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## Episode 74 - Priyam Patel

## Manage episode 320101278 series 1516226

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

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. I almost forgot my name there for a second.

EL: It happens.

KK: I realized, like, I was hesitating, and I was like, “Who am I again?” Yeah, you know — so our listeners don't know, but it's 5:30 where I am, which, you know, doesn't sound late. But I've been at work all day, and now I'm tired.

EL: Yeah. Well, you should have made up something. You know, just tried on a different name for fun just to see.

KK: Well, yeah, so even my parents had the deal that if I was a boy, my dad got to name me. So he went with Kevin Patrick. And if I was a girl, my mother was going to get to name me. And should I tell you what I would have been?

EL: Yeah.

KK: Kandi. Kay Knudson.

EL: Yikes!

KK: Now, I'll let you work out why that would have been terrible for lots of reasons. Already, there are multiple axes along which that is terrible.

EL: Great. Yeah, well, my name if I had been a boy ended up with my younger brother. So it was kind of not that interesting. I mean, if you knew my family, you would be like, Okay, well, that's boring. Anyway, yeah. We are very happy today to have Priyam Patel on the show. So yeah, Priyam, could you introduce yourself a little bit?

Priyam Patel: Sure. So my name is Priyam. I am an assistant professor at the University of Utah, and I have been here for three years. Before that I was around everywhere, it feels like, for my postdoc. I was at UCSB for a few years, before that at Purdue for a few years, And I did my PhD at Rutgers, which now feels like ages ago.

EL: Yeah, you’ve been in, like, every region of the country, though, I guess not central timezone, because Indiana is right on the west edge of Eastern.

KK: That’s right.

PP: Yeah. So I was never in the Central time zone. And that's why — in the summer in Indiana, the sun sets at, like, 10:30pm. It's really bizarre.

KK: You could call that Central Daylight if you wanted to, right?

PP: Yeah. Something like that.

EL: Yeah. And as you mentioned, you've been at Utah for about three years. And you you first got here in fall 2019, and I was gone for most of the fall 2019. And then of course, we all know what happened in 2020. So part of the reason I wanted to invite you is because I feel like I should know you better because you've lived here for three years. But, like, with the weirdness of the past three years, I feel like I haven't gotten to talk with you that much. And so of course, obviously the best way to do this is, like, on a podcast that we want to just broadcast to the entire world.

PP: Yeah, perfect. So no private conversation over drinks. Just put me on the podcast.

EL: Yeah. Excellent. So So yes, I'm excited to get to chat with you. And yeah, hopefully we can do this over drinks in a real venue at some point.

KK: Wait a minute, what happened in 2020?

EL: I tried to block it out.

PP: Nothing at all.

EL: For some parts of it, really nothing.

PP: It feels like a whole blur since then. So

KK: I’m not convinced it isn’t still 2020 somehow.

PP: Yeah, yeah.

KK: Alright. Anyway, I'm being weird today, and I apologize. So let’s get to math. So Priyam, you have a favorite theorem. Which is it?

PP: Yeah. So I chose the Brouwer fixed point theorem, which I learned has been done twice already on this podcast.

EL: Yes, I'm very excited to hear more about it because in our emails, you mentioned some aspects of that I wasn't aware of. And so this is very exciting. And this is when people, when we email with people, they’re always like, “well has this been used?” And we're like, “It doesn't matter if it has, you can use it anyway.” We like to talk about theorems because it is interesting, just the different relationships people have with the same math. So for anyone who hasn't been you know, avidly listening and taking notes on every single episode we've done since 2017, can you tell us what the Brouwer fixed point theorem is?

PP: Yeah, so I'm just going to state it for the closed disk because that's the only context that I'm going to talk about it in. But basically, if you take in the plane in our two if you take the closed unit disk, then the theorem says that every continuous map from the disk to itself necessarily has a fixed point. So should I go into detail about what a continuous map? Would that help?

EL: Yeah. Or at least intuitively.

KK: Sure.

PP: So I actually did listen to a lot of the previous podcast episodes while I was preparing. And I like this idea of if you take the unit desk, and you, like, kind of shake it around a little bit, and everything kind of moves in a nice smooth fashion where things don't get sent, like, really far away — so if in a little neighborhood, you’re wiggling, one point is not just going to pop out and end up somewhere else, right? I like that idea of continuity. So if you're wiggling around the disk, the unit disk, and you use any continuous map, somehow one of the points has to stay fixed, so it gets sent to itself. And that's kind of surprising. It feels like if you just move things around enough, something, everything, should get moved off of itself. But in fact, that can't happen. So that's kind of my interpretation of Brouwer’s fixed point theorem.

EL: Yeah. And it's like I guess I always imagine it made of rubber or something. Because you are allowed to, like, stretch and smush a little bit. It doesn’t — because otherwise, you might think, Oh, the only thing you can do is rotate it. So of course, that central point will be fixed. But you could do a lot of other things.

PP: Yeah, absolutely.

EL: And fix some different point.

PP: Yeah, so I think Evelyn has a great point, like, you can spread things out, like you're making it out of like stretchy fabric or material, you can spread things out in one part of the circle, in the unit disk, and then, you know, string things together in another part and that's okay. It's like, you know, just kind of smoothly moving around is the way I think about it.

KK: Yeah, yeah. But something stays put.

PP: Something stays put, which is kind of strange sometimes, actually. And there's like, so many proofs of this theorem, I feel like, and so many different perspectives for proving it. But I do have a favorite proof of that, actually.

KK: Okay, good. Let’s hear it.

PP: So it's unfair, because it uses some algebraic topology. So o be able to get to this point in this in your math life, where you're like, Yeah, this is the proof I like the best, you have to learn some algebraic topology. But essentially, the idea is that when you're in topology, in the field of topology, you're trying to understand when two objects that are made out of bendy, squishable material that you can stretch and shrink, when two of those are really the same. So if you have, let's say, a circle, or a really oblong wiggly circle, those two are the same. It doesn't really matter if one is really beautiful and perfectly symmetric. It's really the same space in topology. So two things that are not the same topologically are the closed unit disk, and just the outer boundary, which is just a circle. Okay, so there's an a thing called an algebraic invariant that you can compute called the fundamental group, that tells you that topologically, formally, these two spaces really aren't the same. And essentially, there's a proof that says, If there wasn't a fixed point, then you could basically take the entire closed unit disc, and shrink every point in the desk to the boundary. This is called a retract. You’re basically saying like, I'm going to retract the entire closed unit disc to just the circle. And retracts are supposed to give you the same fundamental group. And you already know that those two things aren't the same. And so that's my favorite version of this group. And I can slow down on any part of that if you'd like more details.

EL: Yeah, that's really nice. Well, I think maybe a good way to see this is like, you know, that example of turning the circle around, you know, like a record spinning on a record player or something. Like if you took away that central point, everything else can move. And you can also imagine pulling that rubber all the way to the edge, making it into a bike tire or something else like that. (Which is actually topologically different.)

PP: Right, but as soon as you puncture it. So Evelyn's basically saying, let's just take out the center point. But what corresponds to the origin in R2? Well actually, once you do that, there's no contradiction that you derive, right? You can have every point moving. And in fact, that punctured disk and the circle are the same topologically. That retract that you can use to just pull everything to the boundary shows you, actually, that they're the same topologically. So it's just that one— it’s amazing how much like one point can make such a huge difference, right?

EL: Yeah.

PP: Adding in that one point. But yeah, so that's my favorite proof. It's fancy in some ways, but once you know the basic material that leads up to it, it's like a three-line proof, right? Which is kind of incredible.

EL: Yeah, but it's maybe a little bit like, what is the phrase, like using a sledgehammer to kill a mosquito.

PP: Oh yeah.

EL: Once you’ve built all of this fundamental group, then sure, you could just whack that.

PP: Yeah. And it's so funny because in math, typically I am the opposite of a hammer-striker, right? I never use the hammer. I want to understand the nitty-gritty of why you can just explain this using elementary math or something like that, right? But for some reason, when I saw this proof in, like, Hatcher’s algebraic topology book, I was like, Oh my gosh, that just like makes perfect sense to me, like I totally get why now. So it definitely is one of those use a hammer use a sledgehammer to kill a mosquito type of approaches.

