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Evelyn Lamb: Hello and welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, and I am joined, as always, by our other host. Will you introduce yourself?



Kevin Knudson: Hi, I’m Kevin Knudson. Yeah, I will. I'm Kevin Knudson, professor of mathematics at the University of Florida. It's been a while.




EL: Yeah.




KK: You know, I've actually gotten a few emails from our listeners saying, hey, where, where the hell is My Favorite Theorem? And I always have to reply, you know, we're trying, but everybody's busy.




EL: Yeah, and we're busy.




KK: And we're busy. But here we are. We are still committed. We're still into this. This is — we're going to go to year eight soon enough, which is kind of mind-blowing.




EL: Yes.




KK: I had less gray and more hair in those days. So here we are.




EL: You’re as lovely as ever.




KK: As are you, Evelyn, as are you. Yeah, although I kind of missed the green hair. I wish you would start coloring your hair again.




EL: Yeah. Honestly, like many people, during COVID, I just lost my ability to put forth more than minimal effort into my appearance.




KK: Yeah.




EL: Sorry, that sounds like a smear on other people. I just, I've heard this from other people. I'm not judging based on what I see from other people. But yes, it's just, like, the bleaching, the dyeing, it just, it's more maintenance than I'm willing to put forth right now.




KK: It’s a whole thing. And as one gets older, you just go, who cares?




EL: Yes. But anyway, we are delighted today to have Robin Wilson on the show. Robin, will you introduce yourself? Tell us where you're joining us from, and a little bit about yourself?




Robin Wilson: Yeah, hi everyone. So I am joining you from Los Angeles, currently in my office at Loyola Marymount University, where I'm a professor in the math department. And so I have been a professor since about 2007 and work in topology and math education. It's great to be here today.




KK: What part of town is Loyola in? I don't think I actually know where that is.




RW: Loyola Marymount is right on the coast, near LAX. So if you've ever visited us here and flown into that airport, then you've flown right over campus, and depending on which runway you land, you can actually, like, see the fountain.




KK: Okay. I'll be flying through LAX in December. I will try to take a look.




RW: Come say hello, yeah.




EL: And I have to say, if we were broadcasting a video of this, you have one of the best backgrounds, the beautiful bookshelf, and then the chalkboard behind you has the appropriate level of mathiness. So our listeners can't enjoy that. But I will say that, you know, it looks very math professor.




KK: It does. It could actually go in that book of math chalkboards. Have you seen this? What's it called? I can't remember. Anyway.




RW: I’m so honored.




EL: I think it’s something like Please Do Not Erase, or something. [Editor’s note: Very close! It’s just Do Not Erase by Jessica Wynne.]




KK: Please Do Not Erase. That’s right, yeah.




EL: But yes, we are so happy to have you here. And Robin and I have actually worked together before a few years ago, on a publication for — at at the time, known as MSRI. It has a new name now, the SL Math Institute now, but we worked together putting together a booklet for the math and racial justice workshops that they did in summer 2021, was it? And it was such a delight to work with you, and I'm glad that lo these many years later, we can get to chat about your favorite theorem. So what have you chosen for your favorite theorem today?




RW: Okay, so the theorem that I've chosen for my favorite theorem today, this was a tough one, and so I chose one that was sort of important for me on my journey. So the Poincare-Hopf index theorem.




KK: Oh, I love this theorem.




EL: All right!




RW: So should I tell you more about theorem?




KK: Please.




EL: Please.




RW: Okay, so the theorem, so I'll state the two-dimensional version of the theorem, which is the one that I can kind of see. So you take a surface and you add a smooth vector field on the surface, and there's an extra condition that the vector field has only finitely many critical points. And so it turns out that the sum of the indices of the critical points of the vector field is equal to the Euler characteristic of the surface. And so the theorem connects these two different areas of math, sort of analytical and topology. And so it was one that I encountered kind of at the beginning of my journey as a topologist. And so it's the one that I picked to share today.




