other set, then they must be isomorphic. That’s the Yoneda lemma. Let’s now consider a function from the set A times the disjoint union of B+C to another set X. The first thing I can do with such a function is something called currying, or maybe uncurrying. (I never remember which way these go.) I have a function here of two variables. The domain is the set A times the disjoint union (B+C). So I can instead regard this as a function from the set (B+C), the disjoint union, into the set of functions from A to X. KK: Yes. ER: Rather than have A times (B+C) to X, I have from (B+C) to functions from A to X. There I’ve just transposed the cross and adjunction. That was the adjunction bit. So now I have a function from the disjoint union B+C to the set of functions from A to X. Now when I’m mapping out of a disjoint union, that just means a case analysis. Either I need to define a function like this, I have to define firstly a function from B to functions from A to the X, and also from C to functions from A to the X. So now a single function is given by these two functions. And if I look at the piece, now, which is a function from B to functions from A to the X, by this uncurrying thing, that’s equally just a function from A times B to X…

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Episode 18 - John Urschel May 10, 2018

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Episode 18 - John Urschel Kevin Knudson & Evelyn Lamb Download Kevin Knudson: Welcome to My Favorite Theorem. I’m your host Kevin Knudson, professor of mathematics at the University of Florida. I’m joined by your cohost. Evelyn Lamb: Hi, I’m Evelyn Lamb. I’m a math and science writer in Salt Lake City, Utah, where it is very cold now, and so I’m very jealous of Kevin living in Florida. KK: It’s a dreary day here today. It’s raining and it’s “cold.” Our listeners can’t see me doing the air quotes. It’s only about 60 degrees and rainy. It’s actually kind of lousy. but it’s our department holiday party today, and I have my festive candy cane tie on, and I’m good to go. And I’m super excited. John Urschel: So I haven’t been introduced yet, but can I jump in on this weather conversation? I’m in Cambridge right now, and I must say, I think it’s probably nicer in Cambridge, Massachusetts than it is in Utah right now. It’s a nice breezy day, high 40s, low 50s, put on a little sweater and you’re good to go. EL: Yeah, I’m jealous of both of you. KK: Evelyn, I don’t know about you, but I’m super excited about this one. I mean, I’m always excited to do these, but it’s the rare day you get to talk to a professional athlete about math. This is really very cool. So our guest on this episode is John Urschel. John, do you want to tell everyone about yourself? JU: Yes, I’d be happy to. I think I might actually be the only person, the only professional athlete you can ask high-level math about. KK: That might be true. Emily Riehl, Emily Riehl counts, right? EL: Yeah. KK: She’s a category theorist at Johns Hopkins. She’s on the US women’s Australian rules football team. EL: Yeah, JU: Australian rules football? You mean rugby? KK: Australian rules football is like rugby, but it’s a little different. See, you guys aren’t old enough. I’m old enough to remember ESPN in the early days when they didn’t have the high-end contracts, they’d show things like Australian rules football. It’s fascinating. It’s kind of like rugby, but not really at the same time. It’s very weird. JU: What are the main differences? EL: You punch the ball sometimes. KK: They don’t have a scrum, but they have this thing where they bounce the ball really hard. (We should get Emily on here.) They bounce the ball up in the air, and they jump up to get it. You can run with it, and you can sort of punch the ball underhanded, and you can kick it through these three posts on either end [Editor's note: there are 4 poles on either end.]. It’s sort of this big oval-shaped field, and there are three poles at either end, and you try to kick it. If you get it through the middle pair, that’s a goal. If you get it on either of the sides, that’s called a “behind.” The referees wear a coat and tie and a little hat. I used to love watching it. JU: Wait, you say the field is an oval shape? KK: It’s like an oval pitch, yeah. JU: Interesting. KK: Yeah. You should look this up. It’s very cool. It is a bit like rugby in that there are no pads, and they’re wearing shorts and all of that. JU: And it’s a very continuous game like rugby? KK: Yes, very fast. It’s great. JU: Gotcha. KK: Anyway, that’s enough of us. You didn’t tell us about yourself. JU: Oh yeah. My name is John Urschel. I’m a retired NFL offensive lineman. I played for the Baltimore Ravens. I’m also a mathematician. I am getting my Ph.D. in applied math at MIT. KK: Good for you. EL: Yeah. KK: Do you miss the NFL? I don’t want to belabor the football thing, but do you miss playing in the NFL? JU: No, not really. I really loved playing in the NFL, and it was a really amazing experience to be an elite, elite at whatever sport you love, but at the same time I’m very happy to be focusing on math full-time, focusing on my Ph.D. I’m in my third year right now, and being able to sort of devote more time to this passion of mine, which is ideally going to be my lifelong career. EL: Right. Yeah, so not to be creepy, but I have followed your career and the writing you’ve done and stuff like that, and it’s been really cool to see what you’ve written about combining being an athlete with being a mathematician and how you’ve changed your focus as you’ve left playing in the NFL and moved to doing this full-time. It’s very neat. KK: So, John, what’s your favorite theorem? JU: Yes, so I guess this is the name of the podcast? KK: Yeah. JU: So I should probably give you a theorem. So my favorite theorem is a theorem by Batson, Spielman, and Srivastava. EL: No, I don’t. Please educate us. JU: Good! So this is perfect because I’m about to introduce you to my mathematical idol. KK: Okay, great. JU: Pretty much who I think is the most amazing applied mathematician of this generation, Dan Spielman at Yale. Dan Spielman got his Ph.D. at MIT. He was advised