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.
HK: Maybe I should talk about it mathematically first, maybe the statement?
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?
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.
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.
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 continuity this kind of terrible behavior can’t happen.
KK: Right. In other words, you need additional structure to get your sense that the plane is bigger than the line.
HK: That’s right. Although even taking into account continuity, topology is weird sometimes. There are space-filling curves, so in other words, there are surjective maps from the real line, (well, let’s just stick to intervals) from the unit interval to any-dimensional box that you want. And so somehow that’s really counterintuitive to most people. So it’s not so obvious that maybe what you think of as the reverse, an injection of a large space into a small space, maybe that would be problematic. But thanks to Brouwer’s fixed-point theorem, it’s not.
KK: So what pairs well with Brouwer’s fixed-point theorem?
HK: Well, okay, it has to be a cocktail, right, because I chose the cocktail example and because cocktails are fun. And they’re anti-Brouwer, presumably, as we discussed. So for the overlap of the cocktail description and the map description that I gave of Brouwer’s fixed-point theorem, I’m going to go with a Manhattan.
KK: Is that your favorite cocktail?
HK: It’s one of my favorites. Also Manhattan is almost convex.
HK: Almost convex.
KK: So you’re a whiskey drinker?
HK: I am a whiskey drinker.
KK: All right. I don’t drink too much brown liquor because if I drink too much of it I’ll start fights.
HK: Fortunately being sort of small as a human has prevented me from starting too many fights. I just don’t think I would win.
EL: So in my household I am married to a dynamicist, so I’m a dynamicist-in-law, but I’m more of a geometer, and we have this joke that there are certain chores that I’m better at, like loading the dishwasher because I’m good at geometry and what shapes look like. My spouse is good at dynamics, and he is indeed our mixologist. So do you feel like your dynamical systems background gives you a key insight into making cocktails? It certainly seems to work with him.
HK: Definitely for the first cocktail. Subsequent cocktails, I don’t know.
KK: Well I’m going to happy hour tonight. Maybe I’ll get a Manhattan.
HK: Maybe you should talk about Brouwer’s fixed-point theorem.
KK: With my wife? Not so much.
HK: Doesn’t go over that well?
KK: Well, she would listen and understand, but she’s an artist. Cocktails and math, I don’t know, not so much for her.
HK: I don’t know, that just makes me think of, okay, wow, I’m really going to nerd it out. Do you guys ever watch Battlestar Galactica?
EL: I haven’t. It’s on my list.
KK: When I was a kid I watched the original.
HK: The new one. All right. This is for listeners who are BSG nerds. So there’s this drawing of this vortexy universe, this painting of the vortexy universe that features in the later, crappier seasons. Now that makes me think it’s kind of an illustration of Brouwer’s fixed-point theorem. So maybe you should tell your wife to try and paint Brouwer’s fixed-point theorem for you.
HK: Marital advice from me. Don’t take it.
KK: We’ve been married for almost 26 years. I think we’re okay. We’re hanging in all right. So we always like to give our guests a chance to plug anything they’ve been working on. You’ve been in a bunch of Numberphile videos, right?
HK: Yeah, that’s right, and there will be more in the future, so if anyone hasn’t checked out Numberphile, it’s this amazing YouTube channel where maths is essentially explained to the public. Mathematicians come, and they talk about some interesting piece of mathematics in what is really meant to be an accessible way. I’ve been a guest on there a couple of times, and it’s definitely worth checking out.
EL: Yeah, they’re great. Holly’s videos are great on there. I like Numberphile in general, but I have personally used your videos about the Mandelbrot set, the dynamics of it and stuff, when I’ve written about it, and some other related dynamical systems. They’ve helped me figure out some of the finer points that as not-a-dynamicist maybe don’t come completely naturally to me.
HK: Oh, that’s awesome.
EL: I’ve included them in a few of the posts I’ve done, like my post about the Mandelbrot set.
HK: That’s amazing. That’s good because I’ve used your blog a few times when I’ve tried to figure out things that people might be interested to know about mathematics and things that are accessible to write and talk about to people. So it goes both directions.
KK: It’s a mutual lovefest here.
EL: People can also find you on Twitter. I don’t remember actually what your handle is.
HK: It’s just my name, @hollykrieger.
EL: Thanks a lot for being on the show. It was a pleasure.
HK: Thanks so much for having me. It was great to talk to you guys.
KK: Thanks, Holly.