KK: I’m actually teaching algebraic topology this year, and so I that is the proof I use for the Bouwer fixed-point theorem. But I use the sledgehammer to prove that every polynomial of odd degree with real coefficients has a root. And I use to use the Lefschetz fixed point theorem to do it. Let’s use the biggest sledgehammer we can find!

PP: It’s awesome. I mean, honestly, if you want for this podcast, if you want to talk about a fun theorem that's called theorem at the end, right? Not just some result, but that's actually named, a lot of the ones you come up with in topology are the fixed-point theorems, right? And I was like, Oh, this one's actually my favorite. And there's a reason for it. So yeah, that makes sense.

EL: So what else do you love about this theorem?

PP: Yeah. So this actually was inspired from talking to a few grad students the other day, but I realized that, you know, I gave them this task, which says, Can you classify all of the isometries of hyperbolic 2-space. Now, that's already a fancy sentence to say. So I can break down all of what that means. But in fact, one of the key ingredients for the approach that is my favorite to solving that problem is to use the Brouwer fixed-point theorem. So I can start off by talking about what hyperbolic space is, and like what metric spaces are. And from there, I can explain what an isometry. It’s kind of similar to a continuous map, but it has a lot more structure and preserves a lot more structure.

So let's start off with just hyperbolic space, shall we? Okay, so the way I think about anything that is a non Euclidean geometry, which hyperbolic geometry is one of those, I have to start thinking about, well, Euclidean geometry first, right? And Euclidean geometry, when I think of that, I think axioms, right? There's Euclid axioms and they're written down. You don't need to know what they are. But the last one is the one that people started saying, let's try to break it and see what happens to these models of geometry that we're sort of studying, right? Like, could we come up with a different interpretation than just Euclidean space. And so if you break the parallel postulate, there's a few different types of geometries you can get that satisfy all the other ones, but they don't satisfy the parallel postulate. And hyperbolic geometry is one of them. So what is a geometry, right? It's a space, like we just talked about, let's say the closed unit disk. And to me geometry, you're studying rigid things like distances, angles. And so you want to have a notion of measuring distance on whatever space you choose. So since we're going to talk about the power fixed point theorem, of course, my space is going to be the closed unit disk. In fact, I'll just start off with the open unit disk for now. So let's just get rid of the boundary. So if I start off with the open unit disk, that is my space. And there is a way of measuring distance on that space. So you can say, oh, put in these two points, I want to know the distance between them. There's a way of measuring distance on it where if you want to go from the center point at the origin, out to one of the boundary points, let's say just (0,1), or (1,0) in the plane, it actually takes you an infinite amount of distance to get there. Okay, so in hyperbolic space, this model of hyperbolic space, which is called the Poincaré disk model, the boundary of the disk is sort of off at infinity. And as you get close to that sort of boundary at infinity, points are getting really, really, really far away. That's what it means to get, you know, closer and closer to infinity, is distances go really big. And so that's the idea of what the Poincaré disk model of hyperbolic to space is. And, of course, if it weren't a podcast, I'd be showing like tons of pictures right now.

EL: Yeah, it is quite attractive. It's just a lovely, appealing model.

PP: Yeah. And like, you can look up all these amazing pictures by Escher. There are these famous paintings where Escher uses the upper half plane model or the disc model, and shows how, like, a bat, or whatever the figure is that he uses to tessellate, a bat of the same area drawn in different parts of the hyperbolic plane can look to our eyes, very different, right? And that's again, coming back to this notion of as you move out towards the boundary of the unit disk, distances are getting really big so the bat would have to look really small to your eye to to have the same area as a bat in the center of the disk. So I highly encourage listeners to Google just hyperbolic space or Escher's hyperbolic paintings, right. And you'll come up with so many things.

EL: Yeah, well, fun fact is that my Twitter profile picture is a tiling of the hyperbolic plane with the Poincaré. with like, a picture of me in it.

PP: Yeah. So I love it. I know Evelyn really loves all of these, like, hyperbolic geometry, topology type thing. So yeah, that's also partly why I chose this topic.

EL: Yeah, you’re definitely speaking my language.

PP: Okay, so that's the idea of what hyperbolic space is. There's so many more things you need to do to sort of gain the intuition of what it feels like to live in hyperbolic space, right? And those are the kinds of things that you build over years of studying it in your life. But the real thing I want to talk about is isometries of the hyperbolic disk model.

So what is an isometry? It's a map of the space to itself. So kind of like that jiggling that we were talking about. But where all of the distances between any two pairs of points remain the same. So a great example that Evelyn already talked about was this rotation around the origin, right? If you rotate around the center point, all of the points, pick any two of them, they actually see the same distance apart. And that's not an easy thing to see, partly because I never told you how to measure distances, right? That completely relies on how we decided to define that, which is the metric on the space, that notion of measuring distances. But if you knew it, and you wrote it down explicitly, you could actually calculate that that rotation is distance-preserving in the Poincaré disk model of hyperbolic space. So that's the idea of what an isometry is.

Okay, so now we want to try to get fancy. And usually, what you do when you have a space is you say, I'm going to try to understand all the isometries of this space. So where does this notion come from? In topology, if we're not talking about geometric structure, and we just kind of care about a space and it's all blobby, and can be stretched and shrunken, we think about all of the symmetries, right? All the topological symmetries of the space. When we're talking about isometries, what we're actually talking about is geometric symmetries of the space, all different ways of moving around this space, where distance hasn't really changed, so you're preserving the geometric structure. It turns out that if you take all of the isometries, you end up with a group. It has a really nice structure, you can compose two of them. But that's not really even important for today, what you really care about always is, can I classify all things of this type? And there are infinitely many of them. It's really hard to classify things when there's a whole infinite set.

EL: Right.

PP: I’m not just putting things in bins, like these are red marbles, and these are blue ones and these are green ones, right? So amazingly, it actually turns out that isometries of H2, the hyperbolic plane, or disk model, only fall into three flavors. They're either elliptic, which are very similar to the rotations that we talked about, or they are the rotations basically. There are parabolics, and there are loxadromics, or sometimes called hyperbolics, which doesn't make sense, because it's confusing. When you're talking about hyperbolic space, calling something a hyperbolic isometry, when you mean a certain type is confusing. So I'll just call it loxadromic, right? And so there's a few things you need to know, like what could the isometry even look like? How could I possibly get equations of these isometries? So you have to work a little bit. But it turns out that you can write down a general formula for a generic isometry of the Poincaré disk model, or the upper half plane model of hyperbolic space. So just as an example, I'm going to switch models. And in fact, when I say I'm going to switch a model of hyperbolic space, what I mean is, I'm just going to go to a different space with another metric on it, but it ends up being the same geometrically. There is a nice map between the two of them where all the geometry is preserved. So I like sometimes the upper half-plane model, because it's really easy to like write down what the isometries are.

EL: Yeah.

PP: So what I'll say is, and I, again, I'd write this down on the board, if I could. But imagine it, close your eyes and imagine it. I'll take four real numbers a, b, c, and d. And all, really orientation-preserving, but let's sweep that under the rug. Well, orientation-preserving isometries of the upper half plane model look like (az+b)/(cz+d), where z is a complex number. Okay, the criteria you need to make sure you satisfy is that ad−bc is 1. Okay, there's another interpretation actually in terms of matrices. Put those four things into a matrix [Editor’s note it’s hard to write a matrix in Word. The top row is the numbers a and b. The bottom row is the numbers c and d.] Well, you're seeing I want ad−bc to be equal to 1, that's SL(2,R). If you multiply the top and the bottom by negative one in the top of the bottom, you're not changing the transformation. So you have to mod out by plus or minus the identity. So we're really looking at PSL(2,R) protect devised special linear space.

EL: Yeah. I actually I do love that because the first time you see these hyperbolic isometries, the az plus b, over cz+d,, and then they're like, oh yeah, ad−bc, you just have this almost spidey sense tingling of like, Okay, that's like a determinant. Why are we doing this? There must be some relationship here with linear algebra.