by Mike Sipser, and he was a professor at MIT and eventually moved to Yale. He’s done amazing work in a number of fields, but this paper, it’s a very elegant paper in applied math that doesn’t really have direct algorithmic applications but has some elegance. The formulation is as follows. So suppose you have some graph, vertices and edges. What I want to tell you is that there exists some other weighted graph with at most a constant times the order of the graph number of edges, so linear in number of edges with respect to vertices, that approximates the Laplacian of this original very dense graph, no matter how dense it is. So I’m doing not the very best job of explaining this, but let me put it like this. You have a graph. It’s very dense. You have this elliptic operator on this graph, and there’s somehow some way to find a graph that’s not dense at all, but extremely, extremely sparse, but somehow with the exact, well not exact, but nearly the exact same properties. These operators are very, very close. KK: Can you remind our reader—readers, our listeners—what the Laplacian is? JU: Yeah, so the graph Laplacian, what you can do, the way I like to introduce it, especially for people not in graph theory type things, is you can define a gradient on a graph. You take every edge, directed in some way, and you can think of the gradient as being a discrete derivative along the edge. And now, as in the continuous case, you take this gradient, you get your Laplacian, and the same way you get a Laplacian in the continuous case, this is how you get your graph Laplacian. KK: This theorem, so the problem is that dense graphs are kind of hard to work with because, well, they’re dense? EL: So can I jump in? Dense meaning a lot of edges, I assume? JU: Lots of edges, as many edges as you want. KK: So a high degree on every vertex. JU: Lots of edges, edges going everywhere. EL: And then with the weighting, that might also mean something like, not that many total edges, but they have a high weight? Does that also make it dense, or is that a different property? JU: No, in that case, we wouldn’t really consider it very dense. KK: But the new graph you construct is weighted? JU: And the old graph can be weighted as well. KK: All right. What do the weights tell you? JU: What do you mean? KK: On the new graph. You generate this new graph that’s more sparse, but it’s weighted. Why do you want the weights? What do the weights get you? JU: The benefit of the weights is it gives you additional leeway about how you’re scaling things because the weights actually come into the Laplacian because for weighted graphs, when you take this Laplacian, it’s the difference between the average of each node, of all its neighbors, and the node, in a way, and the weights tell you how much each edge counts for. In that way, it allows you greater leeway. If you weren’t able to weight this very sparse graph, this wouldn’t work very well at all. KK: Right, because like you said, you think of sort of having a gradient on your graph, so this new graph should somehow have the same kind of dynamics as your original. JU: Exactly. And the really interesting thing is that you can capture these dynamics. Not only can you capture them, but you can capture them with a linear number of edges, linear in the order of the graph. KK: Right. JU: So Dan Spielman is famous for many things. One of the things he’s famous for is he was one of the first people to give provable guarantees for algorithms that can solve, like, a Laplacian system of equations in near-linear time, so O(n) plus some logs. From his work there have been many, many different sorts of improvements, and this one is extremely interesting to me because you only use a linear number of edges, which implies that this technique, given this graph you have should be extremely efficient. And that’s exactly what you want because it’s a linear number of edges, you apply this via some iterative algorithm, and you can use this guy as a sort of preconditioner, and things get very nice. The issue is, I believe—and it has been a little bit since I’ve read the paper—I believe the amount of time it takes to find this graph, I think is cubic. EL: Okay. JU: So it’s not a sort of paper where it’s extremely useful algorithmically, I would say, but it is a paper that is very beautiful from a mathematical perspective. KK: Has the algorithm been improved? Has somebody found a better than cubic way to generate this thing? JU: Don’t quote me on that, I do not know, but I think that no one has found a good way yet. And by good I mean good enough to make it algorithmically useful. For instance, if the amount of time it takes to find this thing is quadratic, or even maybe n to the 1.5 or something like that, this is already not useful for anything greater than near-linear. It’s a very interesting thing, and it’s something that really spoke to me, and I really just fell in love with it, and I think what I like about it most is that it is a very sort of applied area, and it is applied mathematics, theoretical computer science type things, but it is very theoretical and very elegant. Though I am an applied mathematician, I do like very clean things. I do like very nice looking things. And perhaps I can be a bad applied mathematician because I don’t always care about applications. Which kind of makes you a bad applied mathematicians, but in all my papers I’m not sure I’ve ever really, really cared about the applications, in the sense that if I see a very interesting problem that someone brings to me, and it happens to have, like some of the things I’ve gotten to do in machine learning, great, this is like the cherry on top, but that isn’t the motivating thing. If it’s an amazing application but some ugly, ugly thing, I’m not touching it. EL: Well, before we actually started recording, we talked a little bit about how there are different flavors of applied math. There are ones that are more on the theoretical side, and probably people who do a lot of things with theoretical computer science would tend towards that more, and then there are the people who are actually looking at a biological system and solving differential