My Favorite Theorem

Episode 95 - Kyne Santos

Kevin Knudson: Be sure to listen to the end for a very special announcement. Eveyn Lamb: Hello and welcome to My Favorite Theorem, the podcast with no quiz at the end. I'm Evelyn Lamb, 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, where it's hot. It’s still hot. I mean, you guys are, you know, you and our guest are in some place not so hot. And I'm, like, I’m in short sleeves. I got sweaty walking to work. EL: Yeah. I've got a sweater and a thick scarf on. And I spent yesterday so cold, just like sitting under a blanket in the house turning up the thermostat by degrees, just not — we had a warm October, so it got cold so fast. Not a fan. My Texas roots are coming out. KK: Yeah. Plus it's, you know, it's November 7. So we'll let our listeners think about what's happened since, you know, in the last couple of days. EL: No need. The problems that existed before November 5 were always still going to exist now. KK: That's accurate. EL: There’s always work to be done, and we are thrilled today. KK: That’s right. EL: To be welcoming Kyne Santos to the show. Kyne, please introduce yourself. Let us know what your deal is, where you're coming from, all that. Kyne Santos: Hi everyone. Thanks for having me on the podcast. My name is Kyne. I am a drag queen from Canada. I'm based about an hour outside of Toronto, in a little town called Kitchener Ontario. I have a Bachelor of mathematics from the University of Waterloo, and I make math videos on social media. You may know me as Online Kyne. I make videos really just about all of my broad interests in math, and I do it all dressed in drag. EL: Yes, gorgeous, amazing videos. I'm gesturing, which our listeners, I know they always appreciate when we do that in this audio only format, but yeah, just really fun. And I think your your videos like really make math inviting in a different way than a lot of people who make math inviting do it, and I think it's really great. And you haven't mentioned it yet, and I'm sure you would get to it, but you do have a book called Math in Drag that I, as I mentioned earlier, I read last week, finally gave myself the push I needed to actually get it off of my ever-growing TBR pile. KS: And what did you think? EL: I really enjoyed it, and I enjoyed, you know, there's some memoir about, like, your experiences as a drag queen and as a math-interested, young queer person, and like, how you you know how you've kind of gotten where you're going, and plus some things that you know, not all — what am I trying to say? I'm trying to say I really, like a few of the ways that you, you bring the intersection of your queer life into math, and kind of help us see it in a different perspective. And see, you know, like your discussion of complex numbers and imaginary numbers, and how like expanding what you think a number can be, and like how you view that expanding, you know what gender or sexuality can mean. And so, yeah, I just really appreciate the overlap of that. And there's a huge intersection of queer people and math enthusiasts, myself included, and you know, I think it's great that there's a book that kind of goes out and explicitly does that. So I’ve talked about your book. KS: Thank you. EL: But do you want to talk about your book and how you decided to write it? KS: Yeah, well, thank you. I appreciate that. And really, when I started making videos online, I just thought that it would be kind of funny and silly to see a drag queen talk about math riddles. I started doing the videos really just to be funny and to be camp, but I didn't imagine that there was such a huge intersection of queer people and math enthusiasts. But after posting, and after the videos started going viral, I would just get messages from people all over the world saying that they felt very seen by by the videos, which gave me the motivation to really just keep sticking with it, because I want to show people that being a math person can look like anything, and it doesn’t matter what you look like or where you come from. I mean, why not wear a big, fabulous wig on your head and a sequined gown? Because it doesn't matter. And I think math should be fun, and one of the big messages of the book is that math has a lot in common with drag, and I think that both fields sort of require you to be creative and to think in abstractions and metaphors, and to be able to see something and understand it in many different ways, whether you're seeing something algebraically and geometrically at the same time. I think that a lot of math can have a fabulous side and maybe a more boring side, right? Just like a drag queen. KK: I mean, drag can be very conceptual. So my, you know, full disclosure, my wife is a huge fan of the whole drag race enterprise. So you're on season one of Canada's drag race, correct? KS: Yes, I was. so. KK: So Priyanka won that season, right? KS: Yes. KK: And Jimbo was on there. Jimbo, of course, is hilarious. KS: A legend. KK: And went on to win an All-Stars later. So yeah, we watch drag roughly four nights a week at my house, because my wife is a huge fan and the franchise has grown. You know, it's in every country of the world, it seems. KS: Well, it's grown quite exponentially, hasn't it? Because it used to just be once a year, and then it really just snowballed on top of it. KK: It kind of never ends now. It's always on. Is there much of a drag scene in Kitchener? Do you have to make your way over to Toronto most of the time? KS: Well, it's a bit different here in Kitchener, because we don't have clubs and gay bars anymore, so it's a lot of drag brunches and, like, drag dinners. So we've had, we've had to expand. But the funny thing is, out here in the smaller towns outside of Toronto, people really are hungry for drag. It's a different audience than like the college students that go out to the gay bars in Toronto, but it’s, like, moms and dads and older people or younger people who don't have a gay bar to go to. And so we all have found each other and found communities. KK: Well, that's great. I mean, drag shows are so much fun. You know, I've never had a bad time at a drag show. And my standard line is, if you're not having fun at a drag show, you just don't know how to have fun. KS: Yes. KK: It's just a blast. So, okay, this is a math podcast. We can talk more about drag, too, but so, do you have a favorite theorem? Why don't you tell us what it is? KS: Yes. So in light of talking about math as a drag queen and believing that math theorems may have a side of them that is in drag and out of drag, my favorite theorem is the fundamental theorem of calculus. EL: Wonderful. KS: Which was introduced to me in school as like a tool for solving integrals. Because really what it says is that integration is like an inverse process of differentiation. And I think when I first learned it, I didn't really appreciate what that meant because when, when you learn it, you sort of learn it as a tool for for solving an integral, which an integral is like, you're dividing — sorry, let me start over. An integration problem is really essentially finding an area of a shape by cutting it up into rectangles and then adding up the areas of those rectangles and taking the limit of that sum as the rectangles get thinner and thinner. But that's not actually how people solve integrals. The way that everybody solves an integral is by finding the function’s antiderivative, which uses the fundamental theorem of calculus. KK: Right. EL: Yeah, I do think this is one that we're introduced to so early in our math journeys a lot of the time. You know, you like, probably all of us took calculus in high school. And if you take it in high school, you — I at least — hadn't really seen the creative side of math and the — I saw it much more as a rule book for how to solve problems, rather than this entire weird, lumpy, creative universe. And I think, you know, you, see it as like, Oh, this is, you know, the fundamental theorem of calculus exists to take integrals of of things. But it's like, it doesn't really, it's, it's much deeper than you realize when you're 16 or whatever, and learning it, than you can understand at that point. KS: Yeah, I think if you really stop and think about what the theorem is saying, aside from just seeing it as a tool for solving a real-world application, as a tool for finding an area or finding an amount of money, if you really think about what the theorem is saying, I think it's it's quite profound, because here you have two separate problems, the area problem, which is about finding the area of some curved shape, and the tangent problem, which is about finding the slope of the tangent at a particular point on a curve. Who could tell at first glance that these problems are in any way related? EL: Yeah. KS: But it turns out that they are. KK: And then, of course, there's the other part of the theorem that that students tend to forget what which mathematicians like the most, which is that what you started with, which is that differentiation and integration are sort of inverse processes, right? If you differentiate the integral, you get the function back. That's the one that that always just sort of goes over students’ heads conceptually, because it's kind of, although it's kind of the more fun part, it's actually the easy, you know, it's not so hard to prove once you think about it in the right way. And I always thought that was pretty remarkable. But when I learned calculus as a high school senior, that went completely past me. I learned how to do those sorts of problems, but I was like, Oh, I'm finding areas by finding antiderivatives and now, as a professional mathematician, it's like, yeah, okay. Yeah, that’s useful. Great. EL: I think I didn't really appreciate either direction of the theorem that much until I actually taught calculus, which I do think this is the thing that happens all the time, is like, you know, teaching these concepts, it gives the teacher such a deeper appreciation, maybe sometimes more for the teacher than the student, although hopefully not entirely. KS: Well, I totally relate to that. I'm not a traditional math teacher. I just make videos on social media. But I enjoy making the videos because it helps me deepen my own understanding of subjects. And I find that it forces me to think of of theorems and concepts in different ways. When I've sat down and thinking, how am I going to explain this to somebody who is only hearing this for the first time? And it gives me a deeper relationship with with a lot of math theorems. KK: Yeah. So you were a student at Waterloo. They have a very strong math department. What was that like for you as a student? I mean, was there anything in particular that you really liked besides the fundamental theorem, of course, was there a particular branch of mathematics that you were drawn to, or anything like that? KS: My major was in mathematical finance, which was, like, half pure math and half finance, like actuarial science. I initially wanted to go down a path of doing statistics and maybe working in like data or with a bank. I ended up taking a very unconventional path, doing Tiktoks and going on Canada's drag race, as one does. KK: Yes. KS: And now I found myself in this world of being a math communicator like yourselves, and just talking about math and enhancing public understanding and engagement with math. EL: Yeah. So getting back to the fundamental theorem of calculus, can you tell us a little bit about you know, maybe your appreciation of it. Was it something that you really saw the profundity of when you first encountered it, or is it something that's kind of grown over time? KS: I think what's great about theorems is that in the beginning you may look at it and just see it as a bunch of words on the page. But once you really wrap your head around it, I think theorems can become obvious, and thinking of integration and differentiation as inverse processes of each other can seem confusing, but the way I like to think about it that makes it obvious to me is I think about integration like you're doing a sum, right? Because when you're finding the area by dividing a region into rectangles, you're adding up those areas. So you're taking a sum, you're adding up the regions of positive area, subtracting the regions of negative area, and finding a total area. The key insight is that if the curve you're dealing with is