PP: Yeah. And your spidey sense is totally on point. So I love I love that connection. And I sort of, you know, I always wonder how much detail to go into with these things, since I'm not writing at the board. But I love that, right? Because anybody listening to the podcast should be like, Okay, wait, that's the determinant, just like you did. So, yeah, so there's a little bit of complex analysis that goes into deriving the fact that these are actually isometries, that they map the hyperbolic plane to itself and so on. But once you have a nice general formula, you start to use Brouwer’s fixed-point theorem. So I, this is part of my life, I go back and forth between the models all the time. So I'm sorry if this is getting annoying, but I'm going to go back to the disk model for a second.

Okay. And well, let's think for a second. Right now, what I have is the open disk. And I have a boundary at infinity. And often, when you're working with spaces, you can sort of complete up the space by adding in the boundary. Okay, this is the sort of completion of the Poincaré disk model of hyperbolic space. So now, if I take the hyperbolic disk model with its boundary, I have the closed unit disk, right? And an isometry is way better than a continuous map, but in particular, it is continuous. And so Brouwer’s fixed-point theorem says no matter what map I'm talking about from the disk to itself, the closed disk to itself, I have a fixed point. So if you have a formula, and you know there has to be a fixed point, you should try to solve for those fixed points, right?

KK: Right.

PP: And so I'm going to pop back to the hyperbolic plane model, because that's where we have our nice formula. So I'm just going to try to solve az+b/cz+d=z, right? This is a function, your input is z. Even though it seems weird, because you're like, wait, I'm multiplying by a, then adding b, dividing by… this is really complicated, right? But there are certain points for which you put it in, and depending on a, b, c, and d, you spit out the same number, right? The same complex number. So this is what I tell people to do. This is what my students did the other day. They said, How am I supposed to approach this? And that's the classic proof, you use the Brouwer fixed-point theorem, and you start to solve for az+b/cz+d=z. The key point that you use after you're doing all the algebraic manipulations is that ad−bc is always 1. So a ton of stuff cancels out as you're solving, right? It becomes a very nice equation. But what am I doing, actually? If I cross-multiply, right, multiply by cz+d on both sides, and then move everything over to one side, I'm getting a degree two polynomial in the z coordinate. How do you solve any degree two polynomial? You use the quadratic equation! So the bane of some people's existence when they’re going through high school, they’re like, I'll never use this? Well, first, if you become a mathematician, you're definitely going to use it.

EL: Yes.

PP: But I love when things that you learn when you're so mathematically young still come into play when you're doing really sophisticated math. I think that's really cool. So okay, we write down the formula. And basically, what it comes down to is what's underneath the square root, right? What is the discriminant? Because when you take the square root of a negative number, you get imaginary things, right? Imaginary numbers, complex numbers with non-trivial imaginary part. If the thing under the under the square root is zero, well, you're just getting one root, right? And then if the thing under the square root is not zero, but it's positive, you're just getting two real roots. Okay, so let's think about that. We have three categories: one real root, two complex roots, or two real roots, right?

The thing is that in the upper half plane model, the thing I never talked about, was that you need the imaginary part to be positive. Okay, so the actual categories are one real fixed point, one complex fixed points in the hyperbolic plane, or two real fixed points. Really, when you're going between the models, what ends up happening is in the upper half plane model, the boundary at infinity is the real number line, it's the x-axis in the complex plane. So that ends up being the boundary of the circle, the unit disk. So we're talking about three cases: one fixed point on the boundary, two fixed points on the boundary, or one fixed point on the interior, which is the complex one. And that's it. That's literally the classification. Because if you have anything more than three fixed points, you can show that your transformation was the Do Nothing transformation. It's the identity.

KK: Right.

PP: So it's not just that the Brouwer fixed-point theorem tells you that you can find fixed points, that there is one, you can actually classify all of the isometries based on these three categories, which I think is like, just incredible. And if you want, I can give you a little bit of a geometric interpretation of what the three isometry is the classes of isometry.

KK: Sure!

EL: Yeah, but I would like to pause and say one of the last classes I taught when I was at the University of Utah was an undergraduate, like introduction to topology class. So we touched on some of this a little bit, and it's like, I kind of want to go back and teach it now and use this for that part of it.

PP: Yeah, it's kind of amazing. Even though it has some sophisticated things going on, you can tell some advanced undergrad students about this stuff, and really show them a lot of beautiful pictures. So when I was in Santa Barbara, I did teach a non-Euclidean geometry class. And, you know, of course, I have to do all the other geometries justice as well.

EL: Eh, do you really?

PP: I know, I know. But I mean, when you get to hyperbolic geometry, though, it's kind of like it's limitless, right? The amount of stuff you can kind of tell and teach students. So I do love that aspect of it. But it's not such an advanced sort of theorem that you're just like, What is this using? Where does it come from? It's like, you need to know the Brouwer fixed-point theorem, you need to know the basics of the model of hyperbolic geometry you're thinking about, and that's basically it, right? Okay, so let's talk about the three sort of classic, I guess I would say, the canonical examples that people give for each of the three categories. So for anybody who's listening that's really into math, and that knows a little bit about algebra, every isometry is actually conjugate to one of these. But they're like the model. They're sort of like, the best-behaved one in each category. What do they look like?

Okay, so for the elliptic ones, we're going to start there, because Evelyn's already told us what they look like, right? In the disk model, they’re all just rotations about the origin. Technically, the fixed point could be anywhere, and it's still kind of a rotation around that fixed point. But again, up to this conjugation, you can move that fixed point to the origin. And then so now you're really just asking if I want to just understand the canonical form of this, I just am going to try to understand isometries of the disk that fix the origin. And I'm going to get that these are rotations, right? You can actually go through and derive the formulas, you know, you say I solved for the root, it's complex. Here it is, let me write down what this might mean. And you can really see which Möbius transformations you're talking about. But that's the canonical way that we think about elliptic ones. And I think the elliptic word has to do with that like sort of rotation, right? But don't quote me on that, because I'm not good at words. I'm good at the math and pictures, but not great at words.

Okay, so let's go to something more interesting. So what are the parabolic ones? These are the ones where you have one fixed point on the boundary. And I'm going to go ahead and use the upper half-plane model again, because they think it's a little bit prettier to see the parabolic one there. So I didn't really say what the upper half-plane model was. So let me go ahead and do that. So the upper half-plane model of hyperbolic space, is you take the entire plane, but then you only think about the upper half part, right? So where the y-coordinate is strictly greater than zero. In the disk model, when we approached the boundary, distances got really big, right? What is the boundary for the upper half plane model? It's the real axis in the complex plane, so the x-axis, and the point all the way out at infinity in the plane itself. So in R2, there's actually an infinity, the one point at infinity, that has to get thrown in there. Okay, so what we can do is say, let's have fun and say that the fixed point in the hyperbolic plane model is infinity.

KK: Sure.

PP: So I'm going to take a straight line, it goes from zero, the origin, (0,0) in the complex plane, and it's just going to go straight up to infinity. Okay, if infinity is fixed, and you have to map these sort of straight lines to straight lines, what you can come up with I mean, I'm waving my hands here, you have to actually do some like algebra and manipulate everything and sort of make sure you're reducing this the right way. But what it turns out to say is that these are all translations. So the maps az+b/cz+d, well, c is 0, d is 1. And it's really just z+b, or something like that. Okay, it's like a translation by some by some number. That's approximately what a parabolic transformation looks like. So translation is something we understand from Euclidean geometry, right? It's just that the way that it affects points in — so what do I want to say? Translation in the Euclidean plane, we understand with that metric. We have the upper half-plane with a different metric. It turns out in this case, that translation is still an isometry. But you have to remember, distances look different. So when you're going to see parabolics in the disk model, things get a little bit more complicated. You have to talk about things called horocycles. And that, I would say, it's better to just look up.

EL: Yeah.