equations or something like this, where they’re really getting their hands dirty. It sounds like you’re more interested in the theoretical side of applied math. JU: Yeah. KK: Applied math needs good theory, though. JU: That’s just true. KK: You’ve got to develop good theory so that you know your algorithms work, and you want them to be efficient and all that, but if you can’t prove that they actually work, then you’re a physicist. JU: There’s nothing I hate more than heuristics. But heuristics do have a place in this world. They’re an important thing, but there’s nothing I dislike more in this world than doing things with heuristics without being able to give any guarantees. EL: So where did you first encounter this theorem? Was it in the research you’ve been doing, the study you’ve been doing for your Ph.D.? JU: Yes, I did encounter this, I think it was when I was preparing for my qualifying exams. I was reading a number of different things on so-called spectral graph theory, which is this whole field of, you have a graph and some sort of elliptic operator on it, and this paper obviously falls under this category. I saw a lecture on it, and I was just fascinated. You know it’s a very nice result when you hear about it and you’re almost in disbelief. KK: Right. JU: I heard about it and I thought I didn’t quite hear the formulation correctly, but in fact I did. KK: And I seem to remember reading in Sports Illustrated — that’s an odd sentence to say — that you were working on some version of the traveling salesman problem. JU: That is true. But I would say, KK: That’s hard. JU: Just because I’m working on the asymmetric traveling salesman problem does not mean you should be holding your breath for me to produce something on the traveling salesman problem. This is an interesting thing because I am getting my Ph.D., and you do want, you want to try to find a research project where yes, it’s tough and it’s challenging you, but at the end of your four or five years you have something to show for it. KK: Right. Is this version of the problem NP-hard? JU: Yes, it is. But this version, there isn’t any sort of inapproximability result as in some of the other versions of TSP. But my advisor Michele Gomez [spelling], who—for the record, I’m convinced I have the single best advisor in the world, like he is amazing, amazing. He has a strong background in combinatorial optimization, which is the idea that you have some set of discrete objects. You need to pick your best option when the number of choices you have is often not polynomial in the size of your input. But you need to pick the best option in some reasonable amount of time that perhaps is polynomial. EL: Yeah, so are these results that will say something like, we know we can get within 3 percent of the optimal… JU: Exactly. These sorts of things are called approximation algorithms. If it runs in polynomial time and you can guarantee it’s within, say, a constant factor of the optimal solution, then you have a constant approximation algorithm. We’ve been reading up on some of the more recent breakthroughs on ATSP. There was a breakthrough this August someone proved the first constant approximation algorithm for the asymmetric traveling salesman problem, and Michele Gomez, who also is the department head at MIT of math, he had the previous best paper on this. He had a log log approximation algorithm from maybe 2008 or 2009, but don’t quote me on this. Late 2000s, so this is something we’ve been reading about and thinking about. EL: Trying to chip away a little bit at that. JU: Exactly. It’s interesting because this constant approximation algorithm that came out, it used this approach that, I think Michele won’t mind me saying this, it used an approach that Michele didn’t think was the right way to go about it, and so it’s very interesting. There are different ways to construct an approximation algorithm. At its core, you have something you’re trying to solve, and this thing is hard, but now you have to ask yourself, what makes it hard? Then you need to sort of take one of the things that makes it hard and you need to loosen that. And his approach in his previous paper was quite different than their approach, so it’s interesting. KK: So the other thing we like to do on this show is to ask our guest to pair their theorem with something. So what have you chosen to pair your theorem with? JU: I still haven’t fully thought about this, but you’ve put me on the spot, and so I’m going to say this: I would pair this with, I think this is a thing, Miller 64. That’s a thing, right? KK: This is a beer? JU: Yeah, the beer. KK: It’s a super low-calorie beer? JU: It’s a beer, and they advertise it on TV. KK: I see, it’s very sparse. JU: People weightlifting, people running, and then drinking a 64-calorie beer. It’s the beer for athletes. EL: Okay. JU: I think it’s a very, very good beer because it at least claims to taste like a beer, be very much like a b…

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Episode 17 - Nalini Joshi Apr 26, 2018

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Episode 17 - Nalini Joshi Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Hello and welcome to My Favorite Theorem. I’m your cohost Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah. And this is your other cohost. Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. I’m looking forward to this because of the time zone issue here. This is taking place on two different days. EL: Yes, yes, we are delighted to be joined by Nalini Joshi, who is joining us from tomorrow in Australia, which we’re getting a kick out of because we’re easily amused. KK: That’s right. EL: Hi, Nalini. Would you like to tell us a little bit about yourself? Nalini Joshi: Sure. My name is Nalini Joshi. I’m a professor of applied mathematics at the University of Sydney. What else can I say except I’m broadcasting from the future? I was born in Burma and I moved to Australia as a child with my parents when they emigrated to Australia, and most of my education has been in Australia except for going to the U.S. to do a Ph.D., which I did at Princeton. EL: Okay, so you’ve spent some time in both hemispheres. I guess in multiple times in your life. NJ: Yeah. EL: So when I was a little kid I had this idea that the world could never end because, you know, in the U.S., there’s always someone who’s a full day ahead, so I know that Thursday would have to happen because if it was Wednesday where I was, someone was already living in Thursday, so the world could never end. NJ: That’s such a deep insight. That’s wonderful. KK: That’s pretty good. EL: Well… KK: I was watching football when I was a kid. NJ: I used to hang out at the back of the school library reading through all the old Scientific American magazines. If only they had columns like yours, Evelyn. Fantastic. I really, really wanted to work out what was happening in the universe, and so I thought about time travel and space travel a lot as a teenager. EL: Oh. So did you start your career wanting to maybe go more into physics, or did you always know you wanted to be a mathematician? NJ: No, I really wanted to become an astrophysicist, because I thought that was the way, surely, to understand space travel. I wanted to be an astronaut, actually. I went to an all-girls school for the first half of my school years, and I still remember going to see the careers counselor and telling her I wanted to be an astronaut. She looked at me and she said, you have to be more realistic, dear. There was no way that somebody like me could ever aspire to it. And nowadays it’s normal almost. People from all different countries around the world become astronauts. But at the time I had to think about something else, and I thought, okay, I’m going to become a scientist, explore things through my own mind, and that was one way I could explore the universe. So I wanted to do physics when I came to university. I studied at the University of Sydney as an undergraduate. When I got to first-year physics, I realized my other big problem, which is that I have no physical intuition. So I thought, I really needed to understand things from a really explicit, literal, logical, analytical point of view, and that’s how I came to know I must be more of a mathematician. EL: Okay. KK: I have the same problem. I was always going to be a math major, but I thought I might pick up a second major in physics, and then I walked into this junior-level relativity class, and I just couldn’t do it. I couldn’t wrap my head around it at all. I dropped it and took logic instead. I was much happier. NJ: Yeah. Oh good. EL: So we invited you on to find out what your favorite theorem is. NJ: Yes. Well that was a very difficult thing to do. It was like choosing my favorite child, which I would never do. But I finally decided I would choose Mittag-Leffler’s theorem because that was something that really I was blown away by when I started reading more about complex analysis as a student. I mean, we all learnt the basics of complex analysis, which is beautiful in itself. But then when you went a little bit further, so I started reading, for example, the book by Lars Ahlfors, which I still have, called Complex Analysis. KK: Still in use. EL: That’s a great one. NJ: Which was first I think published in 1953. I had the 1979 version. I saw that there were so many powerful things you could do with complex analysis. And the Mittag-Leffler theorem was one of the first ones that gave me that perspective. The main thing I loved about it is that you were taking what was a local, small piece of information, around, for example, poles of a function. So we’re talking about meromorphic functions here, that’s the subject of the theorem. EL: Can we maybe set the stage a little bit? So what is a meromorphic function? NJ: A meromorphic function is a function that’s analytic except at isolated points, which are poles. The worst singularities it has are poles. EL: So these are places where the function explodes, but otherwise it’s very smooth and friendly. KK: And it explodes in a controlled way, it’s like 1/zn for some finite n kind of thing. NJ: Exactly. Right. An integer, positive n. When I try to explain this kind of thing to people who are not mathematicians, I say it’s like walking around in a landscape with volcanoes. Well-timed, well-controlled, well-spaced volcanoes. You’re walking in the landscape of just the Earth, say, walking around these places. There are well-defined pathways for you to move along by analytic continuation. You know ahead of time how strong the volcano’s eruption is going to be. You can observe it from a little distance away if you like because there is no danger because you can skirt all of these volcanoes. KK: That’s a really good metaphor. I’m going to start using that. I teach complex variables in the summer. I’m going to start using that. That’s good. NJ: So a meromorphic function, as I say, [cut ~7:29-7:31?] it’s a function that gives you a pathway and the elevation, the smoothness of your path in this landscape. And its poles are where the volcanoes are. EL: So Mittag-Leffler’s theorem, then, is about controlling exactly where those poles are? NJ: Not quite. It’s the other way around. If you give me information about locations of poles and how strong they are, the most singular part of that pole, then I can reconstruct a function that has poles exactly at those points and with exactly those strengths. That’s what the theorem tells you. And what you need is just a sequence of points and that information about the strength of the poles, and you need potentially an infinite number of these poles. There’s one other condition, that the sequence of these poles has a limit at infinity. KK: Okay, so they don’t cluster, in other words. NJ: Exactly. They don’t coalesce anywhere. They don’t have a limit point in the finite plane. Their limit point is at infinity. EL: But there could be an infinite number of these poles if they’re isolated, on integer lattice points in the complex plane or something like that. NJ: Right, for example. KK: That’s pretty remarkable. NJ: If you take your standard trigonometric functions, like the sine function or the cosine function, you know it has periodically spaced zeroes. You take the reciprocal of that function, then you’ve got periodically