My Favorite Theorem

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, a math podcast. I'm one of your hosts, Evelyn Lamb. I'm a freelance math and science writer, usually in Salt Lake City, Utah, currently in Providence, Rhode Island. And this is your other host.
Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics, almost always at the University of Florida these days. How's it going?
EL: All right. We had hours of torrential rain last night, which is something that just doesn't happen a whole lot in Utah but happens a little more often in Providence. So I got to go to sleep listening to that, which always feels so cozy, to be inside when it's pouring outside.
KK: Yeah, well, it's actually finally pleasant in Florida. Really very nice today and the sun's out, although it's gotten chilly—people can't see me doing the air quotes—it’s gotten “chilly.” So the bugs are trying to come into the house. So the other night we were sitting there watching something on Netflix and my wife feels this little tickle on her leg and it was one of those big flying, you know, Florida roaches that we have here.
EL: Ooh
KK: And our dog just stood there wagging at her like, “This is fun.” You know?
EL: A new friend!
KK: “Why did you scream?”
EL: Yeah, well, we’re happy today to invite aBa to the show. ABa, would you like to introduce yourself?
aBa Mbirika: Oh, hello. I’m aBa. I'm here in Wisconsin at the University of Wisconsin Eau Claire. And I have been here teaching now for six years. I tell them where I'm from?
EL: Yeah.
KK: Sure.
aM: Okay. I am from, I was born and raised in New York City. I prefer never to go back there. And then I moved to San Francisco, lived there for a while. Prefer never to go back there. And then I went up to Sonoma County to do some college and then moved to Iowa, and Iowa is really what I call home. I'm not a city guy anymore. Like Iowa is definitely my home.
EL: Okay.
KK: So Southwestern Wisconsin is also okay?
aM: Yeah, it's very relaxing. I feel like I'm in a very small town. I just ride my bicycle. I still don't know how to drive, like all my friends from New York and San Francisco. But I don't need a car here. There's nowhere to go.
EL: Yeah.
aM: But can I address why you just called me aBa, as I asked you to?
EL: Yeah.
aM: Yeah, because maybe I'll just put this on the record. I mean, I don't use my last name. I think the last time I actually said some version of my last name was grad school, maybe? The year 2008 or something, like 10 years ago was the last time anyone's ever heard it said. And part of the issue is that it's It's pronounced different depending on who's saying it in my family. And actually it's spelled different depending on who’s in the family. Sometimes they have different letters. Sometimes there's no R. Sometimes it’s—so in any case, if I start to say one pronunciation, I know Americans are going to go to town and say this is the pronunciation. And that's not the case. I can't ask my dad. He's passed now, but he didn’t have a favorite. He said it five different ways my whole life, depending on context. So he doesn't have a preference, and I'm not going to impose one. So I'm just aBa, and I'm okay with that.
EL: Yeah, well, and as far as I know, you're currently the only mathematician named aBa. Or at least spelled the way yours is spelled.
aM: Oh yeah, in the arXiv. Yeah, on Mathscinet that it’s. Yeah, I'm the only one there. Recently someone invited me to a wedding and they were like, what's your address? And I said, “aBa and my address is definitely enough.”
EL: Yeah, so what theorem would you like to tell us about?
aM: Oh, okay, well I was listening actually to a couple of you shows recently, and Holly didn’t have a favorite theorem, Holly Krieger. I'm exactly the same way. I don't even have a theorem of, like, the week. She was lucky to have that. I have a theorem of the moment. I would like to talk about something I discovered when I was in college, that’s kind of the reason. but can I briefly say some of my like, top hits just because?
EL: Oh yeah.
KK: We love top 10 lists. Yeah, please.
aM: Okay. So I'm in combinatorics, loosely defined, but I have no reason—I don't know why people throw me in that bubble. But that's the bubble that that I've been thrown in. But my thesis—actually, I don’t ever remember the title, so I have to read it off a piece of paper—Analysis of symmetric function ideals towards a combinatorial description of the cohomology ring of Hessenberg varieties.
KK: Okay.
aM: Okay, all those words are necessary there. But my advisor said, “You're in combinatorics.” Essentially, my problem was, we were studying an object and algebraic geometry, this thing called a Hessenberg variety. To study this thing we used topology. We looked at the cohomology ring of this, but that was very difficult. So we looked at this g...