actually a derivative and represents a rate of change, then doing integration is equivalent to adding up a bunch of changes and adding the positive changes, subtracting the negative changes, and just looking at the total change, which is the same thing as just zooming out and looking at the big picture of where the function started and finished and observing the total change. So that's how I like to think about the fundamental theorem of calculus. It's small changes add up to big changes. EL: Nice. KK: Cool. So the other thing on this podcast is we ask our guests to pair their theorem with something, and this is often the most challenging part. What have you chosen to pair with the fundamental theorem? KS: I pair the theorem with hiking up a mountain. So last year, I climbed up Acatenango volcano in Guatemala, which was one of the most thrilling experiences of my life. It was, like, a six-hour hike before we reached the base camp, like one of the hardest things I've ever done in my life. But what I noticed is that you don't climb at a constant slope, right? There are times when the slope is flat, and maybe even some moments where you're going downhill for a bit in order to reach the next bit. So to give an example, imagine you're hiking up a mountain, going from point A to point B, and you want to find out the overall change in elevation. So let's say that point A, the starting point, is 500 meters above sea level. In the first hour, you ascend 100 meters. In the second hour, you descend 50 meters, and in the third and final hour you ascend 200 meters to arrive at point B, which is 750 meters above sea level. The question is, what's the overall change in elevation? Well, there's two ways to go about it. You can find the final elevation, which is 750 meters, and just subtract the starting elevation, which was 500 and the difference between 750 and 500 is 250 meters. Or you can add up the little changes along the way. So in the first hour, we climbed 100 meters, and then we descended 50, and then we climbed another 200 so 100 minus 50 plus 200 is 250 meters. And these two approaches represent the two sides of the equation in the fundamental theorem of calculus, because on the left hand side, you have an integral of a derivative. You're taking a sum of all the changes. That's what we did when we added up the little changes of elevation each hour. Those are technically derivatives, because they're rates of change. On the right hand side, you just have to take the difference of the two endpoints of the function, which is what we did when we took the final elevation minus the starting elevation. So I think that illustrates this idea that you can add up the small changes, or you can just look at the overall change. And I think that the the power of this example is made a bit more clear when you look at some of the higher-dimensional analogs of the fundamental theorem of calculus, like I recently was reading about Stokes’ theorem, which is like the fundamental theorem of calculus on higher-dimensional manifolds. And what it says is that the average of a derivative on the interior of a manifold is equal to the average of a function on the boundary. And when I first read that, I thought, okay, how? What does this have anything to do with the fundamental theorem of calculus? But really, all it's saying is that adding up the little changes on the inside of the function is the same as just looking at the overall change of the function. So in one dimension, which is what we do when we do regular calculus, the boundary of an interval is just the start and end points. So if you know your elevation at the end and at the start, that's all you need to calculate the overall change. But you can also calculate the net change if you know all the little changes that happen in between, aka the derivatives on the interior. KK: This sounds like you just described a really good YouTube video. Have you made this video? KS: I have! If you go on my if you go on my Tiktok, I made a whole series, okay, on calculus. EL: Yeah, nice. Yeah, I must admit, it's probably a failure of imagination on my part, but I did not expect our drag queen guest to have hiking as her example on this. So, yeah, so do you do a lot of hiking? KS: No, and that's why it stuck out as such an experience in my life, because I swear I was not, like, an outdoorsy person, but my husband is British, and we, like started out as a long-distance couple, and he, in many ways, is like the complete opposite of me. And in many ways we're like the same person, but he's like very naturey. He loves the outdoors, and he was the person that that got me into hiking and walking and birdwatching, which, by the way, I love the red-winged blackbird in your background. KK: Thank you. I mean, I like them so much, I’ve even got one of my arm. Oh my gosh, yeah, yeah, yeah. I took that photo at a local place here in Florida. EL: Oh, I just want to sayI live in Utah and didn't grow up. I grew up in Dallas, which doesn't have a lot of hiking opportunities super close by, but now that I live in Utah, it's one of my very favorite things. So if you and your husband ever find yourself in this area, please let me know, and we can go to go on a hike. And there are drag shows here too. So I'm sure we can hook you up with both of those experiences. KS: It’s definitely on our bucket list of places to visit in the US, one of the reasons being that we love the Real Housewives of Salt Lake City. So I think we have to go and meet Heather Gay and Lisa Barlow. And of course, you, Evelyn. EL: Yeah. You know, various famous Utahns, yeah. So one of the things that I don't know, I'm maybe slightly embarrassed about, because it's off-brand for most of the rest of my life, is I do watch Real Housewives of Salt Lake City. I've got a little watch group here. KS: It's, like, the best show on TV. That's what I tell everyone. EL: It is so much. But yeah, I of course, it's because I'm local here, and I get to be like — my watch group, we actually, at the end of each season, we go to one of the restaurants that they went to at some point on the show as a group and like, do our little thing. And, you know, and then remember whatever stupid fight they were having in that restaurant. KK: Do you reenact it? EL: Occasionally. KS: Okay, which housewife do you identify with the most, Evelyn? EL: Oh, gosh, that is hard. I must say it is hard for me to find many points of identification. Honestly, what I'm I'm yelling at the TV all the time is, like, you all need to learn what an apology is. When you say that you're sorry, you'll know what you are actually meaning when you say that and what it means when you accept the apology. KK: I think that's a rule for everybody. EL: I mean, honestly, many, many people in this world could learn what an apology is. KK: It doesn't start with “if.” EL: Yeah, but anyway, yeah, I'm trying to think. I'm not sure. I'm not sure what, who the most mathematical of the housewives is. Although Heather had a storyline where she was putting together a choir that sang hymns in a non-religious setting. And that is actually one of my hobbies. So I guess. KS: Oh, there you go. EL: Yeah, I've come this close to, like, sending Heather Gay an email saying like, hey, come check out our recreational singing group. So Heather, if you're listening to My Favorite Theorem, please, come on, check us out. KK: Yeah, okay. I wonder how many of the Real Housewives listen to us. I’d be curious. KS: So you never know. You had a Drag Race queen that was a fan of the podcast. So you never know who could be listening. KK: So do you tour much, Kyne? Are you on the road in drag much? KS: I just got finished with doing a book tour all across Canada. I drove all the way from Vancouver out west to Halifax out east. I visited, like, 11 different independent bookstores talking about my book Math in Drag. KK: So you drove all of that? So my son lives in Vancouver, and I've driven that bit of the Trans-Canada Highway from Vancouver to Banff. And sometimes it's a little sketchy. I mean, it's, they're still working on it, you know. KS: Oh no, I didn't find that at all. KK: Really? Okay. KS: I mean, yeah, I just really loved it. KK: Oh, I loved it. KS: Because I'm part from the part of Canada that doesn't have as much of the mountains and that natural beauty. KK: Oh, it’s spectacular. KS: I’m near the Great Lakes, which, of course, is beautiful in its own way. But I just loved seeing all of Canada, and listen all the all the crap that I've got with all my drag couldn't fit in a checked suitcase anyway, so I had to load up the car. KK: So I've always wondered that about, like, when you, when you go to compete on drag race, right, where do they film it? In Canada? Is it in Toronto they film it, or they do it, they film it somewhere else? KS: It was one of the cities around, around, like, Hamilton was where I filmed it. I mean, we were able to bring five pieces of luggage, which had to be, like, a certain weight. I just brought it in, like, cardboard boxes. KK: Yeah, I've always wondered about that because, I mean, you see some of these things. I mean, these outfits get very elaborate, and it just seems like they wouldn't fit into a suitcase very well, but you managed to make it work? KS: Oh, yeah. Well, I like to think of drag race as a little bit of its own prisoner’s dilemma and arms race. Because sure, if you go back and watch the earlier seasons of drag race, I mean, the outfits were so simple. You could just buy something from the mall and then go go compete on the show, because that's what drag queens did on stage. But with Drag Race being such a global phenomenon, and drag queens being able to get rich, then every season, queens just raised the bar and started bringing in custom outfits and working with haute couture designers. And each season, it feels like the bar is being raised. And I mean nowadays, like you have to go into debt to get on the show, and there's not even a guarantee that you make that money back. So it's its own economic arms race. KK: Yeah, yeah. I mean, it gets pretty — the most recent one the global All Stars we're watching where Alyssa Edwards won, I mean, some of her outfits are just ridiculous. And you think, I mean, she's spending hundreds of thousands of dollars on this stuff. She has to be. KS: Yeah. KK: It’s pretty nutty. Oh well, yeah. EL: Well, I want to say one of my favorite things in the book is you talking about, like sewing some of your own outfits and the geometry of that. KK: That’s a math problem. EL: One of the videos on your channel that I really enjoyed is sewing this hyperbolic, I don't remember if it was a skirt or a dress. KS: It was a dress. EL: The hyperbolic pentagons. It is pentagons, right? KS: Yeah, yeah. EL: And that's so cool. And I just love that, you know, another of my hobbies is sewing. And, you know, the way that people think of that as, you know, maybe “women's work,” this domestic task that isn't scientific or something, and it's like. KS: My gosh, it's totally mathematical. EL: It’s the most geometrical. KS: Yeah, the most, like, you're constantly, like, splitting an inch down into eight parts and figuring out, okay, if I flip this inside out, will it work? And how to fit it under the sewing machine. A lot of mathematical thinking, way more than I ever thought. EL: I mean, the number of times I've installed a zipper and accidentally made a non-orientable shirt by getting one of the sides wrong. It's not good. KK: Sure. And this is one of these things. You'll mention this to people who are very good at sewing or other — you know, like, I once had a guy who was laying tile, and he said, I'm no good at math. And I'm like, what do you think you're doing? I mean, sewing is, is I can't sew. EL: Applied geometry. KK: That’s right. It is challenging and mathematical. EL: You know, it's a manifold, the human body is a manifold with, like, you know, non constant curvature. Not even constant-signed curvature. You've got positive and negative areas. It's like, yeah, make a, make a two-dimensional thing that fits perfectly on this inconsistently curved manifold. That’s hard! KK: It is hard. Yeah, yeah. Cool. All right, so, Kyne, where can our listeners find you online? You're Online Kyne on all platforms? KS: Yes I am. You can find me at Online Kyne on Instagram, Twitter, Tiktok. I'm mostly active on Instagram and Tiktok, and you can find a