placed poles, and it’s a meromorphic function, and you can work out which trig function it is by knowing those poles. It’s powerful in the sense that you can reconstruct the function everywhere not just at the precise points which are poles. You can work out that function anywhere in between the poles by using this theorem. KK: That’s really remarkable. That’s the surprising part, right? NJ: Exactly. KK: If you knew you had a finite number of poles, you could sort of imagine that you could kind of locally construct the function and glue it together, that wouldn’t be a problem. But the fact that you can do this for infinitely many is really pretty remarkable. NJ: Right. It’s like going from local information that you might have in one little patch of time or one little patch of space and working out what happens everywhere in the universe by knowing those little local patches. It’s the local to global information I find so intriguing, so powerful. And then it struck me that this information is given in the form of a sum of those singular parts. So the function is reconstructed as a series, as an infinite sum of the singular parts of the information you’re given around each pole. That’s a very simple way of defining the function, just taking the sum of all these singular things. KK: Right. EL: Yeah, I love complex analysis. It’s just full of all of these things where you can take such a small amount of local information and suddenly know what has to be happening everywhere. It’s so wonderful. NJ: Right, right. Those two elements, the local to global and the fact that you have information coming from a discrete set of points to give you continuous smooth information everywhere in between, those two elements, I realized much later, feature in a lot of the research that I do. So I was already primed to look for that kind of information in my later work. EL: Yeah, so I was going to ask, I was wondering how this came up for you, maybe not the Mittag-Leffler theorem specifically, but using complex analysis in your work as an applied mathematician. NJ: Right. So what I do is build toolboxes of methods. So I’m an applied mathematician in the sense that I want to make usable tools. So I study asymptotics of functions, I study how you define functions globally, functions that turn out to be useful in various mathematical physics contexts. I’m more of a theoretical applied mathematician, if you like, or I often say to people I’m actually a mathematician without an adjective. KK: Right. Yeah. NJ: You know that there is kind of a hierarchy of numbers in the number system. We start with the counting numbers, and we can add and subtract them. Subtraction leads you to negative integers. Multiplication and division leads you to rational numbers, and then solving polynomial equations leads you to algebraic numbers. Each time you’re building a higher being of a type of number. Beyond all of those are numbers like π and e, which are transcendental numbers, in the sense that they can’t be constructed in terms of a finite number of operations from these earlier known operations and earlier known objects. So alongside that hierarchy of numbers there’s a hierarchy, a very, very closely related hierarchy of functions. So integers correspond to polynomials. Square roots and so on correspond to algebraic functions. And then there are transcendental functions, the exponential being one of them, exponential of x. So a lot of the transcendentality of functions is occupied by functions which are defined by differential equations. I started off by studying differential equations and the corresponding functions that they define. So even when you’re looking at linear differential equations, you get very complicated transcendental functions, things like the exponential being one of them. So I study functions that are even more highly transcendental, in the sense that they solve nonlinear equations, and they are like π in the sense that these functions turn out to be universal models in many different contexts, particularly in random matrix theory where you might be, for example, trying to work out the statistics of how fundamental particles interact when you fire them around the huge loop of the CERN collider. You do that by looking at distributions of entries in infinitely large matrices where the entries are random variables. Now under certain symmetries, symmetry groups acting on, for example, you might have particles that have properties that allow these random matrices to be orthogonal matrices, or Hermitian matrices, or some other kind of matrices. So when you study these ensembles of matrices with these symmetry properties and you study properties like what’s their largest eigenvalue, then you get a probability distribution function which happens to be, by some miracle, one of those functions I’ve studied. There’s kind of a miraculous bridge there that nobody really knows why these happen. Then there’s another miraculous thing, which is that these models, using random matrices, happen to be valid not just for particle physics but if you’re studying bus arrival times in Cuernavaca, or aircraft boarding times, or when you study patient card sorting, all kinds of things are universally described by these models and therefore these functions. So first of all, these functions have this property: they’re locally defined by initial value problems given for the differential equation. KK: Right. NJ: But then they have these amazing properties which allow them to be globally defined in the complex plane. So even though we didn’t have the technology to describe these functions explicitly, not like I could say, take 1 over the sine function, that gives you a meromorphic function, whose formulae I could write down, whose picture I could draw, these functions are so transcendental that you can’t do that very easily, but I study their global properties that make them more predictable wherever you go in the complex plane. So the Mittag-Leffler theorem sort of sets up the baseline. I could just write them as the sum of their poles. And that’s just so powerful to me. There are so many facets of this. I could go on and on. There is