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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, 10:30 pm. 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