bunch of little short math lessons and fun-sized bites over on there. KK: Okay, yeah, cool. EL: Check her out. KK: Yep, this has been a lot of fun. I'm glad we did this. Yeah, I'm glad. Thanks for agreeing to come on. KS: Thank you for having me. Yeah, I'm a big fan of the podcast. I love it, and I was so glad when you guys reached out. KK: Oh, great. Good to know. See, Evelyn is great at this sort of thing. Well take care, Kyne. Thanks. KS: All right. KK: Well, folks, this has been the last episode of My Favorite Theorem, and we want to take a few minutes to say goodbye and some thank yous. So first of all, I started, so I'm going to go first. Evelyn, thank you for saying no and then changing your mind. Yeah, this — we’ve been at this for eight years, and, you know, I think we've become pretty good friends over the years, and I've certainly enjoyed working with you, and you made this podcast better than anything I ever imagined. So I really appreciate all of that. And our guests, of course, have been, you know, real troopers and just so generous and thoughtful in their theorem selections, and pairings especially. And it's just been a lot of fun. So, thank you, and thanks to everybody else. EL: Yes, it has been really fun. We started recording on Emmy Noether’s birthday in 2017 from a little apartment I had in Paris. And since then, Paris has completely like, changed itself. It's become, it's like taking cars out of the whole center. I'd love to go back there and live in a little apartment again, if anyone wants to help me do that. KK: Sounds great. EL: And yeah, it was just so fun to do. And yeah, I mentioned to you, the first time my now-husband asked me on a date, my answer was maybe, so I'm a person who just needs to take a little time to think things over, you know, think about what I want. And I don't know if we've shared this story before, but yeah, you approached me about this, and it was a time where I was really hustling for freelance work and didn't feel like I could take on an uncompensated project. KK: Right. EL: Which this has been. KK: Sure. EL: But it's been so fun. The reason that I said yes later was a few weeks, maybe even just a week later, I was thinking about, like, silly blog article kind of things I could do, and something that popped in my mind was wine pairings for famous theorems. KK: Yup. EL: And I realized, like, this wouldn't be that fun as a little list that I made, especially if it was only wine, because it's like, I don't know anything about wine. It's not that funny. Like, the title is funnier than the content actually could have been. But it made me think about the podcast you had pitched, and the idea of getting people to break out of their math teacher mode and have to talk about their theorem and pair it with something, whether, you know, food, wine, we've had sports, we've had, I think, lots of, some literature, music, just all sorts of things, just make them talk about math in a less, you know, less concrete way, a really impressionistic way, and that was so fun to me that I was like, yes, this uncompensated work sounds like it'll be worth it, with this person that I don't know, because I didn’t know you. KK: We didn’t know each other, right. EL: Yeah, I had seen your writing, but I did not know you as a person. So I was like, and then, of course, I was like, well, if I don't like it, I can just, you know, do it a few times and stop. KK: Stop, yeah. No contract. So yeah, it's been a lot of fun. I really appreciate that you asked me to do it and that you didn't find someone else to do it before I changed my mind. KK: Well, like you know, I had certainly always admired your writing and I hope to see more of that. I mean, I hope you've got a lot of projects going. EL: I’ve got some stuff in the cooker. KK: Good. EL: We’ll see. I hope to be able to share some of that more. I've had a little bit of a lower time in terms of what I'm I'm outputting right now, but I'm working on great things. KK: Quality over quantity. That’s always the thing, yeah, yeah. Well, you know, I'm more in administrative land these days. EL: Yes. KK: Chair of the department for six years. Now I'm in the Dean's office, andit's not that I don't have time for this, but it certainly, it's become a bit of a crunch. And, you know, our listeners have probably noticed that we've been recording less frequently. EL: Yeah. KK: I think both, because both of us have had other things going on, and weirdly, it's been getting more and more difficult, just to get people to say yes. EL: Or to get it actually scheduled once we want to do it. KK: Get it scheduled, yeah. EL: Yeah, everyone’s busy and everyone's a little Zoomed out, and it's very understandable. But we've had so much fun. I love that we've had such a breadth of theorems, from things like the fact that there are an infinite number of prime numbers, or the Pythagorean theorem that you saw in grade school, probably, to things that, like, four people in the world can actually understand. And we've really enjoyed talking to mathematicians about all of these things at all of these different levels, and just see what makes mathematicians excited about their work and and force them to talk about their work in a way that they wouldn't if they were presenting it in a seminar or for a class. KK: Right. There’s lots of hand waving that our listeners can't see. EL: Yeah. They don't have a chalkboard that they can write on. Yeah, so I've really enjoyed that. I've I've loved the repeats of theorems that we've gotten, which people were so afraid to do. And we just love hearing two different, two, three, four, more different perspectives on one theorem, and like, what grabbed one person or what it reminds a different person of just talking about it in a different way. And I think you need to be exposed to math concepts a few times anyway before they really start to stick. That's why teaching is so great. Because when you when you learned it in the class, you probably didn't understand it the way you do when you teach it, because you've seen it more and thought about it in more different ways. So yeah, I’ve loved sharing, sharing the repeats and the one that you know, the unique ones. KK: So yeah, been been great. Yup. It's been great fun. So I think it's time to sign off. EL: Yeah. KK: After eight years, all right, yup. EL: Thanks for listening, everyone. KK: Thanks for listening, and you forgot your little line you were going to use, about the best theorems. EL: That’s right! I think you deserve the right to use it now. KK: It’s yours. EL: Our favorite theorems were the friends we made along the way. KK: That’s correct. That’s right. Well, goodbye, everyone. [outro] In this episode, we were delighted to talk with Kyne Santos, a math communicator and drag queen who competed on Drag Race Canada, about the fundamental theorem of calculus. Find Kyne at her website and Tiktok, or on other social media with the same handle: onlinekyne. Her book is Math in Drag.

Episode 94 - Jeremy Alm

Kevin Knudson: Welcome to my favorite theorem, the math podcast with no quiz at the end. I'm one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host, fabulous as usual, with a really good zoom background. Evelyn Lamb: Yes, I am Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, and I'm celebrating fall with a nice zoom background that none of our listeners can see of a lovely bike trail near me with decked out in fall colors. So I hope everyone appreciates that. KK: So judging from Instagram this weekend, you took a train trip somewhere, and it looked really cool. EL: I did. Yeah, because I'm a freelancer and have quite a bit of schedule flexibility, I do silly things like take the Amtrak for 24 hours to go to Omaha for the weekend and then take it back. And yeah, it was, it was fun. KK: Why Omaha, just out of curiosity? EL: Singing, which shouldn't surprise people who know me. KK: Sure, yeah. Well, I did none of that. I was on an NSF panel last week. That was my big, EL: Slightly different adventure. KK: You know, but it's important work. I mean, it really is. And and our listeners, if you happen to get asked to be on an NSF panel, you should do it. It's very interesting and important work. So anyway, now I'm back home, where I’m doing — no, it was here. It was a Zoom panel, but also that was the extent of my week last week. Now that I'm in the Dean's office, which I don't think we've actually mentioned. So I was chair of my department for six years. Now I am an interim associate dean in our college, and one of my responsibilities is that I'm in charge of the college tenure promotion committee, and that committee meets three days a week, at 8am. EL: Oh, that's great. KK: That is not my jam at all. And then so twice last week I had T & P first thing in the morning, followed by, you know, seven hours of NSF proposals. EL: Yeah. KK: But anyway, I’m glad to be back on Zoom today to welcome our guests. So we're pleased to welcome Jeremy. All Jeremy, why don't you introduce yourself? Jeremy Alm: Thanks, Kevin and Evelyn. My name is Jeremy Alm. I am Associate Dean for programs in the College of Arts and Sciences at Lamar University in Beaumont, Texas, where it is still quite hot, even the last week of October. EL: Yes. JA: Before that, I was department chair at Lamar, and before that, I was department chair at a small college in rural Illinois called Illinois College. KK: Yeah, cool. EL: And what's your field of math? JA: My field of math, so I wrote a dissertation in algebraic logic and universal algebra. Decided I wasn't very good at algebra, started learning combinatorics, so now I solve combinatorial problems that arise in algebraic logic. EL: Nice. I do think it's funny, if I can interrupt for a moment. It is funny how grad school can do this to us, where you literally wrote a dissertation in algebra. And so what this means, in an objective sense, is, like of the billions of people in the world, you're probably in like the top 100th or 1,000th of 1% of people in knowledge of algebra. And yet your conclusion is “I'm not very good at algebra,” so I have had a similar conclusion that I drew about my field of math as well. So just interesting fact about higher — PhD programs in general, I think, JA: Yeah. Well, in my case, there's some further evidence, and that is that my main dissertation result was a conditional result, and about four years after I graduated, a Hungarian graduate student proved that my condition, like my additional hypothesis, held in only trivial cases. EL: Oh, that is a blow, but I'm glad you're using it now in combinatorics. KK: That sounds like one of those apocryphal stories, right? Where that always gets attributed, like, I don't know, somebody's giving their dissertation defense, and somebody like Milnor, probably not, but somebody like Milnor's in the audience, and they go, “The class of examples here is empty. This is — there's nothing here, you know?” JA: Well, fortunately, this was only discovered after I graduated. KK: Right, right, right, right. Well, and, you know, hey, I mean, things like that happen. It's not that big a deal. So, yeah, so Jeremy and I, we go back a little bit. We actually met at an AMS department chairs workshop some number of years ago that I can't even remember anymore. We were both still chairs. I know that. But was it 2019 maybe? JA: I think it was 2018, but one of those two years. KK: It was in DC. Question Mark. I don't know, Baltimore? B-more? Yeah. Anyway. JA: San Diego. Did you go to San Diego? KK: No. It must be in Baltimore. So anyway, we've kind of kept up a virtual friendship since then. So here we are, and I thought he would — he entertains me, so I thought he would entertain our guests. So, so Jeremy, we asked you on for a purpose. What is your favorite theorem? JA: So, my favorite theorem is that the Rado graph has certain properties. EL: Okay, and you know, the next question. JA: Yes, so I have to have a little bit of setup, okay? Okay, so first I want to talk about random graphs. Cool. Okay. Now imagine you've got a bunch of vertices, or I like to call them dots, because that's usually what they literally are. They're just dots, and we're going to connect two dots, or not. We call that putting an edge in and we're going to flip a coin for each potential edge to decide whether it will be present or absent in the graph. Okay? And usually we assume the coin is fair. Today, we