another direction I wanted to insert into our conversation, which is that the next natural level when you go beyond things like trigonometric functions and their reciprocals is to take functions that are doubly periodic, so trigonometric functions have one period. If you take double periodicity in the complex plane, then you get elliptic functions, right? So these also have sums of their poles as an expression for them. Now take any one of these functions. They turn out to be functions that parametrize very nice curves, cubic curves, for example, in two dimensions. And so the whole picture shifts from an analytic one to an algebraic geometric one. There are two sides to the same function. You have meromorphic functions on one side, and differential equations, and on the other side you have algebraic functions and curves, and algebraic properties and geometric properties of these curves, and they give you information about the functions on the other side of that perspective. So that’s what I’ve been doing for the last ten years or so, trying to understand the converse side so I can get more information about those functions. EL: Yeah, so using the algebraic world, NJ: Exactly, the algebro-geometric world. This was a huge challenge at the beginning, because as I said, I was educated as an applied mathematician, and that means primarily the analytic point of view. But to try and marry that to the algebraic point of view is something that turned out to be a hurdle at the beginning, but once you get past that, it’s so freeing and so beautiful and so strikingly informative that I’m now saying to people, all applied mathematicians should be learning algebraic geometry. KK: And I would say the converse is true. I think the algebraic geometers should probably learn some applied math, right? NJ: True, that too. There’s so many different perspectives here. It all started for me with the Mittag-Leffler theorem. EL: So something we like to do on this show is to ask our guest to pair their theorem with something: food, beverage, music, anything like that. So what have you chosen to pair your theorem with? NJ: That was another difficult question, and I decided that I would concentrate on the discrete to continuous aspect of this, or volcanoes to landscapes if you like. As I said, I was born in Burma, and in Burma there are these amazing dishes called le thoke. I’ll send you a Wikipedia link so you can see the spelling and de…

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Episode 16 - Jayadev Athreya Apr 12, 2018

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Episode 16 - Jayadev Athreya Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Hello and welcome to My Favorite Theorem. I’m Evelyn Lamb, one of your hosts. And this is your other host. Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. How are you doing, Evelyn? EL: I’m good. I actually forgot to say what I do. In case anyone doesn’t know, I’m a freelance math and science writer, and I live in Salt Lake City, Utah, where it has been very cold recently, and I’m from Texas originally, so I am not okay with this. KK: Everyone knows who you are, Evelyn. In fact, Princeton University Press just sent me a complimentary copy of the Best Math Writing of 2017, and you’re in it, so congratulations, it’s really very cool. [clapping] EL: Well thanks. And that clapping you heard from the peanut gallery is our guest today, Jayadev Athreya. Do you want to tell us a little bit about yourself? Jayadev Athreya: Yeah, so I’m based in Seattle, Washington, where it is, at least for the last 15 minutes it has not been raining. I’m an associate professor of mathematics at the University of Washington, and I’m the director of the Washington Experimental Mathematics Lab. My work is in geometry, dynamical systems, connections to number theory, and I’m passionate about getting as many people involved in mathematics as a creative enterprise as is possible. KK: Very cool. EL: And we actually met a while ago because my spouse also works in your field. I have the nice privilege of getting to know you and not having to learn too much about dynamical systems. JA: Evelyn and I have actually known each other since, I think Evelyn was in grad school at Rice. I think we met at some conferences, and Evelyn’s partner and I have worked on several papers together, and I’ve been a guest in their wonderful home and eaten tons of great granola among other things. On one incredibly memorable occasion, a buttermilk pie, which I won’t forget for a long time. KK: Nice. I’ve visited your department several times. I love Seattle. You have a great department there. JA: It’s a wonderful group of people, and one of the great things about it is of course all departments recognize research, and many departments also recognize teaching, but this department has a great tradition of public engagement with people like Jim Morrow, who was part of the annual [ed. note: JA meant inaugural; see https://sites.google.com/site/awmmath/awm-fellows] class of AWM fellows and runs this REU and this amazing event called Math Day where he gets two thousand high school kids from the Seattle area on campus. It’s just a very cool thing for a research math department to seriously recognize and appreciate these efforts. I’m very lucky to be here. KK: Also because I’m a topologist, I have to take a moment to give, well, I don’t know what the word is, but you guys lost a colleague recently. JA: We did. KK: Steve Mitchell. He was a great topologist, but even more, he was just a really great guy. Sort of unfailingly kind and always really friendly and helpful to me when I was just starting out in the game. My condolences to you and your colleagues because Steve really was great, and he’s going to be missed. JA: Thank you, Kevin. There was a really moving memorial service for Steve. For any of the readers who are interested in learning more about Steve, for the last few years of his life he wrote a really wonderful blog reflecting on mathematics and life and how the two go together, and I really recommend it. It’s very thoughtful. It’s very funny, even as he was facing a series of challenges, and I think it really