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 Instagram account, clearly.

EL: We Rate Theorems.

RW: 10 out of 10.

KK: That’s right.

EL: Yeah. So another thing we like to do on this podcast is we ask our mathematicians, as if it weren't hard enough to choose a theorem, to choose a pairing for their theorem. You know, be it art, music, food, wine, any delight in life. What have you chosen to pair with the Poincare-Hopf [index] theorem?

RW: So I think I might have actually started with the food and then went back to the theorem. But there was this example that also really like captivated me, captured my attention as a student, and that's the hot fudge flow. So it's a vector field over a surface. And so the idea is to imagine a ball of ice cream, and you do what you do with ice cream. You take the hot fudge and you drizzle it on top of the ice cream, and you try and hit the center. And then what happens to the fudge? It sort of, you want it to expand and wrap around and then come back as a source and drip out of the bottom, if this was, you know, suspended in the air. So that's the hot fudge flow. And you can compute the sum of the indices of the critical points of that vector field, and it'll match of the Euler characteristic of the sphere. So the pairing is a hot fudge sundae.

KK: Okay.

EL: Excellent.

KK: That’s exactly perfect. Yeah.

EL: Of course we have to ask. What is your number one ice cream flavor for a hot fudge sundae?

RW: I was actually hoping you wouldn't ask that I'm the most boring ice cream person. Vanilla is my favorite.

KK: Look, you can't go wrong.

RW: Yeah.

EL: I will say, it is very unfair to vanilla that it has become this word in in our our language, for something that's boring, or pedestrian, because, like, it is an incredibly complex flavor, like, if you get an actual vanilla bean, it's like, there's so much going on. And I don't, I don't know the the history of how vanilla became “boring,” but, you know it is, it is anything but boring. Justice for vanilla.

KK: And so complicated to grow, right? It only grows in very specific places.

EL: A few places. And it’s expensive. Isn’t it, like, the second or third most expensive spice after definitely saffron.

KK: Saffron, I think, is number one.

EL: Maybe something like cardamom. Cardamom is up there too, I think.

KK: It’s not cheap.

EL: No hate to vanilla.

KK: It’s not cheap, because one little pod of vanilla, one little pod at the store is like, $4 or something. You know, it's like, it's really, really absurd. But it's an orchid, right? I mean, so, I live in Florida. We can actually get orchids to grow here, but it's still not easy.

EL: Right. Do you know if the vanilla orchid can grow there?

KK: I doubt it. If it could, they would be cultivating it left and right. I actually think it's too hot here. It's not humid enough, somehow, yeah, so some orchids will work.

EL: Because I think, like Madagascar, Tahiti and maybe Mexican? Is it grown in Mexico also?

KK: I think there might be some spots in Mexico, yeah, like, maybe in southern Mexico, Oaxaca or something. But, yeah, anyway, okay, all right, this is not a vanilla podcast.

EL: Yeah, three mathematicians speak extemporaneously on vanilla cultivation. Tune in next week for the exciting conclusion.

KK: That’s right. Yeah, so Robin, we always like to give our guests a chance to plug anything they're doing. Where can we find you online, what sort of, any big projects you're working on that people might be interested in, or anything like that?

EL: Or have done recently?

RW: Yeah, so I have a really bad online presence right now. At the moment, the website could use some dusting off. But one of the projects that I'm working on that I'm excited about right now is in math education. So we've been making videos of Black mathematicians talking about their work, their educational experiences, and giving advice to young people. And so these are for K to 12 students, but also, I think they're going to be of interest to lots of folks. And so we do have a website, but the URL isn't in on my mind to pass on to you right now. Maybe I could share it with you afterwards.

EL: Yeah, we'll, we'll get that from you and put it in the show notes, so it’ll be easy for people to get.

RW: That’ll be fantastic, but thanks for letting me make that plug.

EL: Yeah, well, and I remember seeing recently, you did a talk at the Museum of mathematics, right with and was that a conversation with Ingrid Daubechies?