PP: This is where a picture would be very, very useful, right? Okay, and now, the queen of them all is the loxodromic isometries. So this is where we start to see a lot of the connections between hyperbolic geometry and dynamics. So when you have a loxodromic isometry, there are two fixed points on the boundary. With a little bit of work, again, what you can see is that if you take those two fixed points on the boundary, there's actually a, sort of like a shortest line segment going from one to the other. Lines look different in hyperbolic space because the metric is different. But essentially, the way that this isometry acts on the disk model is that one of those fixed points acts as a source. The other one acts as a sink, and everything in the disk model is getting taken away from the source and being pulled towards the sink, actually along the axis — that geodesic axis that you have between the two fixed points, it’s acting as translation along that. So this is a phenomenon in dynamics, more generally called north-south dynamics. You have this source and a sink, and things are moving from the source to the sink in a north-south sort of way. And, yeah, that's my favorite one. Of course, it's the most complicated one. It's the one that comes up the most when you're studying surfaces. So yeah, that that sort of is my — I know, that's just my favorite. That's my favorite type of isometry. I think it makes sense when you're working in hyperbolic geometry, because it comes up all the time.

EL: Yeah, well, that's so fun. So another thing we like to do on the podcast is force our guests to pair their theorem with something in the real world.

KK: “Force.”

EL: So what have you chosen as your pairing today? “Invite” our guests to do that.

PP: Yeah, I actually didn't think about this one so hard, because I have a natural pairing in my life, which is climbing. So I love to climb. So I got into rock climbing when I was a postdoc in Santa Barbara.

KK: Good place.

PP: In fact, one of my most favorite, yeah, it is a good place for climbing. And it just so happened that my mathematical grandfather, Mike Freedman actually, is in Santa Barbara as well. He is a very good rock climber. And he took me on my first like, sort of ropes climbing outside adventure. It was a lot more intense than I thought it was going to be. But he's an intense guy. So I kind of knew what I was getting myself into. But it was such a moment of growth for me. And there are a lot of mathematicians that are attracted to climbing. And there's a reason, right? It's problem-solving. But like, with the physical component put in there, right? So when I'm not problem-solving in my office, or at my home office, I'm usually in the gym, problem-solving climbing problems with my friends.

EL: Yeah, well, and that's, I have, I've like, done one little rock climbing thing. I've never done it, but they actually call them problems, right? Like, figuring out a route is called a problem.

PP: Yeah, absolutely. Yeah, it's very heady you know. Of course, if you ask a mathematician to choose a physical sport to get into, they're like, oh climbing, then I can still use my brain all the time.

KK: When I was a postdoc, I did some. There was a climbing gym in Evanston. And yeah, and you're right. It's very good for your brain. And I always thought that too, but man, my fingers just, it hurt so bad. And I was mostly a cyclist at the time, which was also good because it doesn't require you to really use your brain, like riding a bike is so automatic that I could go out for a ride and think about math while I was riding. I mean, I know you want to get away from the math sometimes, but it was actually a good thing to do for me.

PP: Oh, yeah, no, I totally agree. I was really into running for a while. And I loved that sort of time to decompress and add space to your brain, right? So with climbing, I don't have that sort of flow, I don't reach flow quite as quickly as I do in other exercise. But it's very good for getting out the work stress, I’ve got to say. You just, like, work really, really, really hard physically. And it is very rewarding. I think it's the type of thing where, just like math, in my opinion, you know, “natural ability” is not actually the thing I think that determines how well somebody does in math over time. I think it has a lot to do with how hard you work, right? And if you're in training, taking care of your body, learning things, watching climbers, being very observant, you tend to pick it up pretty quickly. And there's a lot of big, burly guys in the gym who struggle, and then they're sort of surprised when, like, petite women get up and sort of just crush the problem, which I mean, I don't mind that. You know, I think it's a good lesson for everybody. So yeah, I also think, you know, for me, climbing and math are both very dominated by certain genders and races, it's a very white dominated sport, it's a very white male dominated sport, often like math is. Math spaces are very dominated in those same ways. And a lot of my work in math has been to promote diversity and equity and justice, really, in my math communities wherever I exist. And it turns out that that extends to my climbing communities as well, because I co-founded a group called Color the Wasatch, which is an affinity group for people of color climbers in the Wasatch Valley. So it's been really great, and I've learned a lot from that. And it's very similar. When you have people around you that have similar experiences to you, it can be so enriching, and it can be such a relief to just sort of feel yourself relax, and feel just comfortable in your own skin wherever you might exist. And that's a big one that I sort of learned very recently. I’ve sort of always been into sort of activism in the math community. But this was a real first big thing I did in my community outside of my work as a university professor, and it's incredibly rewarding. And it's kind of taken off. It's been great.

KK: Cool.

EL: And we also like to give our guests a chance to talk about anything, you know, you'd like to plug. And actually, I think I just saw on Twitter, some people tweeting about the Roots of Unity conference that is happening. Is that — I don't know if conference is the right word for it — happening this summer. And I think this episode will probably be published in time for people who learned about it to apply,

KK: I think next week, this will be out.

PP: Yeah, that’s great. Because the application deadline for the Roots of Unity workshop is actually February 15, which is great. As you said, it's going to be out in time, the podcast will be out in time. So the Roots of Unity workshop, I am actually co-organizing with phenomenal women in math who I really looked up to, actually. And we designed this workshop to sort of support people who might not see people that look like them at their home institutions early on in their graduate career. So there's amazing programs out there, like EDGE, the EDGE network for people going into grad school, there are amazing research-focused conferences, like the Women in Numbers group, we actually have a Women in Geometry, Groups and Dynamics group now. And we felt that there was a sort of gap in between. And it's very hard, speaking from personal experiences, to be the only woman of color in your department or in your graduate cohort. And so we're really aiming to support anybody who would benefit from this kind of kind of program, but especially gearing it towards people who don't get that same training or preparation or encouragement at their home institutions, especially women of color. So it's going to be a professional development and research development workshop. And one of the things that we're doing is we're sort of helping grad students learn how to read papers, because like, gosh, we just are given papers and said, Go read this, right? And there is a skill to reading a paper well, I think. And so that's one of the big things that we're focusing on, is that transition between early coursework that feels very much like an extension of the undergraduate curriculum, and then into this whole new world that really requires a big pivot mentally, of reading papers and coming up with research problems and having a good network of support and collaborators as you do that. So that's occurring in June of this year at the IMA, so the program is you know in the process, the schedule is in the process of being set. But yeah, applications are open, and we would love to see lots and lots of applications.

KK: Excellent. All right. Where can we find you elsewhere online?

PP: So I am on Twitter. What is my handle? Evelyn, do you know what my handle is?

EL: Um…

PP: I think it might be priyam886. [Editor’s note: It is!]

EL: Maybe?

PP: That might be it. But you know, y'all can post it if you want on the website, eventually. My website actually has a link to the Roots of unity workshop. So that's patelp.com. And I also have a little page actually about Color the Wasatch there as well. So if people are interested in the climbing aspect of things also, that's that's all on my website.

KK: Oh, cool. All right. Well, this has been a lot of fun. I always like learning new things about the Brouwer fixed-point theorem. You know, I've taught complex analysis, and so I've thought about — I think I've shown my students sort of how linear fractional transformations, which are these isometries, are acting on the upper half plane, but I never thought about it in terms of isometries. So this is good for the next time I do it.

PP: Yeah, absolutely. So those linear fractional transformations are like the basic ingredient you need, right? Once you know those, you say which ones are actually going to map the upper half plane to itself, and then which ones are going to be distance-preserving, and everything falls out from there, so it's really nice.

EL: Yeah, this is a lot of fun.

KK: Yeah. Thanks.

[outro]

On this episode of My Favorite Theorem, we're revisiting the popular Brouwer fixed-point theorem with Priyam Patel of the University of Utah. Below are some links you might enjoy after you listen.

Patel's website and Twitter profile

Our previous episodes about the Brouwer fixed point theorem with Francis Su and Holly Krieger

A pdf of Allen Hatcher's algebraic topology book (available, legally, for free!)

The Lefschetz fixed-point theorem

Douglas Dunham's page about Escher and hyperbolic geometry

A blog post Evelyn wrote about putting pictures into the hyperbolic plane

Information about the Roots of Unity workshop (application deadline: February 15, 2022; if you're listening to this in later years, poke around and see if it's happening again!)