Evelyn Lamb: Hello and welcome to My Favorite Theorem, a podcast where we ask mathematicians to tell us about their favorite theorems. I'm Evelyn Lamb. I'm one of your hosts. I am a freelance math and science writer in Salt Lake City, Utah. Here's your other host.
Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. How's it going, Evelyn?
All right. I am making some bread right now, and it smells really great in my house. So slightly torturous because I won't be able to eat it for a while.
KK: Sure. So I make my own pizza dough. But I always stop at bread. I never that extra step. I don't. I don't know why. Are you making baguettes? Are you doing the whole...
EL: No. I do it in the bread machine.
KK: Oh.
EL: Because I'm not going to make, yeah, I'm not going to knead and shape a loaf. So that's the compromise.
KK: Oh. That that's the fun part. So I've had a sourdough starter for, the same one for at least three or four years now. I've kept this thing going. And I make my own pizza crust, but I'm just lazy with bread. I don't eat a lot of bread.
EL: So yeah. And joining us to talk about--on BreadCast today--I'm very delighted to welcome Cynthia Flores. So hi. Tell us a little bit about yourself.
Cynthia Flores. Oh, thanks for having me on the show. I'm so grateful to join you today, Kevin and Evelyn. Well, I'm an assistant professor of mathematics and applied physics at California State University Channel Islands in the Department of Math and Physics. The main campus is not located on the Channel Islands.
EL: Oh, that's a bummer.
CF: It's actually located in Camarillo, California. It's one hour south of Santa Barbara, one hour north of downtown Los Angeles, roughly.
But the math department does get to have an annual research retreat at the research station located in the Santa Rosa Island. So that's kind of neat.
KK: Oh, how terrible for you.
EL: Yeah. That must be so beautiful.
KK: I was in Laguna Beach about a week and a half ago, which is, of course, further south from there, but still just spectacularly beautiful. Really nice.
CF: Yeah, I feel really fortunate to have the opportunity to stay in the Southern California area. I did my PhD at UC Santa Barbara, where I studied the intersections of mathematical physics, partial differential equations, and harmonic analysis and has motivated what I'm going to talk about today.
KK: Good, good, good.
EL: Yeah. Well, and Cynthia was on another podcast I host, the Lathisms podcast. And I really enjoyed talking with her then about the some of the research that she does. And she had some fun stories. So yeah, what is your favorite theorem? What do you want to share with us today?
CF: I'm glad you asked. I have several favorite theorems, and it was really hard to pick, and my students have heard me say repeatedly that my favorite theorem is the fundamental theorem of calculus.
EL: Great theorem.
KK: Sure.
CF: It's also a very, I find, intimidating theorem to talk about on this series, especially with so many creative individuals pairing their favorite theorems with awesome foods and activities. And so I just thought that one was maybe something to live up to. And I wanted to start with something that's a little closer to to my research area. So I found myself thinking of other favorites, and there was one in particular that does happen to lie at the intersection of my research area, which is mathematical physics, PDEs and harmonic analysis. And it's known as Heisenberg's Uncertainty Principle. That's how it's really known by the physics community. And in mathematics, it's most often referred to as Heisenberg's Uncertainty Inequality.
EL: Okay.
CF: So, is it familiar? I don't know.
EL: I feel like I've heard of it, but I don't--I feel like I've only heard of it from kind of the pop science point of view, not from the more technical point of view. So I'm very excited to learn more about it.
KK: So I actually have a story here. I taught a course in mathematics and literature a couple years back with a friend of mine in the in the foreign languages department. And we watched A Serious Man, this Coen Brothers movie, which, if you haven't seen is really interesting. But anyway, one of the things I made sure to talk about was Heisenberg's Uncertainty Principle, because that's sort of one of the themes, and of course now I forgotten what the inequality is. But I mean, I remember it involves Planck's constant, and there's some probability distribution, so let's hear it.
CF: Mm hmm. Yeah, yeah. So I was, I was like, this is what I'm going to pair it with. Like, I'm going to pair the conversation, like the mathematics description, physical description, with, basically I was thinking of pairing it with something Netflix and chill-like. I'm really glad that you brought that up, and I'll tell you mo...