will assume the coin is fair, although you can not assume that, you can make it whatever probability you want. EL: Yeah. JA: But if you assume the coin is fair, then you get the uniform distribution on the class of all graphs on some fixed number of vertices, so it's a convenient assumption that the coin be fair. EL: Right. JA: Now there are other random graph models. One of the ones that Erdős looked at early on was the one where you sample uniformly from the set of all graphs with a fixed number of vertices and a fixed number of edges, but then you lose independence of edges being in or not, and it's hard to prove things about that model. Even harder is the Barabási-Albert random graph model, where you start with some vertices, and then every time you add a vertex, you attach it to existing vertices, but preferentially, with preference for the the vertices with large degree. KK: Okay. JA: And if you do that, instead of getting a sort of binomial degree distribution, like you do with the standard model of random graphs, you get a power law. EL: Okay, yeah, rich get richer. JA: Yeah, yes, it's the rich get richer, right? You see this power law in, oh, like, the Facebook graph, the social network graphs, right? Most people are unpopular, and then there are some extremely popular people, but very few of them. And that was a great result in 2000 that showed how a power law degree distribution arises, and it's through preferential attachment and growth, right? But for the rest of this little talk, I just want to talk about the the coin flip model. EL: Yeah, any graph on that number of vertices is equally likely to any other one. JA: Correct. KK: Right, okay. JA: Ignoring isomorphism, right? KK: Yeah, okay, sure. JA: We’ve got, we've got labeled vertices so we can distinguish between two isomorphic graphs. EL: Yeah, I guess that's kind of important. JA: Yeah, it's very important. I don’t — I’m not sure what would happen if you worked up to isomorphism. EL: Sounds hard. KK: Yeah, let’s ignore that. JA: Okay, so we're going to connect combinatorics and logic here in just a minute. So I want to briefly talk about first-order formulas. What does that mean? A first-order formula in the language of graphs is built as follows. You have one binary relation symbol that you might think of as a tilde [~]. So x ~ y means that the vertex x is adjacent to the vertex y, and then you have logical symbols — and, or, not, implies. And then you have quantification: for all x, there exists y. Okay, any sentence you can write with those symbols and variables is a first-order sentence in the language of graph theory. Okay. So for example, you could say, for all x, there exists y, x is adjacent to y, and what that says is that no vertex is isolated, right? For all x, there's some y adjacent to it. You could also say there exists x for all y, x is adjacent to y. So that would mean x is adjacent to everything, including itself — which, we're not going to allow loops today. So imagine all the things you can say with first-order formulas in the language of graphs. Well, it turns out that the first- order theory of graphs obeys what's called a zero-one law. And what that means is that any first-order sentence is either almost surely true in all finite graphs or almost surely false in all finite graphs. EL: That is very strange. KK: Yeah. JA: It is. EL: I think, I think, as a non graph theorist. JA: Yes. So, for example, almost all finite graphs are connected. EL: That’s funny. When you first introduced random graphs, I was I almost asked you, like, are those usually connected or not? But then I decided to just wait a moment. JA: Yup, they are. They are usually connected. In fact, they almost surely have diameter two, which is how you prove they're connected. It turns out, and this still kind of blows my mind. One thing you cannot say with a first order sentence in the language of graphs is “this graph is connected.” EL: Oh, yeah. Okay. I mean, I, for some reason, I don't know why, maybe it's because I feel like graphs are very tangible and, like, I should be able to understand them quickly. What I want to do right now is first of all, find a way to say that this graph is connected in this first order logic. And second of all, find some proposition that 50% of graphs are going to have, and 50% are not, just to — I don't know why. I don't dislike you, but I want to prove you wrong somehow. JA: Well, it wouldn't be me you're proving wrong. Whoever proved this theorem, and I actually don't know off the my head, who proved it. So yeah, so we have this zero-one law. So to give an example of a statement that's almost surely false, “this graph is complete,” right? You can say that with a first order sentence, right? For all x, for all y, x not equal to y, implies x adjacent to y. KK: Sure. JA: Obviously that's true of few graphs, right? EL: Yeah. JA: So that's a good example of the zero part of it. It's almost surely false. Okay, so what do we mean by almost surely true anyway? Well, what we mean is that, if you look — so take the set of all graphs on n vertices and calculate the fraction that satisfy the property, then let n go into infinity. What's the limit? And it turns out, not only does the limit exist, it's always zero or one. KK: Right. JA: Okay, that's not true in general. In fact, maybe the simplest example of an intermediate property is for finite groups, the probability of being abelian is, think it's roughly a third or something. So that's a property, you can say that with a first order sentence, x times y equals y times x, and its probability, asymptotically, is intermediate between zero and one. Okay, so this is special to have this zero-one property. Okay, so now I want everyone to imagine we're going to, you know, n is going to infinity, right? And we're getting more and more graphs and the number of edges is, you know, the distribution of number of edges is converging to the normal distribution, and all this nice statistical stuff is going on. Well, what if we sort of take it to the limit and say, okay, n reaches infinity. N is countable. What happens? Well, it turns out that you get, as n goes to infinity, you get more and more of these graphs. But then something changes. When you actually go to the countable random graph, there’s one, and it's called the Rado graph. And what I mean by there's one is that there exists this graph called the Rado graph. And if you actually generate via this coin flip random process, a countably infinite,random graph, with probability one, you get the Rado graph. KK: Okay, okay, up to isomorphism, right? Or whatever, yeah, or no? Actually, there's just the one, okay. EL: Yeah, what does the Rado graph mean? JA: Okay, so there, there are two ways — well, actually, there are a bunch of ways to approach it. I'm only going to talk about two. One is that it's the almost sure result of the countably infinite random process. KK: Okay. EL: So, so we're thinking like, let's just say we've got a vertex for every whole number and we want to and then with probability 1/2, we connect, you know, n to m for all n not equal to m. That is that what we mean? JA: Yes. EL: Okay. JA: So that doesn't sound like a definition, though, right, right, EL: Yeah, because I feel like I could make two different things. You know, in one of them there's a vertex between 2 and 3, and in the other one there isn’t. But somehow this is the same thing? JA: Oh, well, yes. I mean, you could get different isomorphic copies of the autograph, right? But with right the probability of you getting a graph that is not isomorphic to the one I get is zero. EL: Yeah. Wild! JA: Yes. And more cool stuff. Going back to that zero-one law, if you have a first-order formula, it has asymptotic probability 1 if and only if it's true in the Rado graph. EL: Wait. Can you say that again? JA: Yes, okay, take a first-order formula in the language of graphs like this graph is complete. That formula is true for almost all finite graphs, if and only if it holds in the Rado graph. EL: Okay, okay, thank you. So the just takes me an extra time through. JA: So the Rado graph, in some sense, tells us what all the true first-order statements are in the language of graph theory. KK: Is it easier to prove these things in the Rado graph? I mean, it's not complete. I get that, but I mean, you know… JA: I don't think so. KK: Okay, just kind of a fun fact. EL: But it might be differently hard and sometimes that's helpful. KK: Sure. JA: Yes, so, yeah, I don't, I don't actually know that. EL: Okay, so, but I feel like I'm marginally, or, you know, provisionally okay with the Rado graph, so yeah. JA: Here’s something that will make you okay-er with it. Okay. The result of this random process actually has a simple construction that is not random at all. EL: Okay, great. JA: Here’s what we do. Our set of vertices is the set of primes that are equivalent to 1 mod 4. EL: Okay. JA: Okay, and here's how we determine whether to put an edge in. So you've got two vertices labeled p and q. Because p and 1 are equivalent to 1 mod 4, by quadratic reciprocity, they’re either both quadratic residues modulo each other, or neither. Okay, so you put the edge in if they are quadratic residues modulo each other, and you don't if they aren’t. EL: Okay. JA: And that gives you something isomorphic to the Rado graph. KK: Oh, okay, no way. JA: I know! It's just ridiculous, right? It's ridiculous. I guess. I mean, the primes are sort of pseudorandom. KK: Yeah. JA: You know, I to my very limited understanding, this is essentially how Green and Tao proved that the primes contain arbitrarily long arithmetic progressions. EL: Yeah, right. JA: Like, if they were random, then that would have to be true. And they are random enough, right? Even though they're in some sense, not random at all, they’re pseudorandom. EL: They’re like, yeah, they are, by definition, not random in the least, right? But they act like they are to, like, any way of looking at them, it’s so wild. JA: Yes, fascinating. Okay, so that that is the Rado graph, yeah. And my theorem is that the Rado graph is, well, in logic, we call it omega-categorical, which means up to isomorphism there's only one countably infinite model of the first-order of graphs, right? KK: Yeah, so… JA: Go ahead. KK: This seems to intersect perfectly for you, right? I mean logic and combinatorics, right? I mean, this is, this is like, if you were going to define something to be like an expert in for you, this is it, right? JA: Yeah. I mean, I really probably should have done a degree in computer science instead of math, because I like combinatorics, yeah. I like doing machine computations. In a computer science department, I would be, oh, the heavy theory guy, whereas in mathematics, I'm that guy who cheats with a computer. EL: Hey, we can all get along. We don't need to have factions here. So how does one even begin to go about proving something like this, that your quadratic reciprocity graph construction thing that you just told us is isomorphic to any other random construction I can come up with? JA: I’m glad you asked that. Okay, so here's the idea of what you do. So it turns out that another way to, sort of obliquely define the Rado graph is the following way. I’m going to define a property and any graph, any countably infinite graph with that property, is isomorphic to the Rado graph. Okay. So given two disjoint subsets of vertices, R and S, there exists a vertex v that is adjacent to every vertex in R and no vertex in S. EL: So any two disjoint sets of vertices? JA: Correct. Okay, so you name any set of vertices and then Kevin names a disjoint set of vertices. I have to find a vertex v that is adjacent to all of Evelyn's vertices and none of Kevin's vertices. And if I can always do that, then my graph is isomorphic to the Rado graph. EL: Okay. We're both skeptical, but okay. KK: This feels like a weird graph, but okay, all right, EL: It’s weird property to try to… KK: It’s because you've got an infinite collection of vertices, right? If everything were finite, this would be bad news, like this would be hard to do. JA: Right, right. It would be impossible. KK: But I guess, because you have an infinite number vertices — yeah, right, yeah — for every disjoint pair, you couldn't do it. But because, okay, yeah. JA: I mean, it's weird. KK: Yeah, okay. JA: Another way of thinking of it is that the Rado graph contains