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

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. I almost forgot my name there for a second.

EL: It happens.

KK: I realized, like, I was hesitating, and I was like, “Who am I again?” Yeah, you know — so our listeners don't know, but it's 5:30 where I am, which, you know, doesn't sound late. But I've been at work all day, and now I'm tired.

EL: Yeah. Well, you should have made up something. You know, just tried on a different name for fun just to see.

KK: Well, yeah, so even my parents had the deal that if I was a boy, my dad got to name me. So he went with Kevin Patrick. And if I was a girl, my mother was going to get to name me. And should I tell you what I would have been?

EL: Yeah.

KK: Kandi. Kay Knudson.

EL: Yikes!

KK: Now, I'll let you work out why that would have been terrible for lots of reasons. Already, there are multiple axes along which that is terrible.

EL: Great. Yeah, well, my name if I had been a boy ended up with my younger brother. So it was kind of not that interesting. I mean, if you knew my family, you would be like, Okay, well, that's boring. Anyway, yeah. We are very happy today to have Priyam Patel on the show. So yeah, Priyam, could you introduce yourself a little bit?

Priyam Patel: Sure. So my name is Priyam. I am an assistant professor at the University of Utah, and I have been here for three years. Before that I was around everywhere, it feels like, for my postdoc. I was at UCSB for a few years, before that at Purdue for a few years, And I did my PhD at Rutgers, which now feels like ages ago.

EL: Yeah, you’ve been in, like, every region of the country, though, I guess not central timezone, because Indiana is right on the west edge of Eastern.

KK: That’s right.

PP: Yeah. So I was never in the Central time zone. And that's why — in the summer in Indiana, the sun sets at, like, 10:30pm. It's really bizarre.

KK: You could call that Central Daylight if you wanted to, right?

PP: Yeah. Something like that.

EL: Yeah. And as you mentioned, you've been at Utah for about three years. And you you first got here in fall 2019, and I was gone for most of the fall 2019. And then of course, we all know what happened in 2020. So part of the reason I wanted to invite you is because I feel like I should know you better because you've lived here for three years. But, like, with the weirdness of the past three years, I feel like I haven't gotten to talk with you that much. And so of course, obviously the best way to do this is, like, on a podcast that we want to just broadcast to the entire world.

PP: Yeah, perfect. So no private conversation over drinks. Just put me on the podcast.

EL: Yeah. Excellent. So So yes, I'm excited to get to chat with you. And yeah, hopefully we can do this over drinks in a real venue at some point.

KK: Wait a minute, what happened in 2020?

EL: I tried to block it out.

PP: Nothing at all.

EL: For some parts of it, really nothing.

PP: It feels like a whole blur since then. So

KK: I’m not convinced it isn’t still 2020 somehow.

PP: Yeah, yeah.

KK: Alright. Anyway, I'm being weird today, and I apologize. So let’s get to math. So Priyam, you have a favorite theorem. Which is it?

PP: Yeah. So I chose the Brouwer fixed point theorem, which I learned has been done twice already on this podcast.

EL: Yes, I'm very excited to hear more about it because in our emails, you mentioned some aspects of that I wasn't aware of. And so this is very exciting. And this is when people, when we email with people, they’re always like, “well has this been used?” And we're like, “It doesn't matter if it has, you can use it anyway.” We like to talk about theorems because it is interesting, just the different relationships people have with the same math. So for anyone who hasn't been you know, avidly listening and taking notes on every single episode we've done since 2017, can you tell us what the Brouwer fixed point theorem is?

PP: Yeah, so I'm just going to state it for the closed disk because that's the only context that I'm going to talk about it in. But basically, if you take in the plane in our two if you take the closed unit disk, then the theorem says that every continuous map from the disk to itself necessarily has a fixed point. So should I go into detail about what a continuous map? Would that help?

EL: Yeah. Or at least intuitively.

KK: Sure.

PP: So I actually did listen to a lot of the previous podcast episodes while I was preparing. And I like this idea of if you take the unit desk, and you, like, kind of shake it around a little bit, and everything kind of moves in a nice smooth fashion where things don't get sent, like, really far away — so if in a little neighborhood, you’re wiggling, one point is not just going to pop out and end up somewhere else, right? I like that idea of continuity. So if you're wiggling around the disk, the unit disk, and you use any continuous map, somehow one of the points has to stay fixed, so it gets sent to itself. And that's kind of surprising. It feels like if you just move things around enough, something, everything, should get moved off of itself. But in fact, that can't happen. So that's kind of my interpretation of Brouwer’s fixed point theorem.

EL: Yeah. And it's like I guess I always imagine it made of rubber or something. Because you are allowed to, like, stretch and smush a little bit. It doesn’t — because otherwise, you might think, Oh, the only thing you can do is rotate it. So of course, that central point will be fixed. But you could do a lot of other things.

PP: Yeah, absolutely.

EL: And fix some different point.

PP: Yeah, so I think Evelyn has a great point, like, you can spread things out, like you're making it out of like stretchy fabric or material, you can spread things out in one part of the circle, in the unit disk, and then, you know, string things together in another part and that's okay. It's like, you know, just kind of smoothly moving around is the way I think about it.

KK: Yeah, yeah. But something stays put.

PP: Something stays put, which is kind of strange sometimes, actually. And there's like, so many proofs of this theorem, I feel like, and so many different perspectives for proving it. But I do have a favorite proof of that, actually.

KK: Okay, good. Let’s hear it.

PP: So it's unfair, because it uses some algebraic topology. So o be able to get to this point in this in your math life, where you're like, Yeah, this is the proof I like the best, you have to learn some algebraic topology. But essentially, the idea is that when you're in topology, in the field of topology, you're trying to understand when two objects that are made out of bendy, squishable material that you can stretch and shrink, when two of those are really the same. So if you have, let's say, a circle, or a really oblong wiggly circle, those two are the same. It doesn't really matter if one is really beautiful and perfectly symmetric. It's really the same space in topology. So two things that are not the same topologically are the closed unit disk, and just the outer boundary, which is just a circle. Okay, so there's an a thing called an algebraic invariant that you can compute called the fundamental group, that tells you that topologically, formally, these two spaces really aren't the same. And essentially, there's a proof that says, If there wasn't a fixed point, then you could basically take the entire closed unit disc, and shrink every point in the desk to the boundary. This is called a retract. You’re basically saying like, I'm going to retract the entire closed unit disc to just the circle. And retracts are supposed to give you the same fundamental group. And you already know that those two things aren't the same. And so that's my favorite version of this group. And I can slow down on any part of that if you'd like more details.

EL: Yeah, that's really nice. Well, I think maybe a good way to see this is like, you know, that example of turning the circle around, you know, like a record spinning on a record player or something. Like if you took away that central point, everything else can move. And you can also imagine pulling that rubber all the way to the edge, making it into a bike tire or something else like that. (Which is actually topologically different.)

PP: Right, but as soon as you puncture it. So Evelyn's basically saying, let's just take out the center point. But what corresponds to the origin in R2? Well actually, once you do that, there's no contradiction that you derive, right? You can have every point moving. And in fact, that punctured disk and the circle are the same topologically. That retract that you can use to just pull everything to the boundary shows you, actually, that they're the same topologically. So it's just that one— it’s amazing how much like one point can make such a huge difference, right?

EL: Yeah.

PP: Adding in that one point. But yeah, so that's my favorite proof. It's fancy in some ways, but once you know the basic material that leads up to it, it's like a three-line proof, right? Which is kind of incredible.

EL: Yeah, but it's maybe a little bit like, what is the phrase, like using a sledgehammer to kill a mosquito.

PP: Oh yeah.

EL: Once you’ve built all of this fundamental group, then sure, you could just whack that.

PP: Yeah. And it's so funny because in math, typically I am the opposite of a hammer-striker, right? I never use the hammer. I want to understand the nitty-gritty of why you can just explain this using elementary math or something like that, right? But for some reason, when I saw this proof in, like, Hatcher’s algebraic topology book, I was like, Oh my gosh, that just like makes perfect sense to me, like I totally get why now. So it definitely is one of those use a hammer use a sledgehammer to kill a mosquito type of approaches.