Kevin Knudson: Welcome to My Favorite Theorem. I’m your host Kevin Knudson, professor of mathematics at the University of Florida. I’m joined by your cohost.
Evelyn Lamb: Hi, I’m Evelyn Lamb. I’m a math and science writer in Salt Lake City, Utah, where it is very cold now, and so I’m very jealous of Kevin living in Florida.
KK: It’s a dreary day here today. It’s raining and it’s “cold.” Our listeners can’t see me doing the air quotes. It’s only about 60 degrees and rainy. It’s actually kind of lousy. but it’s our department holiday party today, and I have my festive candy cane tie on, and I’m good to go. And I’m super excited.
John Urschel: So I haven’t been introduced yet, but can I jump in on this weather conversation? I’m in Cambridge right now, and I must say, I think it’s probably nicer in Cambridge, Massachusetts than it is in Utah right now. It’s a nice breezy day, high 40s, low 50s, put on a little sweater and you’re good to go.
EL: Yeah, I’m jealous of both of you.
KK: Evelyn, I don’t know about you, but I’m super excited about this one. I mean, I’m always excited to do these, but it’s the rare day you get to talk to a professional athlete about math. This is really very cool. So our guest on this episode is John Urschel. John, do you want to tell everyone about yourself?
JU: Yes, I’d be happy to. I think I might actually be the only person, the only professional athlete you can ask high-level math about.
KK: That might be true. Emily Riehl, Emily Riehl counts, right?
EL: Yeah.
KK: She’s a category theorist at Johns Hopkins. She’s on the US women’s Australian rules football team.
EL: Yeah,
JU: Australian rules football? You mean rugby?
KK: Australian rules football is like rugby, but it’s a little different. See, you guys aren’t old enough. I’m old enough to remember ESPN in the early days when they didn’t have the high-end contracts, they’d show things like Australian rules football. It’s fascinating. It’s kind of like rugby, but not really at the same time. It’s very weird.
JU: What are the main differences?
EL: You punch the ball sometimes.
KK: They don’t have a scrum, but they have this thing where they bounce the ball really hard. (We should get Emily on here.) They bounce the ball up in the air, and they jump up to get it. You can run with it, and you can sort of punch the ball underhanded, and you can kick it through these three posts on either end [Editor's note: there are 4 poles on either end.]. It’s sort of this big oval-shaped field, and there are three poles at either end, and you try to kick it. If you get it through the middle pair, that’s a goal. If you get it on either of the sides, that’s called a “behind.” The referees wear a coat and tie and a little hat. I used to love watching it.
JU: Wait, you say the field is an oval shape?
KK: It’s like an oval pitch, yeah.
JU: Interesting.
KK: Yeah. You should look this up. It’s very cool. It is a bit like rugby in that there are no pads, and they’re wearing shorts and all of that.
JU: And it’s a very continuous game like rugby?
KK: Yes, very fast. It’s great.
JU: Gotcha.
KK: Anyway, that’s enough of us. You didn’t tell us about yourself.
JU: Oh yeah. My name is John Urschel. I’m a retired NFL offensive lineman. I played for the Baltimore Ravens. I’m also a mathematician. I am getting my Ph.D. in applied math at MIT.
KK: Good for you.
EL: Yeah.
KK: Do you miss the NFL? I don’t want to belabor the football thing, but do you miss playing in the NFL?
JU: No, not really. I really loved playing in the NFL, and it was a really amazing experience to be an elite, elite at whatever sport you love, but at the same time I’m very happy to be focusing on math full-time, focusing on my Ph.D. I’m in my third year right now, and being able to sort of devote more time to this passion of mine, which is ideally going to be my lifelong career.
EL: Right. Yeah, so not to be creepy, but I have followed your career and the writing you’ve done and stuff like that, and it’s been really cool to see what you’ve written about combining being an athlete with being a mathematician and how you’ve changed your focus as you’ve left playing in the NFL and moved to doing this full-time. It’s very neat.
KK: So, John, what’s your favorite theorem?
JU: Yes, so I guess this is the name of the podcast?
KK: Yeah.
JU: So I should probably give you a theorem. So my favorite theorem is a theorem by Batson, Spielman, and Srivastava.