every finite graph as an induced subgraph. Okay, anything you can dream up you can find in the Rado graph. EL: Okay, that sounds like a property of a random graph for sure. JA: So to go back to the proof about the the construction with the primes, we just have to show that given any two disjoint set of primes, there is some other prime that is a quadratic residue modulo every prime in the first set and no prime in the second set. And if I recall correctly, it's Dirichlet’s theorem on prime progressions, right? Something, you know, wave your hands and it all works. KK: It’s fine, year. EL: Call a number theorist or something. JA: Right. I am not a number theorist. KK: Right. JA: I have used some number theory in my work, but I'm definitely not a number theorist. Don't ask me hard number theory questions. KK: All right. Well, okay, I guess I could kind of see now how one might fall in love with the Rado graph, right? I mean, where'd you first come across this? JA: I don't remember it. It was definitely not in grad school. It was later just learning about stuff. Oh, actually, I think I was asked to — yeah, I think this is what happened. I had to referee this paper that was a bit of a stretch for me, and I had to look some stuff up. And I encountered this as I was trying to figure out what was going on in this paper that I probably should have not actually refereed, but it looked interesting. KK: Sure. JA: Bit of a stretch. These days I would say no to that. KK: Well, as a journal editor, let me say to all of our listeners, I edit a journal, please accept referee requests. Please? Nobody wants to referee papers anymore. You might learn something! EL: You might learn something favorite, graph or theorem. KK: That’s right, that's right, yeah, it's a service to the community and it's good for your brain. So this would be really nice. EL: Yeah, so I always like to ask if this was kind of a love at first sight theorem or construction. I'm not quite sure you know what part of it would be, love at first sight, but yeah, how did you feel? Did your your love for it grow? Or develop over time? JA: Oh, it was definitely love at first sight because I just couldn't believe the result. And it actually turns out that the Rado graph is a special case of a more general phenomenon that we don't need to get into. But this notion of a limiting structure that encapsulates all of the properties of the finite structures, that happens in other spaces too. KK: Cool. All right. Part two, the pairing. So what pairs well with, with the Rado graph? JA: Okay, well, the Rado graph is something taken to the extreme, right, right. Okay, so my pairing is called Huntsman cheese. EL: Huntsman cheese. Okay, I'm a little scared. JA: Okay, so this is something my my parents bring me sometimes when they visit from Wisconsin. It's a big wheel of age cheddar, except inside — it's a pretty tall stack — inside are two layers of blue cheese. Okay, so when you cut into it, it looks like a layer cake. EL: Yeah! JA: All right. It's really pretty and it's intense. EL: Yeah, you’ve got two different kinds of intense cheese flavor, right? JA: Yes, it's delicious, though. EL: Yeah, no, I was a little worried there might be, like, organ meats involved or something. In some way, I'm sort of a typical American in that I'm a little not into the offal. So just wasn't sure what these huntsmen were doing with the cheese. JA: No, I don't know why it's called that. KK: Yeah. EL: Oh, that's that sounds good, although maybe hard to like, just a large amount of it would be intimidating. JA: Oh, yes. I mean, it's very rich, so you don't want to eat very much, yeah? EL: Good party cheese. Like, get a lot of people together to help you go through it. JA: Yes, but half of them will refuse to try it. EL: Yeah, cool. Well, then, I mean, that's great too, because then, once you've got your party, you can start to make graphs of who was already friends and who didn't know each other when they came. Then you can start to do other graph theory. You can find some Ramsey kind of theorem examples, or Ramsey theory kind of stuff. And so you could just like, take this towards Rado graphs and towards Ramsey and other, whatever your your graph theorist heart desires. KK: I’ve got to try this now. I mean, I was born in Wisconsin. I haven't been back in many, many years. I do not know the Huntsman's cheese, but we’ll have to find some of this. EL: Put in a special order of the cheese monger. KK: Yeah, right, yeah, the Florida cheese monger. Actually, we do have a local liquor store that that also does cheese, and they like all these weird — I say weird. I shouldn't say weird, unusual imported cheeses from from England and, you know, the really stinky ones and all of that. I'll have to go there. Maybe they have this Huntsman's cheese. EL: Yeah. JA: We like to give our guests a chance to plug anything that they're working on. Or where can we? Can we find you online somewhere? I actually do know some things about Jeremy that, again, our listeners can't see, but he's got this collection of instruments hanging on the wall. EL: I noticed that. Yeah, do you? So it seems like you've got a variety of stringed instruments behind you. So do you record or, like, publish what you play? JA: I do. I play several instruments poorly. KK: I play one poorly. JA: But yeah, I really like the songwriting process. It's sort of, you know, it scratches an itch that mathematics doesn't necessarily. EL: Well, mathematicians are creative people, but this is using your creativity in a different direction. JA: Yes, now that I'm in full time administration, I'm often too tired in the evening to think about mathematical research, but I can strum a ukulele. So it gives me a sort of outlet for creativity that was sort of missing for a while when I went into administration. EL: Yeah. KK: So you, you can be found, are you on Spotify? JA: Yes, yes, right. The band name is the Unbegotten Brothers. And actually, a new single came out just, like, four days ago or something. EL: Oh, congratulations. JA: So if you want to hear yet another 12 bar blues, check it out. KK: Yeah, Jeremy and I have an unpublished 12 bar blues too, that we had a third person lined up to do the singing, and that person never, and we won't out that person, but never followed through with the recording of the vocals. JA: So we will just, we will shame them in private, not in public. KK: And I only play rhythm guitar, and again, not especially well, but well enough for 12 bar blues, right? So. EL: Yeah, it's about enjoying the music-making process. KK: That’s right. JA: Yes. I mean, I write it for myself, not for anyone else. KK: Sure. JA: But the thing I really want to plug for everyone is an open problem called the chromatic number of the plane. EL: Ah, I’ve written about this. JA: Ah, good. There was some shocking, at least shocking to me, progress about four years ago. EL: Maybe might even be a little longer i could find the date on that article, because I yeah, maybe 20. Yeah, I'm not going to hazard a guess. Time kind of gets weird for me before 2020. Or my memory. JA: I think it was pre pandemic, though, so yeah. I was absolutely shocked by that result, because I was convinced that the correct answer was four. At least four in Zermelo Fraenkel set theory with choice. I had convinced myself that the answer to the question depended on which axioms of set theory you adopt. So I was shocked when somebody came up with a finite graph with chromatic number five. EL: Yeah. JA: I was just like, oh, I couldn't believe it. EL: Yeah, yeah. Well, that is — yeah, especially if you were really convinced of this and yeah, that it would you, you would require looking at that kind of set theoretic aspect of it in order to eventually prove it, I assume, then, yeah, you got your socks knocked off. JA: Yes, yeah, I had drawn this analogy. There's an object in mathematical logic that's pretty important, called a non-principal, ultrafilter. Yep, and you can't really construct it. You have to appeal to Zorn's lemma, so you you can't write down. I mean, there are literally no examples. KK: Right. JA: But they exist, right? And I kind of thought that there was a four coloring, but we would never be able to describe it. EL: Right. JA: And maybe, maybe in different versions of set theory, the answer would be different. In fact, the thing I was originally going to talk about as my favorite theorem is, just very briefly, there exists an infinite, a graph with continuum many vertices, such that in Zermelo Fraenkel set theory with choice, ZFC, the chromatic number is 2. And in ZF plus countable choice, plus the axiom that all subsets of the reels are Lebesgue measurable, the chromatic number is uncountable. EL: Yeah, I'm glad you didn’t. We would have kicked you off! Yeah, that is wild. So we did maybe skip a little bit for our listeners. So what is the question of the chromatic number of the plane? JA: Ah. So imagine you are coloring all the points of the plane individually. Okay? And what we're trying to do is not have any two points that are exactly unit distance apart the same color. And it turns out that you need at least four colors. That's an exercise you could assign to undergraduates. Seven suffices. You just, there's a nice little pattern with EL: Hexagons? JA: Pentagons? Must be hexagons. KK: That sounds right. JA: I haven't thought about this in a little while. But then until 2019, or so, that was all we knew. I mean, we had some conditional results. Like someone showed that if the color classes are all measurable, then you need at least five colors. EL: Okay. JA: And then somebody else showed that if you use, like, even nicer sets than measurable sets — I can't remember what it was. Basically like, if you're using sort of rectilinear shapes or something like that, then you need six colors. KK: Okay. JA: But no unconditional results, until fairly recently. EL: So yeah, if you're you know, needing a problem to either put you to sleep or keep you up at night, depending, that's a good one to just kind of try to try to roll around in your head. KK: Right. Cool. All right. Well, this has been great fun, Jeremy. thanks for joining us. EL: Yeah, thanks for joining us. JA: Thank you for having me. KK: Yeah. It was great. I learned some stuff today. All right, all right. Take care, man. JA: Okay. Bye. [outro] On this episode, we enjoyed talking with Jeremy Alm, a math professor and associate dean at Lamar University, about the Rado graph. Here are some links you might find interesting after you listen.Alm's website and his band the Unbegotten BrothersThe Rado graph on Wikipedia and the Visual Math Youtube ChannelOmega-categorical theory on WikipediaEvelyn's 2018 article about recent progress on the chromatic number of the plane

Episode 93 - Robin Wilson

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 MathematicianMore on the Poincare-Hopf index theoremThe 2021 SLMath Workshop on Mathematics and Racial Justice and its follow-up, to be held in May 2025Storytelling for MathematicsThe Algebra ProjectThe 2025 Critical Issues in Mathematics Education workshop, to be held in April 2025, focusing on mathematical literacy for citizenship

Episode 92 - Kate Stange