KK: I’m actually teaching algebraic topology this year, and so I that is the proof I use for the Bouwer fixed-point theorem. But I use the sledgehammer to prove that every polynomial of odd degree with real coefficients has a root. And I use to use the Lefschetz fixed point theorem to do it. Let’s use the biggest sledgehammer we can find!

PP: It’s awesome. I mean, honestly, if you want for this podcast, if you want to talk about a fun theorem that's called theorem at the end, right? Not just some result, but that's actually named, a lot of the ones you come up with in topology are the fixed-point theorems, right? And I was like, Oh, this one's actually my favorite. And there's a reason for it. So yeah, that makes sense.

EL: So what else do you love about this theorem?

PP: Yeah. So this actually was inspired from talking to a few grad students the other day, but I realized that, you know, I gave them this task, which says, Can you classify all of the isometries of hyperbolic 2-space. Now, that's already a fancy sentence to say. So I can break down all of what that means. But in fact, one of the key ingredients for the approach that is my favorite to solving that problem is to use the Brouwer fixed-point theorem. So I can start off by talking about what hyperbolic space is, and like what metric spaces are. And from there, I can explain what an isometry. It’s kind of similar to a continuous map, but it has a lot more structure and preserves a lot more structure.

So let's start off with just hyperbolic space, shall we? Okay, so the way I think about anything that is a non Euclidean geometry, which hyperbolic geometry is one of those, I have to start thinking about, well, Euclidean geometry first, right? And Euclidean geometry, when I think of that, I think axioms, right? There's Euclid axioms and they're written down. You don't need to know what they are. But the last one is the one that people started saying, let's try to break it and see what happens to these models of geometry that we're sort of studying, right? Like, could we come up with a different interpretation than just Euclidean space. And so if you break the parallel postulate, there's a few different types of geometries you can get that satisfy all the other ones, but they don't satisfy the parallel postulate. And hyperbolic geometry is one of them. So what is a geometry, right? It's a space, like we just talked about, let's say the closed unit disk. And to me geometry, you're studying rigid things like distances, angles. And so you want to have a notion of measuring distance on whatever space you choose. So since we're going to talk about the power fixed point theorem, of course, my space is going to be the closed unit disk. In fact, I'll just start off with the open unit disk for now. So let's just get rid of the boundary. So if I start off with the open unit disk, that is my space. And there is a way of measuring distance on that space. So you can say, oh, put in these two points, I want to know the distance between them. There's a way of measuring distance on it where if you want to go from the center point at the origin, out to one of the boundary points, let's say just (0,1), or (1,0) in the plane, it actually takes you an infinite amount of distance to get there. Okay, so in hyperbolic space, this model of hyperbolic space, which is called the Poincaré disk model, the boundary of the disk is sort of off at infinity. And as you get close to that sort of boundary at infinity, points are getting really, really, really far away. That's what it means to get, you know, closer and closer to infinity, is distances go really big. And so that's the idea of what the Poincaré disk model of hyperbolic to space is. And, of course, if it weren't a podcast, I'd be showing like tons of pictures right now.

EL: Yeah, it is quite attractive. It's just a lovely, appealing model.

PP: Yeah. And like, you can look up all these amazing pictures by Escher. There are these famous paintings where Escher uses the upper half plane model or the disc model, and shows how, like, a bat, or whatever the figure is that he uses to tessellate, a bat of the same area drawn in different parts of the hyperbolic plane can look to our eyes, very different, right? And that's again, coming back to this notion of as you move out towards the boundary of the unit disk, distances are getting really big so the bat would have to look really small to your eye to to have the same area as a bat in the center of the disk. So I highly encourage listeners to Google just hyperbolic space or Escher's hyperbolic paintings, right. And you'll come up with so many things.

EL: Yeah, well, fun fact is that my Twitter profile picture is a tiling of the hyperbolic plane with the Poincaré. with like, a picture of me in it.

PP: Yeah. So I love it. I know Evelyn really loves all of these, like, hyperbolic geometry, topology type thing. So yeah, that's also partly why I chose this topic.

EL: Yeah, you’re definitely speaking my language.

PP: Okay, so that's the idea of what hyperbolic space is. There's so many more things you need to do to sort of gain the intuition of what it feels like to live in hyperbolic space, right? And those are the kinds of things that you build over years of studying it in your life. But the real thing I want to talk about is isometries of the hyperbolic disk model.

So what is an isometry? It's a map of the space to itself. So kind of like that jiggling that we were talking about. But where all of the distances between any two pairs of points remain the same. So a great example that Evelyn already talked about was this rotation around the origin, right? If you rotate around the center point, all of the points, pick any two of them, they actually see the same distance apart. And that's not an easy thing to see, partly because I never told you how to measure distances, right? That completely relies on how we decided to define that, which is the metric on the space, that notion of measuring distances. But if you knew it, and you wrote it down explicitly, you could actually calculate that that rotation is distance-preserving in the Poincaré disk model of hyperbolic space. So that's the idea of what an isometry is.

Okay, so now we want to try to get fancy. And usually, what you do when you have a space is you say, I'm going to try to understand all the isometries of this space. So where does this notion come from? In topology, if we're not talking about geometric structure, and we just kind of care about a space and it's all blobby, and can be stretched and shrunken, we think about all of the symmetries, right? All the topological symmetries of the space. When we're talking about isometries, what we're actually talking about is geometric symmetries of the space, all different ways of moving around this space, where distance hasn't really changed, so you're preserving the geometric structure. It turns out that if you take all of the isometries, you end up with a group. It has a really nice structure, you can compose two of them. But that's not really even important for today, what you really care about always is, can I classify all things of this type? And there are infinitely many of them. It's really hard to classify things when there's a whole infinite set.

EL: Right.

PP: I’m not just putting things in bins, like these are red marbles, and these are blue ones and these are green ones, right? So amazingly, it actually turns out that isometries of H2, the hyperbolic plane, or disk model, only fall into three flavors. They're either elliptic, which are very similar to the rotations that we talked about, or they are the rotations basically. There are parabolics, and there are loxadromics, or sometimes called hyperbolics, which doesn't make sense, because it's confusing. When you're talking about hyperbolic space, calling something a hyperbolic isometry, when you mean a certain type is confusing. So I'll just call it loxadromic, right? And so there's a few things you need to know, like what could the isometry even look like? How could I possibly get equations of these isometries? So you have to work a little bit. But it turns out that you can write down a general formula for a generic isometry of the Poincaré disk model, or the upper half plane model of hyperbolic space. So just as an example, I'm going to switch models. And in fact, when I say I'm going to switch a model of hyperbolic space, what I mean is, I'm just going to go to a different space with another metric on it, but it ends up being the same geometrically. There is a nice map between the two of them where all the geometry is preserved. So I like sometimes the upper half-plane model, because it's really easy to like write down what the isometries are.

EL: Yeah.

PP: So what I'll say is, and I, again, I'd write this down on the board, if I could. But imagine it, close your eyes and imagine it. I'll take four real numbers a, b, c, and d. And all, really orientation-preserving, but let's sweep that under the rug. Well, orientation-preserving isometries of the upper half plane model look like (az+b)/(cz+d), where z is a complex number. Okay, the criteria you need to make sure you satisfy is that ad−bc is 1. Okay, there's another interpretation actually in terms of matrices. Put those four things into a matrix [Editor’s note it’s hard to write a matrix in Word. The top row is the numbers a and b. The bottom row is the numbers c and d.] Well, you're seeing I want ad−bc to be equal to 1, that's SL(2,R). If you multiply the top and the bottom by negative one in the top of the bottom, you're not changing the transformation. So you have to mod out by plus or minus the identity. So we're really looking at PSL(2,R) protect devised special linear space.

EL: Yeah. I actually I do love that because the first time you see these hyperbolic isometries, the az plus b, over cz+d,, and then they're like, oh yeah, ad−bc, you just have this almost spidey sense tingling of like, Okay, that's like a determinant. Why are we doing this? There must be some relationship here with linear algebra.