EL: No, I don’t. Please educate us.
JU: Good! So this is perfect because I’m about to introduce you to my mathematical idol.
KK: Okay, great.
JU: Pretty much who I think is the most amazing applied mathematician of this generation, Dan Spielman at Yale. Dan Spielman got his Ph.D. at MIT. He was advised by Mike Sipser, and he was a professor at MIT and eventually mo...

Kevin Knudson: Welcome to MFT. I'm Kevin Knudson, your host, professor of mathematics at the University of Florida. I am without my cohost Evelyn Lamb in this episode because I'm on location at the Banff International Research Station about a mile high in the Canadian Rockies, and this place is spectacular. If you ever get a chance to come here, for math or not, you should definitely make your way up here. I'm joined by my longtime friend Justin Curry. Justin.
Justin Curry: Hey Kevin.
KK: Can you tell us a little about yourself?
JC: I'm Justin Curry. I'm a mathematician working in the area of applied topology. I'm finishing up a postdoc at Duke University and on my way to a professorship at U Albany, and that's part of the SUNY system.
KK: Contratulations.
JC: Thank you.
KK: Landing that first tenure-track job is always
JC: No easy feat.
KK: Especially these days. I know the answer to this already because we talked about it a bit ahead of time, but tell us about your favorite theorem.
JC: So the theorem I decided to choose was the classification of regular polyhedra into the five Platonic solids.
KK: Very cool.
JC: I really like this theorem for a lot of reasons. There are some very natural things that show up in one proof of it. You use Euler's theorem, the Euler characteristic of things that look like the sphere, R=2.
There's duality between some of the shapes, and also it appears when you classify finite subgroups of SO(3). You get the symmetry groups of each of the solids.
KK: Oh right. Are those the only finite subgroups of SO(3)?
JC: Well you also have the cyclic and dihedral groups.
KK: Well sure.
JC: They embed in, but yes. The funny thing is they collapse too because dual solids have the same symmetry groups.
KK: Did the ancient Greeks know this, that these were the only five? I'm sure they suspected, but did they know?
JC: That's a good question. I don't know to what extent they had a proof that the only five regular polyhedra were the Platonic solids. But they definitely knew the list, and they knew they were special.
KK: Yes, because Archimedes had his solids. The Archimedean ones, you are allowed different polygons.
JC: That's right.
KK: But there's still this sort of regularity condition. I can never remember the actual definition, but there's like 13 of them, and then there's 5 Platonics. So you mentioned the proof involving the Euler characteristic, which is the one I had in mind. Can we maybe tell our listeners how that might go, at least roughly? We're not going to do a case analysis.
JC: Yeah. I mean, the proof is actually really simple. You know for a fact that vertices minus edges plus faces has to equal 2. Then when you take polyhedra constructed out of faces, those faces have a different number of edges. Think about a triangle, it has 3 edges, a square has 4 edges, a pentagon is at 5. You just ask how many edges or faces meet at a given vertex? And you end up creating these two equations. One is something like if your faces have p sides, then p times the number of faces equals 2 times the number of edges.
KK: Yeah.
JC: Then you want to look at this condition of faces meeting at a given vertex. You end up getting the equation q times the number of vertices equals 2 times the number of edges. Then you plug that into Euler's theorem, V-E+F=2, and you end up getting very rigid counting. Only a few solutions work.
KK: And of course you can't get anything bigger than pentagons because you end up in hyperbolic space.
JC: Oh yeah, that's right.
KK: You can certainly do this, you can make a torus. I've done this with origami, you sort of do this modular thing. You can make tori with decagons and octagons and things like that. But once you get to hexagons, you introduce negative curvature. Well, flat for hexagons.
JC: That's one of the reasons I love this theorem. It quickly introduces and intersects with so many higher branches of mathematics.
KK: Right. So are there other proofs, do you know?
JC: So I don't know of any other proofs.
KK: That's the one I thought of too, so I was wondering if there was some other slick proof.
JC: So I was initially thinking of the finite subgroups of SO(3). Again, this kind of fails to distinguish the dual ones. But you do pick out these special symmetry groups. You can ask what are these symmetries of, and you can start coming up with polyhedra.