reflects Steve really well. KK: His biography that he wrote was really interesting too. JA: Amazing. He came with a background that was very different to a lot of mathematicians. EL: I’ll have to check it out. KK: Enough of that. Let’s talk about theorems. EL: Would you like to share your favorite theorem? JA: Sure. So now that I’m in the northwest, and in fact I’m even wearing a flannel shirt today, I’m going to state the theorem from the perspective of a lumberjack. EL: Okay. JA: So when trees are planted by a paper company, they’re planted in a fairly regular grid. So imagine you have the plane, two number lines meeting at a 90 degree angle, and you have a grid, and you plant a tree at each grid point. So from a mathematician’s perspective, we’re just talking about the integer lattice, points with integer coordinates. So let’s say where I’m standing there’s a center point where maybe there’s no tree, and we call that the origin. That’s maybe the only place where we don’t plant a tree. And I stand there and I look out. Now there are a lot of trees around me. Let’s say I look around, and I can see maybe distance R in any direction, and I say, hm, I wonder how many trees there are? And of course you can do kind of a rough estimate. Now I’m going to switch analogies and I’ll be working in flooring. I’m going to be tiling a floor. So if you think about the space between the trees as a tile and say that has area 1, you look out a distance R and say, well, the area of the region that you can see is about πR2, it’s the area of the circle, and each of these tiles has size 1, so maybe you might guess that there are roughly πR2 trees. That’s what’s called the Gauss circle problem or the lattice point counting problem. And the fact that that is actually increasingly accurate as your range of vision gets bigger and bigger, as R gets bigger and bigger, is a beautiful theorem with an elementary proof, which we could talk about later, but what I want to talk about is when you’re looking out, turning around in this spot, you can’t see every tree. EL: Right. JA: For instance, there’s a tree just to the right of you. You can see that tree, but there’s a tree just to the right of that tree that you can’t because it’s blocked by the first tree that you see. There’s a tree at 45 degrees that would have the coordinate (1,1), and that blocks all the other trees with coordinates (2,2) or (3,3). It blocks all the other trees in that line. We call the trees that we can see, the visible trees, we call those primitive lattice points. It’s a really nice exercise to see that if you label it by how many steps to the right and how many steps forward it is, call that that the integer coordinates (m,n), or maybe since we’re on the radio and can’t write, we’ll call it (m,k), so the sounds don’t get too confusing. EL: Okay. JA: A point (m,k) is visible if the greatest common divisor of the numbers m and k is 1. That’s an elementary exercise because, well maybe we’ll just talk a little bit about it, if you had m and k and they didn’t have greatest common divisor 1, you could divide them by their greatest common divisor and you’d get a tree that blocks (m,k) from where you’re sitting. EL: Right. JA: We call these lattice points, they’re called visible points, or sometimes they’re called primitive points, and a much trickier question is how many primitive points are there in the ball of radius R, or in any kind of increasingly large sequence of sets. And this was actually computed, I believe for the first time, by Euler KK: Probably. Sure, why not? JA: Yeah, Euler, I think Cauchy also noticed this. These are names, anything you get at the beginning of analysis or number theory, these names are going to show up. KK: Right. JA: And miraculously enough, we agreed that in the ball of radius R, the total number of trees was roughly the area of the ball, πR2. Now if you look at the proportion of these that are primitive, it’s actually 6/π2. KK: Oh. JA: So the total number of primitive lattice points is actually 6/π2 times πR2. And now, listeners of this podcast might remember some of their sequences and series from calc 1, or 2, or 3, and you might remember seeing, probably not proving, but seeing, that if you add up the following series: 1 plus 1/4 plus 1/9 plus 1/16 plus 1/25, and so on, and you can actually do this, you can write a little Python script to do this. You’ll get closer and closer to π2/6. Now it’s amazing, now there is of course this principle that there aren’t enough small numbers in mathematics, which is why you have all these coincidences, but this isn’t a coincidence. That π2/6 and our 6/π2 are in a very real mathematical sense the same object. So that’s my favorite mathematical theorem. So when you count all lattice points, you get π showing up in the numerator. When you count primitive ones, you get π showing up in the denominator. KK: So the primitive ones, that must be related to the fact that if you pick two random integers, the probability that they’re relatively prime is this number, 6/π2. JA: These are essentially equivalent statements exactly. What we’re saying is, look in the ball of radius R. Take two integers sort of randomly in between, so that m2+n2 is less than R squared, what’s the proportion of primitive ones is exactly the probability that they’re relatively prime. That’s a beautiful reformulation of this theorem. KK: Exactly. And asymptotically, as you go off to infinity, that’s 6/π2. JA: Yeah, and what’s fun is, if a listener does like to do a little Python programming, in this case, infinity doesn’t even have to be so big. You can see 6/π2 happening relatively quickly. Even at R=100, you’re not far off. EL: Well the squares get smaller so fast. You’re just adding up something quite small in not too long. JA: That’s right. That’s my favorite mathematical theorem for many reasons. For one, this number, 6/π2, it shows up in so many places. What I do is at the intersection of many fields of mathematics. I’m interested in how objects change. I’m interested in counting things, and I’m interested in the geometry of things. And all of these things come into play when you’re thinking about this