PP: Yeah. And your spidey sense is totally on point. So I love I love that connection. And I sort of, you know, I always wonder how much detail to go into with these things, since I'm not writing at the board. But I love that, right? Because anybody listening to the podcast should be like, Okay, wait, that's the determinant, just like you did. So, yeah, so there's a little bit of complex analysis that goes into deriving the fact that these are actually isometries, that they map the hyperbolic plane to itself and so on. But once you have a nice general formula, you start to use Brouwer’s fixed-point theorem. So I, this is part of my life, I go back and forth between the models all the time. So I'm sorry if this is getting annoying, but I'm going to go back to the disk model for a second.

Okay. And well, let's think for a second. Right now, what I have is the open disk. And I have a boundary at infinity. And often, when you're working with spaces, you can sort of complete up the space by adding in the boundary. Okay, this is the sort of completion of the Poincaré disk model of hyperbolic space. So now, if I take the hyperbolic disk model with its boundary, I have the closed unit disk, right? And an isometry is way better than a continuous map, but in particular, it is continuous. And so Brouwer’s fixed-point theorem says no matter what map I'm talking about from the disk to itself, the closed disk to itself, I have a fixed point. So if you have a formula, and you know there has to be a fixed point, you should try to solve for those fixed points, right?

KK: Right.

PP: And so I'm going to pop back to the hyperbolic plane model, because that's where we have our nice formula. So I'm just going to try to solve az+b/cz+d=z, right? This is a function, your input is z. Even though it seems weird, because you're like, wait, I'm multiplying by a, then adding b, dividing by… this is really complicated, right? But there are certain points for which you put it in, and depending on a, b, c, and d, you spit out the same number, right? The same complex number. So this is what I tell people to do. This is what my students did the other day. They said, How am I supposed to approach this? And that's the classic proof, you use the Brouwer fixed-point theorem, and you start to solve for az+b/cz+d=z. The key point that you use after you're doing all the algebraic manipulations is that ad−bc is always 1. So a ton of stuff cancels out as you're solving, right? It becomes a very nice equation. But what am I doing, actually? If I cross-multiply, right, multiply by cz+d on both sides, and then move everything over to one side, I'm getting a degree two polynomial in the z coordinate. How do you solve any degree two polynomial? You use the quadratic equation! So the bane of some people's existence when they’re going through high school, they’re like, I'll never use this? Well, first, if you become a mathematician, you're definitely going to use it.

EL: Yes.

PP: But I love when things that you learn when you're so mathematically young still come into play when you're doing really sophisticated math. I think that's really cool. So okay, we write down the formula. And basically, what it comes down to is what's underneath the square root, right? What is the discriminant? Because when you take the square root of a negative number, you get imaginary things, right? Imaginary numbers, complex numbers with non-trivial imaginary part. If the thing under the under the square root is zero, well, you're just getting one root, right? And then if the thing under the square root is not zero, but it's positive, you're just getting two real roots. Okay, so let's think about that. We have three categories: one real root, two complex roots, or two real roots, right?

The thing is that in the upper half plane model, the thing I never talked about, was that you need the imaginary part to be positive. Okay, so the actual categories are one real fixed point, one complex fixed points in the hyperbolic plane, or two real fixed points. Really, when you're going between the models, what ends up happening is in the upper half plane model, the boundary at infinity is the real number line, it's the x-axis in the complex plane. So that ends up being the boundary of the circle, the unit disk. So we're talking about three cases: one fixed point on the boundary, two fixed points on the boundary, or one fixed point on the interior, which is the complex one. And that's it. That's literally the classification. Because if you have anything more than three fixed points, you can show that your transformation was the Do Nothing transformation. It's the identity.

KK: Right.

PP: So it's not just that the Brouwer fixed-point theorem tells you that you can find fixed points, that there is one, you can actually classify all of the isometries based on these three categories, which I think is like, just incredible. And if you want, I can give you a little bit of a geometric interpretation of what the three isometry is the classes of isometry.

KK: Sure!

EL: Yeah, but I would like to pause and say one of the last classes I taught when I was at the University of Utah was an undergraduate, like introduction to topology class. So we touched on some of this a little bit, and it's like, I kind of want to go back and teach it now and use this for that part of it.

PP: Yeah, it's kind of amazing. Even though it has some sophisticated things going on, you can tell some advanced undergrad students about this stuff, and really show them a lot of beautiful pictures. So when I was in Santa Barbara, I did teach a non-Euclidean geometry class. And, you know, of course, I have to do all the other geometries justice as well.

EL: Eh, do you really?

PP: I know, I know. But I mean, when you get to hyperbolic geometry, though, it's kind of like it's limitless, right? The amount of stuff you can kind of tell and teach students. So I do love that aspect of it. But it's not such an advanced sort of theorem that you're just like, What is this using? Where does it come from? It's like, you need to know the Brouwer fixed-point theorem, you need to know the basics of the model of hyperbolic geometry you're thinking about, and that's basically it, right? Okay, so let's talk about the three sort of classic, I guess I would say, the canonical examples that people give for each of the three categories. So for anybody who's listening that's really into math, and that knows a little bit about algebra, every isometry is actually conjugate to one of these. But they're like the model. They're sort of like, the best-behaved one in each category. What do they look like?

Okay, so for the elliptic ones, we're going to start there, because Evelyn's already told us what they look like, right? In the disk model, they’re all just rotations about the origin. Technically, the fixed point could be anywhere, and it's still kind of a rotation around that fixed point. But again, up to this conjugation, you can move that fixed point to the origin. And then so now you're really just asking if I want to just understand the canonical form of this, I just am going to try to understand isometries of the disk that fix the origin. And I'm going to get that these are rotations, right? You can actually go through and derive the formulas, you know, you say I solved for the root, it's complex. Here it is, let me write down what this might mean. And you can really see which Möbius transformations you're talking about. But that's the canonical way that we think about elliptic ones. And I think the elliptic word has to do with that like sort of rotation, right? But don't quote me on that, because I'm not good at words. I'm good at the math and pictures, but not great at words.

Okay, so let's go to something more interesting. So what are the parabolic ones? These are the ones where you have one fixed point on the boundary. And I'm going to go ahead and use the upper half-plane model again, because they think it's a little bit prettier to see the parabolic one there. So I didn't really say what the upper half-plane model was. So let me go ahead and do that. So the upper half-plane model of hyperbolic space, is you take the entire plane, but then you only think about the upper half part, right? So where the y-coordinate is strictly greater than zero. In the disk model, when we approached the boundary, distances got really big, right? What is the boundary for the upper half plane model? It's the real axis in the complex plane, so the x-axis, and the point all the way out at infinity in the plane itself. So in R2, there's actually an infinity, the one point at infinity, that has to get thrown in there. Okay, so what we can do is say, let's have fun and say that the fixed point in the hyperbolic plane model is infinity.

KK: Sure.

PP: So I'm going to take a straight line, it goes from zero, the origin, (0,0) in the complex plane, and it's just going to go straight up to infinity. Okay, if infinity is fixed, and you have to map these sort of straight lines to straight lines, what you can come up with I mean, I'm waving my hands here, you have to actually do some like algebra and manipulate everything and sort of make sure you're reducing this the right way. But what it turns out to say is that these are all translations. So the maps az+b/cz+d, well, c is 0, d is 1. And it's really just z+b, or something like that. Okay, it's like a translation by some by some number. That's approximately what a parabolic transformation looks like. So translation is something we understand from Euclidean geometry, right? It's just that the way that it affects points in — so what do I want to say? Translation in the Euclidean plane, we understand with that metric. We have the upper half-plane with a different metric. It turns out in this case, that translation is still an isometry. But you have to remember, distances look different. So when you're going to see parabolics in the disk model, things get a little bit more complicated. You have to talk about things called horocycles. And that, I would say, it's better to just look up.

EL: Yeah.