KK: Sure, sure. Maybe we should remind our readers about-readers-I read too much on the internet-our listeners about duality. Can you explain how you get the dual of a polyhedral surface?
JC: Yeah, it's really simple and beautiful. Let's start with something, imagine you have a cube in your mind. Take the center of every face and put a vertex in. If you have the cube, you have six sides. So this dual, this thing we're constructing, has six ver...

Evelyn Lamb: Welcome to My Favorite Theorem, the show where we ask mathematicians what their favorite theorem is. 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: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. I had to wear a sweater yesterday.
EL: Oh my goodness! Yeah, I’ve had to wear a sweater for about a month and a half, so.
KK: Yeah, yeah, yeah.
EL: Maybe not quite that long.
KK: Well, it’ll be hot again tomorrow.
EL: Yeah. So today we’re very glad to have our guest Henry Fowler on. Henry, would you like to tell us a little bit about yourself?
Henry Fowler: I’m a Navajo Indian. I live on the Navajo reservation. I live by the Four Corners in a community, Tsaile, Arizona. It’s a small, rural area. We have a tribal college here on the Navajo Nation, and that’s what I work for, Diné College. I’m a math faculty. I’m also the chair for the math, physics, and technology. And my clan in Navajo is my maternal clan is Bitterwater and my paternal clan is Zuni Edge Water.
EL: Yeah, and we met at the SACNAS conference just a couple weeks ago in Salt Lake City, and you gave a really moving keynote address there. You talked a little bit about how you’re involved with the Navajo Math Circles.
HF: Yes. I’m passionate about promoting math education for my people, the Navajo people.
EL: Can you tell us a little bit about the Navajo Math Circles?
HF: The Navajo Math Circles started seven years ago with a mathematician from San Jose State University, and her name is Tatiana Shubin. She contacted me by email, and she wanted to introduce some projects that she was working on, and one of the projects was math circles, which is a collection of mathematicians that come together, and they integrate their way of mathematical thinking for grades K-12 working with students and teachers. Her and I, we got together, and we discussed one of the projects she was doing, which was math circles. And it was going to be here on the Navajo Nation, so we called it Navajo Math Circles. Through her project and myself here living on the Navajo Nation, we started the Navajo math circles.
KK: How many students are involved?
HF: We started first here at Diné College, we started first with a math summer camp, where we sent out applications, and these were for students who had a desire or engaged themselves to study mathematics, and it was overwhelming. Over 50 students applied for only 30 slots that were open because our grant could only sustain 30 students. So we screened the students and with the help of their regular teachers from junior high or high school, so they had recommendation letters that were also presented to us. So we selected the first 30 students. Following that we expanded our math circle to the Navajo Nation public school system, and there’s also contract schools and grant schools. Now we’re serving, I would say over 1,000 students now.
KK: Wow. That’s great. I assume these students have gone on to do pretty interesting things once they finish high school and the circle.
HF: Yes. We sort of strategized. We wanted to work with lower grades a little bit. We wanted to really promote a different way of thinking about math problems. We started off with the first summer math camp at the junior high or the middle school level, and also the students that were barely moving to high school, their freshman year or their 10th grade year. That cohort, the one that we started off with, they have a good rate of doing very well with their academic work, especially in math, at their high school and junior high school. We have four that have graduated recently from high school, and all four of them are now attending a university.
KK: That’s great.
EL: And some of our listeners may have seen there’s a documentary about Navajo math circles that has played on PBS some, and we’ll include a link to that for people to learn a little bit about that in the show notes for the episode. We invited you here to My Favorite Theorem, of course, because we like to hear about what theorems mathematicians enjoy. So what have you selected as your favorite theorem?
HF: I have quite a few of them, but something that is simple, something that has been an awe for mathematicians, the most famous theorem would be the Pythagorean theorem because it also relates to my cultural practices, to the Navajo.
KK: Really?
HF: The Pythagorean theorem is also how Navajo would construct their traditional home. We would call it a Navajo hogan. The Navajo would use the Pythagorean theorem charting how the sun travels in the sky, so they would open their hogan door, which is always constructed facing east. So once the sun comes out, it projects its energy, the light, into the hogan. The Navajo began to study that phenomenon, how that light travels in space in the h...