EL: I love that, and I've got to say, I love how many people on our podcast come in and say, it connects this and this. And like mathematicians, we just love these theorems that connect, you know, calculus and topology, or, you know, algebraic geometry and topology, or something like that. It's just something, mathematicians just all love these bridges here. And so I, yeah, can you talk a little bit? You said you encountered it first as a kind of starting out as a beginning graduate student. You know? Can you tell us, bring us back to that moment, tell us about its importance in your life?




RW: Sure. So I was an undergraduate student, and there was a graduate student that was a TA for one of my classes at the time, and I asked him about what type of math he studied, and he drew this picture of a trefoil knot on the board. So shout to Aaron Abrams if you're out there, Aaron. And so I asked him, you know, how could I learn more about this? And he pointed me to a class, an algebraic topology class, that was being offered the next semester. So I signed up for it, and the class was real challenging, but I really enjoyed it. And then the following semester, I got a chance to do a senior thesis, and so I ended up going back to the instructor for that class, and he said yes. So also want to shout out Charlie Pugh for saying yes. And he chose the project that I would work on, the proof of this theorem. And so I'd never, I'd never heard of Poincare. I didn't know much about the historical context of the theorem, but it was — looking back, it was pretty significant that I got a chance to try and think about how to come up with my own proof of something that Poincare had proven, who was right there at the beginning of the field.




KK: And it's a surprisingly tricky theorem to prove. I mean, I was reading this actually, last year sometime. I was trying to remember how this proof goes, because what's remarkable about it is, there's lots of things involved, right? Because no matter which vector field — and it could have, like you said, it can only have finitely many zeros or singularities, but it might have 10 million of them, or it might only have two, but that number, the Euler characteristic, of course, is an invariant of the surface. So no matter how you wiggle this vector field, if you create something, some weird swirl on one side, well, some sort of opposite swirl has to happen somewhere else, effectively, to cancel it out. It’s really remarkable.




RW: Yeah, yeah, that's right. So I was trying to remember what we did to prove the theorem. We, you know, we really were wrestling with a lot of the tools that we used in the course, like, a lot of the details about triangulations of surfaces and trying to find the right ones and paths of vector fields and things that build mature for a very young mathematician back then.




KK: Yeah, yep, yep. So. And I guess one of my favorite corollaries is what, it's the hairy ball theorem, right?




RW: Yeah, that’s right. I was thinking about this. It's got to have, like, one of the worst names in all of mathematics.




KK: Yeah.




RW: So yeah, you can't comb a hairy ball flat without a cowlick, also known as the you can't comb a coconut theorem.




KK: Okay, that’s a little better.




EL: Yeah.




KK: A little little less innuendo, right?




RW: Yeah. That was a close, a close second for my favorite theorem. It's nice that they're connected.




EL: Yeah. We were talking earlier a little bit that, you know, it is hard for people to pick favorite theorems, and, you know, what does it mean if a theorem doesn't immediately leap to mind as a favorite? I just want everyone to know that we are, we might be mathematicians, but we are very not pedantic and mathematician-esque about definition of your favorite. We are very loose, and you know, it can be your favorite of the moment or your favorite for what it meant for your mathematical development. And we’re very imprecise with the definition of favorite on this podcast. All favorites are welcome.




RW: I must say that I had not thought about this theorem for years, until I was asked to find a favorite theorem.




KK: Well, it's sort of like on Instagram, there's this account we rate dogs. Do you know this one? So anyway, basically the guy rates dogs, but the lowest possible rating is 10.




EL: 10 out of 10.




KK: 10 out of 10. Theorems are sort of that way too.




EL: I don't know. I would say, I'm not going to name any theorem names. But I would say there are some theorems that I would put below 10 out of 10. Cancel me if you must. But you know, I’m going to put that out there.




KK: This is it. So we have to start our new Instagr