PP: This is where a picture would be very, very useful, right? Okay, and now, the queen of them all is the loxodromic isometries. So this is where we start to see a lot of the connections between hyperbolic geometry and dynamics. So when you have a loxodromic isometry, there are two fixed points on the boundary. With a little bit of work, again, what you can see is that if you take those two fixed points on the boundary, there's actually a, sort of like a shortest line segment going from one to the other. Lines look different in hyperbolic space because the metric is different. But essentially, the way that this isometry acts on the disk model is that one of those fixed points acts as a source. The other one acts as a sink, and everything in the disk model is getting taken away from the source and being pulled towards the sink, actually along the axis — that geodesic axis that you have between the two fixed points, it’s acting as translation along that. So this is a phenomenon in dynamics, more generally called north-south dynamics. You have this source and a sink, and things are moving from the source to the sink in a north-south sort of way. And, yeah, that's my favorite one. Of course, it's the most complicated one. It's the one that comes up the most when you're studying surfaces. So yeah, that that sort of is my — I know, that's just my favorite. That's my favorite type of isometry. I think it makes sense when you're working in hyperbolic geometry, because it comes up all the time.

EL: Yeah, well, that's so fun. So another thing we like to do on the podcast is force our guests to pair their theorem with something in the real world.

KK: “Force.”

EL: So what have you chosen as your pairing today? “Invite” our guests to do that.

PP: Yeah, I actually didn't think about this one so hard, because I have a natural pairing in my life, which is climbing. So I love to climb. So I got into rock climbing when I was a postdoc in Santa Barbara.

KK: Good place.

PP: In fact, one of my most favorite, yeah, it is a good place for climbing. And it just so happened that my mathematical grandfather, Mike Freedman actually, is in Santa Barbara as well. He is a very good rock climber. And he took me on my first like, sort of ropes climbing outside adventure. It was a lot more intense than I thought it was going to be. But he's an intense guy. So I kind of knew what I was getting myself into. But it was such a moment of growth for me. And there are a lot of mathematicians that are attracted to climbing. And there's a reason, right? It's problem-solving. But like, with the physical component put in there, right? So when I'm not problem-solving in my office, or at my home office, I'm usually in the gym, problem-solving climbing problems with my friends.

EL: Yeah, well, and that's, I have, I've like, done one little rock climbing thing. I've never done it, but they actually call them problems, right? Like, figuring out a route is called a problem.

PP: Yeah, absolutely. Yeah, it's very heady you know. Of course, if you ask a mathematician to choose a physical sport to get into, they're like, oh climbing, then I can still use my brain all the time.

KK: When I was a postdoc, I did some. There was a climbing gym in Evanston. And yeah, and you're right. It's very good for your brain. And I always thought that too, but man, my fingers just, it hurt so bad. And I was mostly a cyclist at the time, which was also good because it doesn't require you to really use your brain, like riding a bike is so automatic that I could go out for a ride and think about math while I was riding. I mean, I know you want to get away from the math sometimes, but it was actually a good thing to do for me.

PP: Oh, yeah, no, I totally agree. I was really into running for a while. And I loved that sort of time to decompress and add space to your brain, right? So with climbing, I don't have that sort of flow, I don't reach flow quite as quickly as I do in other exercise. But it's very good for getting out the work stress, I’ve got to say. You just, like, work really, really, really hard physically. And it is very rewarding. I think it's the type of thing where, just like math, in my opinion, you know, “natural ability” is not actually the thing I think that determines how well somebody does in math over time. I think it has a lot to do with how hard you work, right? And if you're in training, taking care of your body, learning things, watching climbers, being very observant, you tend to pick it up pretty quickly. And there's a lot of big, burly guys in the gym who struggle, and then they're sort of surprised when, like, petite women get up and sort of just crush the problem, which I mean, I don't mind that. You know, I think it's a good lesson for everybody. So yeah, I also think, you know, for me, climbing and math are both very dominated by certain genders and races, it's a very white dominated sport, it's a very white male dominated sport, often like math is. Math spaces are very dominated in those same ways. And a lot of my work in math has been to promote diversity and equity and justice, really, in my math communities wherever I exist. And it turns out that that extends to my climbing communities as well, because I co-founded a group called Color the Wasatch, which is an affinity group for people of color climbers in the Wasatch Valley. So it's been really great, and I've learned a lot from that. And it's very similar. When you have people around you that have similar experiences to you, it can be so enriching, and it can be such a relief to just sort of feel yourself relax, and feel just comfortable in your own skin wherever you might exist. And that's a big one that I sort of learned very recently. I’ve sort of always been into sort of activism in the math community. But this was a real first big thing I did in my community outside of my work as a university professor, and it's incredibly rewarding. And it's kind of taken off. It's been great.

KK: Cool.

EL: And we also like to give our guests a chance to talk about anything, you know, you'd like to plug. And actually, I think I just saw on Twitter, some people tweeting about the Roots of Unity conference that is happening. Is that — I don't know if conference is the right word for it — happening this summer. And I think this episode will probably be published in time for people who learned about it to apply,

KK: I think next week, this will be out.

PP: Yeah, that’s great. Because the application deadline for the Roots of Unity workshop is actually February 15, which is great. As you said, it's going to be out in time, the podcast will be out in time. So the Roots of Unity workshop, I am actually co-organizing with phenomenal women in math who I really looked up to, actually. And we designed this workshop to sort of support people who might not see people that look like them at their home institutions early on in their graduate career. So there's amazing programs out there, like EDGE, the EDGE network for people going into grad school, there are amazing research-focused conferences, like the Women in Numbers group, we actually have a Women in Geometry, Groups and Dynamics group now. And we felt that there was a sort of gap in between. And it's very hard, speaking from personal experiences, to be the only woman of color in your department or in your graduate cohort. And so we're really aiming to support anybody who would benefit from this kind of kind of program, but especially gearing it towards people who don't get that same training or preparation or encouragement at their home institutions, especially women of color. So it's going to be a professional development and research development workshop. And one of the things that we're doing is we're sort of helping grad students learn how to read papers, because like, gosh, we just are given papers and said, Go read this, right? And there is a skill to reading a paper well, I think. And so that's one of the big things that we're focusing on, is that transition between early coursework that feels very much like an extension of the undergraduate curriculum, and then into this whole new world that really requires a big pivot mentally, of reading papers and coming up with research problems and having a good network of support and collaborators as you do that. So that's occurring in June of this year at the IMA, so the program is you know in the process, the schedule is in the process of being set. But yeah, applications are open, and we would love to see lots and lots of applications.

KK: Excellent. All right. Where can we find you elsewhere online?

PP: So I am on Twitter. What is my handle? Evelyn, do you know what my handle is?

EL: Um…

PP: I think it might be priyam886. [Editor’s note: It is!]

EL: Maybe?

PP: That might be it. But you know, y'all can post it if you want on the website, eventually. My website actually has a link to the Roots of unity workshop. So that's patelp.com. And I also have a little page actually about Color the Wasatch there as well. So if people are interested in the climbing aspect of things also, that's that's all on my website.

KK: Oh, cool. All right. Well, this has been a lot of fun. I always like learning new things about the Brouwer fixed-point theorem. You know, I've taught complex analysis, and so I've thought about — I think I've shown my students sort of how linear fractional transformations, which are these isometries, are acting on the upper half plane, but I never thought about it in terms of isometries. So this is good for the next time I do it.

PP: Yeah, absolutely. So those linear fractional transformations are like the basic ingredient you need, right? Once you know those, you say which ones are actually going to map the upper half plane to itself, and then which ones are going to be distance-preserving, and everything falls out from there, so it's really nice.

EL: Yeah, this is a lot of fun.

KK: Yeah. Thanks.

[outro]

On this episode of My Favorite Theorem, we're revisiting the popular Brouwer fixed-point theorem with Priyam Patel of the University of Utah. Below are some links you might enjoy after you listen.

Patel's website and Twitter profile

Our previous episodes about the Brouwer fixed point theorem with Francis Su and Holly Krieger

A pdf of Allen Hatcher's algebraic topology book (available, legally, for free!)

The Lefschetz fixed-point theorem

Douglas Dunham's page about Escher and hyperbolic geometry

A blog post Evelyn wrote about putting pictures into the hyperbolic plane

Information about the Roots of Unity workshop (application deadline: February 15, 2022; if you're listening to this in later years, poke around and see if it's happening again!)

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