Evelyn Lamb: Hello and welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb, a freelance writer in Salt Lake City, Utah, where it is gorgeous spring weather, perfect weather to be sitting in my basement talking to people on Zoom. This is your other host. Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. I don't know, Evelyn, I saw the pictures on Instagram over the weekend and it looked cold in Utah. You wrote that you rode a century, right? EL: Metric century. KK: Okay. Metric. EL: Just in case — you know, I don’t want people to think I'm quite that hardcore. Yeah, at least at this point in the season. Yeah, I hadn't managed to ride more than about 25 miles since last fall because weather, travel, just things conspiring against me. The week before I was like, I really need to get 30 or 40 miles in on Monday. And then it was, like, 20 mile an hour winds and sleet and I was like, well, I guess I'm just going into this cold, but it was fine. It was actually gorgeous weather. It was a little chilly at the start, but better than being too hot. KK: Well, you know, the muscle memory takes over, right? So you can do — I mean, 62 miles isn't that much more than 25, really, once you have the legs, so congratulations. EL: Yes, thank you. Well, we are delighted today to be joined by Kate Stange, who is in the Mountain Time Zone, something that I always feel thrilled about because I'm constantly converting time zones when I'm talking to people, and finally, someone I didn't have to do that for! So Kate, other than being in the Mountain Time Zone, what would you like to tell us about yourself? Kate Stange: Oh, geez. Um, well, I'm also a cyclist. And so I'm jealous hearing about your rides. EL: Wonderful! KS: Here in Colorado we have we have this ride called the The Buff Classic. And so it has a 100 mile option where they close Boulder Canyon so that you can bike up the canyon without any cars. EL: Oh, wow. KS: Then you bike along the peak to peak highway. It's just wonderful. EL: Yeah. KK: Yeah. That sounds great. EL: So you're at CU Boulder? KS: Yes. And it's run from the campus. It starts right outside the math department. EL: Oh, perfect. Yeah, just drop your stuff in your office and hop on and ride it? KS: Yeah. EL: Yeah, great. Well, we are thrilled to have you here today. And I guess we can just dive right in. What you're on what do you like to tell us about? KS: My favorite theorem, at least for today, is the bijection between quadratic forms and ideal classes. KK: That’s a lot of words. EL: Yeah, and I'm so excited to hear about this, because I am honestly a little nervous about both quadratic forms and ideal classes, and a little embarrassed about being nervous about quadratic forms, not so much with ideal classes. So yeah, can you tell us a little bit about what that all means? KS: Yeah, sure. So quadratic forms is probably what sort of comes first in the story, at least sort of the way that the mathematics tells it, and also probably the historical way. And so a quadratic form is just a polynomial with an x2, a y2 and an xy. So it's like 3x2 + 7xy − y2. So that's quadratic form. And, as number theorists, one of the things that we're most interested in studying is what are the integer solutions to polynomial equations? And so first you start with linear equations. And there's a wonderful story to do with Euclidean algorithm and stuff there. And then you move on to quadratic. And really, these are sort of some of the first equations that you would start studying next, I guess. And so they go back to the classical days of number theory, Gauss and Euler and everybody. And, yeah, so they come in, what happens is that they come together in families. So different quadratic forms, you can actually just do a change of variables. And it'll look different, but it won't really be too different, particularly if you're interested in what numbers it can represent when you put in integers. So say I take x2 + y2, which is the simplest one, if I put in various different integers to that I get various different integers out. And then if I do a change of variables on that, just a little change of variables — like maybe I change x to x + y, but I leave y alone — the formula will look different after I do that change of variables, but as I put in all integers and look at all the stuff I get out, those two sets, the in and out, they're going to look the same. And so we kind of want to mod out, we want to ignore that difference. So I'm really thinking of equivalence classes of quadratic forms. So that's the first object. EL: And that change of variables is kind of the only equivalence class thing that happens with them? KS: Yeah. Yeah. Because they could really behave differently between the different classes. KK: And you only allow a linear change of variables, right? KS: Yes, exactly. Yes. Thank you. EL: Yeah. Okay. So now, ideal classes. KS: Now ideal classes. So this is an interesting one, to describe where it comes from, I think. So there's sort of — if you think about the history of math, I would say there are sort of two versions, there are sort of two histories of math at the same time. There's one, which is sort of the human history, which is fascinating and human and quirky. And then there's sort of the way that the math would like to unfold to human understanding the way that as any human coming to it, they might discover the pieces of the mathematics. And I don't know too much about the details of the human history. But in terms of how you might discover this, if you're just looking at the integers, you are interested in how they behave, you discover things like prime numbers, you've got addition, you've got multiplication, you've got powers, you might ask how these things interact. And at some point, maybe when complex numbers are discovered, you think about whether there are possibly other collections, or other number systems, other collections of numbers in which you could do the same kind of thing. And so one of the first examples of this would be what's called the Gaussian integers, which is where you take complex numbers, I don't know whether I should dive into complex numbers, but you take complex numbers with integer coefficients. EL: So that means things like 1 + 2i or something. So the i and the 1 both have integers in front of them. KK: Right. KS: Yeah, exactly. And so this is a collection of integers, kind of, right? And we ask things like, okay, are there prime numbers? And so it turns out that there are in that system, there are Gaussian prime, so, like, 1 + i is a prime number. And so you kind of start to develop this whole theory that you have for the integers. But what you find is that in some of these systems, you lose unique factorization. So we love unique factorization in the integers, right? Every integer, up to reordering the factors and maybe putting on a minus sign, you have always a unique factorization into prime numbers. And in the Gaussian integers, that's true. But in some of these other systems, you lose that. And so what people tried to do is to try to fix it. And it turns out, the way to fix it is to add in what were I think, originally called ideal numbers. They were thought of as numbers that should be in the system that weren't in the system. And what they actually were were collections of numbers. They were sets of numbers instead of individual numbers. And the idea here is that, say, you were to take — like in the integers, if you took the number two, you could replace that idea, that idea of two-ness with just the collection of even integers. And so that's an ideal now. Instead of a number, it’s an ideal, and it's really carrying the same information. But now it's a subset. And so by moving from individual elements of the ring, of the collection of numbers, you move to subsets of them. Now you have more things, and so now you can recover unique factorization in that world. So those are ideals. EL: Yeah. And so the Gaussian integers do have unique factorization. KS: They do. Yeah. EL: So this — actually, I kind of forgot, but recently, this came up in something I was writing where I wanted the example to be the Gaussian integers so bad because it doesn't have any square roots in it. But then it didn't work because it isn't true for that. I was trying to show how unique factorization could fail, but I didn't want to have to use square roots. But as far as I know, you can't do that. So then I fixed it by putting a square root of negative five in there and hoping that people would be patient with me about it. But yeah. KK: So that's the example of one where you don't get unique factorization, right? So you take the integers and you join the square root of minus five. That's one example. KS: That’s one of them. Yeah. EL: And then it's like two and three are no longer primes. KS: So if you multiply (1+ √ −5) × (1− √ −5) KK: You get six. Yeah. KS: You get six, which is also two times three. And those are two different prime factorizations of six. KK: Right. EL: Yeah, but it's so fun that you can do that, and I like your way of putting it where regardless of how these ideas actually formed, you can as a human, looking at some of these basic pieces like primes and then or primes and integers and square roots and things, you can kind of come up with this, like, what happens if I do this? And create this new thing where this this property that I know I always assumed — like unique factorization, when you're growing up, you know, when you take math classes in school and stuff, it just seems like so basic, like, how could you even prove that there's unique factorization? Because how else could you factor anything? KS: Yeah, exactly. EL: It feels so basic. KS: Yeah. And this is what happened, I think, historically, too, is that people didn't expect it to fail. And so they were running into problems and it took a while to figure out that that's what was going wrong. KK: Wasn’t this part of, was it Kummer who had a reported proof of Fermat's Last Theorem, and he just assumed unique factorization? KS: That’s what I've heard, although I never trust my knowledge of history. Yeah. KK: It’s probably true. EL: Well, and there are a lot of good stories. And they may or may not be true sometimes. But yeah, okay. So we've got these, these two things. KK: Yep. EL: The quadratic forms and the ideal classes. So yeah, I guess either historically or mathematically, what is this connection? And how do you know that these two things are going to be related? KS: Yeah, so they seem like different things. So I think quadratic forms were studied earlier. And at some point, people noticed that quadratic forms had an interesting property, which is that sometimes you could multiply them together and get another quadratic form, which is kind of hard to explain. But like, if you actually wrote out (x2 + y2) × (z2 + w2) and you multiplied that all out, you'd have a big jumble. But then you could factor it out. So it looked like, again, a square with some stuff inside, z’s and w's and whatever inside the brackets, plus a square. And so this meant that sometimes if you picked your forms correctly, and they had this sort of relationship, then if you looked at the values they represented, the numbers that can come out, when you're putting integers in, you would take that set of things the first one represents and the set of things the second one represents, and then you’d look at what the third one represents, and it would represent all of the products of those things. So there was this definite relationship, but the way I'm describing it to now is a little awkward, because it's a lot of algebra. But this is, I think, what was noticed first, somehow. And again, I might be mixing the human story with how math tends to want to unfold. I don't know exactly the history. But anyway, so you notice that there's this relationship. And that's kind of reminiscent of an operation, like a multiplication law. And what happens is that, in fact, that's coming from the fact that these ideal classes, each one of them — sorry, my mistake — so it's from the fact that each of these equivalence classes of binary quadratic forms, each one of them is associated to an ideal. And the ideals as the sort of generalization of the idea of number, they can be multiplied together to get new ones. And so on the ideal side, it makes sense that there's an operation because you're already living in a number ring where you've gotten an operation. But on the quadratic forms side, it's a surprise. And so that's one of things I like about this theorem is that you see some structure and you want to understand why. And the reason to understand why is just to change your perspective and realize these objects can be viewed as a different kind of object where that behavior is completely natural. Yeah, so that's one thing that I like about it. EL: And does this theorem have a name or an attribution that you know? KS: Oh, it's such a classical theorem that no, I don't know. KK: Right. It's just the air you breathe, right? So what's the actual explicit bijection? So you've taken a quadratic form. What's the corresponding ideal? KS: Well, actually, the other way is a little bit easier to figure it out. KK: Yeah, let's go that way. KS: So let's take the Gaussian integers, okay. And in the Gaussian integers, you've got — for your ideal, so think of it as a subset of the Gaussian integers. But because it's an ideal, it has the property that it has the same shape as the Gaussian integers. I actually usually like to draw a picture. So I'm going to try to draw a picture just out loud. So