RW: It was so much fun. It was a conversation.

EL: Do you know if that is available in video form somewhere? I meant to look for that before we got on. But of course, I didn’t.

RW: You know, I had the same question cross my mind as I was approaching this as well. And I think it might be available, but it could be, like, for museum members.

EL: Okay.

RW: I need to check.

EL: Yeah, I remember seeing your saying your name in my inbox, and thought, well, that's cool. And you've also, do you mind talking a little bit about the Algebra Project and and Bob Moses?

RW: Sure.

EL: Because I know that's something that you've — I know I've talked with you about it before, and Bob Moses passed away around the time we were putting that book together.

KK: Yeah, it was a couple years ago.

EL: So, yeah, do you mind talking a little bit about it? I thought it was really interesting.

RW: Yeah, sure. That's something that I could talk about for a long time. So just just check me if I start going on too long. I met Bob Moses as a graduate student, and I think I was kind of wrestling with some identity issues about my interest in math, but also, you know, interest in social issues, and kind of wanted to make a difference in my community, and trying to figure out how these two things came together, and if I was doing one, did that mean that I couldn't do the other? And so I came across his book, Radical Equations. It was about math literacy and the civil rights movement, and he brought his work in the civil rights movement together with his work as a math teacher in in Boston, and it really kind of spoke to me. And so I got a chance to meet him, and ended up staying connected with him and Ben Moynihan at the Algebra Project, and so I worked with them in different ways, attending teacher professional development. We helped spearhead an effort in Los Angeles, where the Algebra Project curriculum was used in four different high schools supported by an NSF grant. We had a second effort here, where we've been running some summer programs for students through the Algebra Project. And recently I joined the board of directors, and so I’ve been involved with them since I was in my 20s, and so it was a real honor to be asked to kind of be a part of that, that part of the leadership for the project.

KK: Bob Moses really, really impressive man. And then, this idea that you know, that every you know, things are really important. You know, education is so important to advancing, you know, civil rights and things like that. I mean, Bob Moses was really spectacular. Our listeners, if they don't know much about him, should just look him up, because he was really impressive and influential, and by all accounts, a very kind man. Like I said, I've never met him, but just a really great human being.

RW: And I think what people, a lot of people don't know about him, is he was a math teacher first, he was teaching math and and the sit-in movements happened, and he got drawn into the sit-ins. And then when, when things kind of settled down, he went right back the math classroom. And so kind of think of him as one of us.

EL: Yeah. I think reading, reading radical equations a few years ago, I remember, you know, it's just like sometimes when you're a mathematician, especially if you're really involved in the academic math world you get so, you know, drawn into these very abstract questions that you feel like have nothing to do with, you know, anything resembling reality, or anything resembling social issues, and just the way that he writes about how access to good math education, like is so important for people to be prepared to, you know, have careers that they want, be able to have financial stability in their lives then, and just the, you know, the doors that it opens to have access to math at, you know, the middle school, high school level, really reminds you as a mathematician, like, oh, yeah, we are part of this society.

RW: Yeah, that's right, and we do have a really important role to play. That's one of my biggest takeaways from him that as mathematicians, we do have a really important role to play in how this whole thing turns out.

EL: Well, thank you so much for joining us. Really great to talk with you again.

RW: Thank you so much.

[outro]

On this episode of My Favorite Theorem, we had the pleasure of talking with Robin Wilson, a mathematician at Loyola Marymount University, about the Poincare-Hopf index theorem and the importance of math education. Below are some links you may enjoy after the episode.
An interview with Wilson for Meet a Mathematician
More on the Poincare-Hopf index theorem
The 2021 SLMath Workshop on Mathematics and Racial Justice and its follow-up, to be held in May 2025
Storytelling for Mathematics
The Algebra Project
The 2025 Critical Issues in Mathematics Education workshop, to be held in April 2025, focusing on mathematical literacy for citizenship