theorem and thinking about various incarnations of this theorem. EL: Yeah, I was a little surprised when you told us this was going to be your theorem because I was thinking it was going to be some kind of ergodic theorem for flows or something because the stuff I know about your field is more what my spouse does, which is more related to dynamical systems. I actually think of myself as a dynamicist-in-law. JA: That’s right. The family of dynamicists actually views you as a favorite in-law, Evelyn. You publicize us very nicely. You write about things like billiards with a slit, which is something that we’ve been telling the world about, but until you did. EL: And that was a birthday gift for my spouse. He had been wanting me to write about that, and I just thought it was so technical, I don’t feel like it. Finally, it’s a really cool space, but it’s just a lot to actually go in and write about that. But yeah, I was surprised to see something I think of as more number theory related show up here. That number 6/π2, or π2/6, whichever way you see it, it’s one of those things where the first time you see it, you wonder why would you ever square π? It comes as an area thing, so something else is usually being squared when you see it. Strange thing. JA: So now what I’m going to say is maybe a little bit more about why I picked it. For me, that number π2/6 is actually the volume of a moduli space of abelian differentials. KK: Ah! EL: Of course! JA: Of course it is. It’s what’s called a Siegel-Veech constant, or a Siegel constant. Can I say just a couple words about why I love π2/6 so much? EL: Of course. JA: Let’s say that instead of planting your trees in a square grid, you have a timber company where they wanted to shoot an ad where they shot over the forest and they wanted it to look cool, and instead of doing a square grid, they decided to do a grid with parallelograms. Still the trees are planted in a regular grid, but now you have a parallelogram. So in mathematical terms, instead of taking the lattice generated by (1,0) and (0,1), you just take two vectors in the plane. As long as they’re linearly independent, you can generate a lattice. You can still talk about primitive vectors, which are the ones you can see from (0,0). There are some that are going to be blocked and some that aren’t going to be blocked. In fact, it’s a nice formulation. If you think of your vectors as (a,c) and (b,d), then what you’re essentially doing is taking the matrix (ab,cd)[ed. note: this is a square array of numbers where the numbers a and b are in the top row and c and d are in the bottom row] and applying it to the integer grid. You’re transforming your squares into parallelograms. KK: Right. JA: And a vector in your new lattice is primitive if it’s the image of a primitive vector from the integer lattice. EL: Yeah, so there’s this linear relationship. You can easily take what you know about the regular integer lattice and send it over to whatever cool commercial tree lattice you have. JA: That’s right. Whatever parallelogram tiling of the plane you want. What’s interesting is even with this change, the proportion of primitive guys is still 6/π2. The limiting proportion. That’s maybe not so surprising given what I just said. But here’s something that is a little bit more surprising. Since we care about proportions of primitive guys, we really don’t care if we were to inflate our parallelograms or deflate them. If they were area 17 or area 1, this proportion wouldn’t change. So let’s just look at area 1 guys, just to nail one class down. This is the notion of an equivalence class essentially. You can look at all possible area 1 lattices. This is something mathematicians love to do. You have an object, and you realize that it comes as part of a family of objects. So we started with this square grid. We realized it sits inside this family of parallelogram grids. And then we want to package all of these grids into its own object. And this procedure is usually called building a moduli space, or sometimes a parameter space of objects. Here the moduli space is really simple. You just have your matrices, and if you want it to be area 1, the determinant of the matrix has to be 1. In mathematical terms, this is called SL(2,R), the special linear group with real coefficients. There’s a joke somewhere that Serge Lang was dedicating a book to his friend R, and so he inscribed it “SL2R,” but that’s a truly terrible joke that I’m sorry, you should definitely delete from your podcast. KK: No, that’s staying in. JA: Great. EL: You’re on the record with this. JA: Great. That’s sort of all possible deformations, but then you realize that if you hit the integer lattice with integer matrices, you just get it back. Basically the space of all lattices you can basically think of as 2 by 2 matrices with real entries and determinant 1 up to 2x2 matrices with integer entries. What this allows you to do is allows you to give a notion of a random lattice. There’s a probability measure you can put on this space that tells you what it means to choose one of these lattices at random. Basically what this means is you pick your first vector at random, and then you pick your second vector at random as uniformly as possible from the ones that make determinant 1 with it. That’s actually accurate. That’s actually a technically accurate statement. Now what that means is you can talk about the average behavior of a lattice. You can say, look, I have all of these lattices, I can average. And now what’s amazing is you can fix your R. R could be 1. R could be 100. R could be a million. And now you can look at the number of primitive points divided by the number of total points in the lattice. You average that, or let me put it a slightly different way: you average the number of primitive points and divide by the average number of total points. KK: Okay. JA: That’s 6/π2. EL: So is that… JA: That’s not an asymptotic. That’s, if you average, if you integrate over the space of lattices, you integrate and you look at the number of primitive points, you divide by the average number of total points, it’s…

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