if you think of the Gaussian integers in the complex plane, they fill out a grid, right? It's all the integer coordinates in that plane. So that's a grid. And if you want to see what the ideals are, they’re subsets that are square grids as well, but fit inside that grid that we started with, maybe rotated or scaled out. EL: Okay. KS: But they're square again. EL: Okay. KS: And so, what you can do is with this example, specifically, you can take the norm of each of these elements in the Gaussian integers. So the norm of a complex number, usually I think of it as the length from the origin. But I don't want to do the square root part. So if I have a Gaussian integer x + iy, I'm going to take x2 + y2, and that's the norm. KK: Okay. KS: All right. And so if I take the whole Gaussian integers, which is itself an ideal, that's one of the subsets that is valid as an ideal, then if I take all of the values, all the norms of all those elements, that's all the values of x2 + y2. So from my collection of integers, I take all of the values and that's actually a quadratic form. KK: Okay. KS: Okay? KK: Okay. KS: And so you can do this with the other ideals as well. So for each one, you look at the norms of all of its elements, and that is a quadratic form and the values of that quadratic form? KK: Right. So the Gaussian integers are Euclidean, right? So it's PID, right? KS: It is. It’s a principal ideal domain. KK: So everything's generated by one element, basically every ideal? KS: That’s right. KK: So that makes your life a little simpler, I suppose. KS: Yeah. So the ideals, in that case, really, they're not so different than the numbers themselves. This is one of those ones where you don't have to go to ideals. But by going to it, you think about instead of just, say, 1+i the number, you think about all the multiples of 1+i and you take all of those, and you take their norms. EL: Okay. And I told you, when we were emailing earlier, that you'd have to hold my hand a little bit on this. So yeah, sorry, if this is a too simple question or something. But like, what is the quadratic form like the x2 + whatever xy +whatever y2 that you get from the the Gaussian integers that you just said? KS: Right. So if we take the Gaussian integers, if I take x+iy as a Gaussian integer, its norm is x2 + y2. That’s the form right there. EL: Okay. Yeah. All right. KS: And then if I were to take a subset, like all the multiples of 1+i, I'm not plugging in all x's and y's. I'm plugging in only multiples of 1+i, so you end up with a slightly different form popping out. EL: Yeah, so I guess it's kind of like x+x then. KK: 2x2 squared basically, right? KS: Yeah. Yeah. You could have Yeah, various things in various different situations, but yeah. EL: Okay, thank you. Yeah. And so, yeah, can you talk a little bit about how you encountered this theorem? If it was something that like you really loved to start out with? Or if your appreciation has grown as you have continued as a mathematician? KS: Yeah, well, it's one of these things, so I think everybody has things that they're attracted to mathematically, they all have a mathematical personality. And there's some sort of particular kinds of things that attract you. And for me, one of the things is the sort of projection theorems that tell you that a particular structure, if you look at it a different way, has a whole different personality. And it's actually the same thing, but it has just become totally different. So I really love those things. And I've always loved number theory, because it has such simple questions. But then when you dig into them, you always run into such fascinating, complex structure hidden. And so this is one of those things that if you have that kind of personality thing you, just keep bumping into. And so for me, and all of the research that I've done and things I've been interested in, I keep coming back to this theorem and bumping into it in different places. It shows up when you study complex multiplication of elliptic curves, it shows up when you study continued fractions, it shows up all over the place. And it just seems so fundamental. And it's sort of like maybe one of the most fundamental examples of this special kind of theorem that I really enjoy. EL: Okay. KK: Cool. EL: All right, well, then the next portion of the podcast is the pairing. So yeah, as you know, we like to ask our guests to pair their theorem with something that helps you appreciate the theorem even more. What have you chosen for that? KS: So, when I think about this theorem, I just it's a treat. So I think the only thing that comes to mind really over and over again is just chocolate. I love chocolate. And that's what you should enjoy this theorem with because maybe you should just be happy enjoying it. KK: I mean, chocolate pairs with everything. KS: That’s true. It's a bit of a cop out. KK: No, no, that’s okay. So our most recent favorite chocolate is Trader Joe's has this stuff. And it's got, I don’t even know what’s in it, pretzels and something else crunched up in these like bark of chocolate. And it’s a dark chocolate I really recommend it. So you must have a Trader Joe's in Boulder, right? EL: Are you a dark, milk, or white chocolate person? KS: Oh, definitely dark. Yeah, I have a dark chocolate problem, actually. EL: Yeah, the Trader Joe's. For me the dark chocolate peanut butter cups are are always purchased when I go to Trader Joe’s. KK: Dark chocolate feels healthier, right? It's got more antioxidants and a little less sugar. So you're like this is fine, less milk. Okay. All right. It's actually it's a fruit, right? EL: It’s a bean. You’re having a black bean pate right there. KK: That’s right. EL: Yeah, well, Salt Lake is actually a hub of craft chocolate. We have some really wonderful chocolate makers here, like single origin, super fancy kind of stuff. So if either of you are here, we'll have to pick up some and enjoy together. And yeah, along with quadratic forms and ideal classes. KS: Sounds wonderful. EL: Yeah. So something I meant to talk about this earlier in the episode, but you mentioned that you'd like to illustrate things, and that is how we first met is, through mathematical illustration. So I don't know, maybe it's a failure of imagination on my part, but I always, I'm always fascinated by like, number theorists who are really into illustration as well, because I think of, like, geometry, as you know, it shapes it as the more naturally illustrate-y parts of math. But would you talk a little bit about it, you know, illustrating number theory? And if if you've done anything related to this particular theorem, or if there's something else you want to talk about with your mathematical illustration? KS: Oh, yeah, that's a that's a great idea. Yeah. So there's actually building up gradually a wonderful community of people who are interested in illustrating mathematics. And so that's maybe one of the things that you could add a link for is the website for the community. EL: Definitely. KS: Yeah. And so I've always found that the way I think about mathematics is very visual. I mean, I think as human beings, we have access to this whole facility for visual thinking, because we're embedded in this three-dimensional world that we're living in. And another way that we think about mathematics, I think often, is we're using another one of our natural facilities, which is our sort of social understanding facility, where we imagine characters interacting with each other and having motivations and stuff like that. But for me, it was always a very visual thing. And so even though it wasn't taught in that way, in my mind, somehow these things were always very visual things. And so I've always been really attracted to situations where you can see some hidden geometry in number theory. And with this particular theorem, there is a little bit of nice hidden geometry. I mean, the first hint of this is that when I talked about ideals in the Gaussian integers, I visualized them as a lattice. EL: Yeah. KS: And in all of these number rings, you can do this, you can you can think about lattices. And you're really talking about lattices, and lattices have things like shape. And you know, there's lengths and angles and stuff like that to talk about. And so one of the really cool things that you can do is you can think about, for example, with the Gaussian integers or with some other ring of interest that you can put in the plane like this, into the complex plane, then you can ask this question, it's a natural question that people ask: how can I study the collection of objects instead of the individual objects themselves? So if you want to study the collection of lattices, say, two-dimensional lattices in the plane, then one way to do it would be okay, how do I decide on a lattice? Well, I have one vector that's generating it, and then another vector that's generating it. So let's put the first one, let's sort of ignore issues of scaling and rotation, let's put the first one down pointing from like zero to one. And then the other one is somewhere, but now you don't have any choice anymore. No more freedom. And so you can think of the plane itself as a sort of moduli space, as a parameter space for the collection of lattices. And this space has a lot of beautiful properties. So you might as well order your vectors so that we're just talking about the upper half plane. So the first vector is from zero to one, and the other one is an angle less than 180 degrees from that. And so when you start looking at the geometry of this plane, and you want to talk about well, okay, I want to look at lattices, but maybe I don't care so much about what basis I'm using for the lattice, you start to divide the plane up in certain ways. And what you discover is that the natural way to talk about this plane is using hyperbolic geometry, actually. And so all of a sudden, you're doing hyperbolic geometry. And I find myself doing hyperbolic geometry sometimes when I'm doing number theory, because when I want to look at these these objects and stuff, that's just the natural world in which they live. I mean, the mathematics kind of tells you what you have to do you. You know? EL: Yeah. KS: And so those are moments that I really enjoy, because you're doing something that you think is just some algebra, but all of a sudden, it turns out it's geometry. EL: Very cool. Yeah. So we will include a link to your website, which I know has some of the cool illustrations that you've done available there. And to the illustrating math, there's an online seminar that meets monthly that is really nice to go to, if you can. And, yeah, it's a lot of fun. And I, yeah, just so many different fields of math represented with that in ways that I never would have guessed. KS: That’s true. That's one of the nice things about that community is that there's people from all different areas that you wouldn't normally interact with, because usually you have a pretty narrow research circle, if you're doing research in mathematics. But there, you're talking to everybody. And it has a much more creative feel for that reason, you get surprised by people's ideas, because they come from just a little bit farther from your home base, you know? EL: Yeah. And I think it also kind of pushes people to really think about how they're explaining things, where you have a shorthand when you're working with someone who is in, or talking to someone who's in such a close field, and since you don't necessarily have that same common background, people, I think, it seems like are very thoughtful about how they describe things and what they assume that you already know. KS: Yeah, exactly. It's just good to get out of your little corner. EL: Yeah. KK: All right. Well, this has been great. I definitely learned something today. I did not know this connection between ideal classes and quadratic forms. KS: Oh, I thought of one more, one book I'd like to plug. KK: Okay. Please do. EL: Great. Yes. KS: Yeah, so Martin Weissman has written a book called — I'm going to get the title slightly wrong. It's An Illustrated Theory of Numbers, maybe? Oh, you have it. Oh, I got it right. EL: Yeah. An Illustrated Theory of Numbers. It's been holding up my laptop, after I read it, I will say. KS: Yeah, and so you were asking about illustrating number theory, and this is just a beautiful book. It's completely accessible. I used it when I was teaching an introduction in number theory for undergraduates. But it doesn't require any particular background because it starts from, you know, we’ve got the integers, we’ve got addition, we’ve got multiplication, let's do some stuff. And and he really looks hard for ways to turn theorems which are usually completely algebraic into something visual, and they're just lovely. EL: Yeah, and really amazing illustrations, and full color, like everywhere, which I know is more expensive to make books, and that's why books so often have the color in the middle, like in a little section and not the whole thing. But I do think this is just much more pleasurable to read because it it is does use that aspect, too. And it's not as stark as every page being black and white. KS: Yeah, it's so inviting. It's a wonderful book. EL: Yeah. Great recommendation. Thank you so much for joining us. I really enjoyed talking with you. KS: Yeah, me too. Thank you so much for having me on. [outro] For this episode, we were excited to talk to Kate Stange from the University of Colorado, Boulder about the bijection between quadratic forms and ideal classes. Below are some links you might find interesting as you listen.Stange's websiteThe Illustrating Mathematics website and seminar, which meets monthly on the second FridayAn Illustrated Theory of Numbers by Martin WeissmanThe Buff Classic bike ride in Boulder

Episode 91 - Karen Saxe

Kevin Knudson: Welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I am joined, as always, by my fabulous co-host. Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, trying to remember how to do this. It's been a minute since we've recorded one of these. We kind of went dormant for the winter. KK: Yeah, a little bit, a little bit. Yeah. But Punxsutawney Phil told us — I don’t, what did he say? Let's pretend he said six more weeks of winter. EL: I think he usually does. I don’t know. KK: I mean, objectively, there are always six more weeks of winter. Like, the calendar says so, right? EL: Yeah. KK: Anyway, yeah. EL: And, you know, he probably is pretty good at seeing shadows if he's a prey animal because he'd be used to seeing, like, a bird coming overhead. KK: That’s an interesting question. EL: Do birds eat groundhogs? KK: That’s what I was going to wonder. I mean, like, eagles, maybe, but groundhogs are pretty large, right? I mean, EL: Yeah. What eats groundhogs? KK: Well, that's something to investigate later. EL: Yeah. KK: So it is Pi Day, right? EL: It is! Well… KK: We’re actually, we're recording this on Pi Day. When our listeners hear this, it won't be, but we're recording. EL: And, I always have to put in a plug for my calendar. KK: That’s right. EL: The AMS math page-a-day calendar on which Pi Day does not occur on this day. KK: That’s right. EL: There are other Pi days on this calendar, none of which is this day, my little joke here. So you can find that in the AMS bookstore. KK: Right. Are you Team Pi or Team Tau? EL: I’m Team whichever one works for the calculation that you’re doing. It’s not that big a deal. KK: That’s right. That's right. Okay. All right. Enough of us, enough of our useless banter, although we did discuss what's the ratio of banter to actual talk, right, that there's, there's like a perfect ratio. But we are pleased today to welcome Karen Saxe. Karen, why don't you introduce yourself and let us know all about you? Karen Saxe: Hi, there, everybody. So first of all, happy Pi Day. If listeners know who I am, I was a professor at Macalester College for about for over 25 years. And then about seven years ago came to work at the American Mathematical Society, where I am very happy to be the director of the Government Relations Office. So I work in DC with Congress and federal agencies. And could quite a bit about this. I'm also happy to be here because it's Women's History Month. And it will be appropriate that it is Pi Day when you hear what my favorite theorem is. KK: Okay, good to know. So, I'm curious to know more about this government relations business. So I mean, I know that the AMS does a lot of work on Capitol Hill, but maybe some of our listeners don’t. Can you explain a little more about what your office does? KS: Yeah, so we do a lot of things. So first of all, we communicate — I sort of view the work of our office as going two ways. One is to communicate to Congress why mathematics is important to almost everything they make decisions about, you know, our national security, health care, you know, modeling epidemics, thinking, like you’re in Florida, thinking about how to model severe weather and things they care about, and then why they should fund fundamental research in mathematics and all sciences. And then also you know, how they make decisions about education. So we tell Congress, we give them advice and feedback on our view about what they should do in those realms. And then on the sort of flip side, I tell the AMS community, the whole math community about what Congress is doing and what's happening at the agencies like the NSF, and Department of Defense and Department of Energy, that that they might care about things, things that would affect their lives. So that’s sort of it in a nutshell. I spend a lot of time on the hill. I just came this morning, I went to a briefing put on by the National Science Board, which is the presidentially-appointed board that oversees the NSF. And they put out a congressionally mandated report every few years on the state of, it's called the indicators report. I'm sure I found it more interesting than everybody else, but it's pretty fascinating. You know, it covers everything from publications around the world, like which countries are are putting out the most science publications, what the collaborator network looks like around the world, and that to sort of US demographic information about education, you know, who's getting undergraduate degrees? Who's getting two year degrees? Who's getting PhDs, that that sort of thing. It covers a lot, actually. Pretty interesting. KK: Yeah, yeah. All that in like two hours, right, and then it's over. KS: Yeah, all that in two hours. And then they give you the big report that you can. And I've got them sitting in front of me. But given that this is a podcast, showing things doesn't work. KK: Well, we do it all the time. KS: Here’s one of the reports I picked up this morning. Actually, one really, so they're, you know, they're one thing. And you might end up cutting this, but one thing that's sort of fascinating to me is they always list barriers for getting into STEM degrees. And you know, there are things listed, like college accessibility, things that — and even going back. So like, you know, school kids who say they don't have science teachers in their schools, they don't have math teachers, but they've added to this list. “I can't support my family on a graduate student stipend.” So this is something. EL: Yeah. KK: That’s real. KS: And we are, we've endorsed a bill in Congress that would look that would help to improve the financial stability, I guess, you would say, or the ability to be a grad student or a postdoc. So it's looking at stipends, it's looking at benefits, you know, leave time, all that sort of stuff, making it a job that you can choose to take when you're 23, and have a family to support and could make a hell of a lot more money doing something else with a math undergraduate degree. EL: Yeah, and not see it as something where it's like, you're kind of putting off real life for a little longer, which I think maybe in the past was more of the model, like, oh, yeah, you'll have a real career later. But you know, in your mid-20s, you'll just keep being a student and not have kids or, you know, things, you know, not have parents to support or things like that. KS: Exactly. KK: Yeah. Okay. That's, that's good to know. Thank you for all that hard work you do, Karen. So but this is a math podcast. KS: Right. KK: So what’s your favorite theorem? KS: Okay, so first, I'm going to tell you about the three theorems that I didn't choose. KK: Cool. EL: Great. KS: So — I'm sure everybody goes through this — and thinking about my research, it would probably have to be the Riesz-Thorin interpolation theorem, which basically tells you that if you've got a bounded linear operator on two Lp spaces, then it's bounded on every Lp space in between those two values of p, so I used that all the time when I did research on that sort of thing. Then, but I was primarily a teacher of undergraduates, and kind of my two favorite theorems to teach are always Liouville’s theorem and, and then the uncountability of the real numbers. EL: Yeah. KS: And Liouville, they’re the one that says, you know, that there's a bounded — if you have a bounded entire function function, it's got to be constant. And the result is so stunning, and it gives a great proof of the fundamental theorem of algebra, that every non-constant polynomial has a root. So I always love teaching that. And then of course, like, Cantor’s diagonal argument about the real numbers, nothing beats that proof in terms of like, cool proof, in my opinion. EL: Yeah. All-time great. KS: Yeah, all-time great, right. And I think it's been mentioned on your podcast before. But what I picked was this theorem that says that if you have a given fixed perimeter, then the circle maximizes the two-dimensional shape you can make, so the isoperimetric theorem. EL: Nice! And as you said, very appropriate for Pi Day. KS: Yeah, which, I hadn’t even thought about that, which is sort of also embarrassing. But until we started acknowledging Pi Day, I hadn't thought about that. So another way to say it, or the way you might see it in a textbook, is if you have a perimeter P and an area A, then P2−4πA is greater than or equal to 0, with equality if and only if you have a circle. So this theorem has a very long, fascinating history. Lots of great applications. And for all those reasons, I love it. I love history. KK: Yeah. KS: I love math. KK: Yeah. Do you have a favorite proof of this theorem? KS: I do, actually. Yeah. Well, I didn't know you'd ask that. So there are a lot of proofs. And the one that I like, and this comes from being an analyst probably, is in the early 1900s. Hurwitz gave a proof using Fourier series. I love that proof. And proofs are quite old, going back thousands of years to the Greeks. And then in 1995, Peter Lax actually gave a new short calculus-based proof. But I like the Fourier series proof, just because I like Fourier series. EL: Yeah, that's a topic that I wish I understood better. Somehow I kind of missed really, ever feeling like I've really got my teeth into Fourier series. Maybe that's a little embarrassing to admit on a math podcast. KK: I don’t know. I took that one PDEs class as an undergrad and, like, that's where you see it, you know, doing the — whichever, the wave or the heat equation, whichever one it is — maybe both? I don't know. And then that’s it, that shows you how much I remember, too. KS: Yeah. Good. So you're not gonna dare ask me to give you that proof or anything? EL: Yeah, generally, a proof like that on audio is not the ideal medium. KS: It doesn’t work. EL: Actually, you brought up these ancient proofs. So yeah. Yeah, I guess how long has humanity known this fact, do you think, or do you know? KS: So it's considered that the Greeks knew the proof. And then it was proved around 200 BCE. It even features in Virgil's version of the tale of Dido, Queen Dido. EL: Oh, that’s right. KS: So yeah, I think that was around 50 or 100 BCE, after the Greeks knew the theorem. So can I say what that story is? EL: Yeah. KK: Yeah, please. KS: So she apparently fled her home after her brother had killed her husband. Okay, so we're already in an interesting phase. She somehow ended up on the north coast of Africa after that, and she was bargaining to get some land. And they told her, oddly, that that somehow she could get as much land as she could enclose with an oxhide. KK: Okay. KS: And so she took this oxide and cut it into very thin strips, and then enclosed an area, that was the largest she could conceive of, with the given per perimeter. KK: Okay. KS: So there's that. So it appeared, like, 2000 years ago, or more, and then you sort of we sort of jumped into the early 1800s when Steiner gave geometric proofs. But what's kind of fascinating is his proofs all assumed that a solution existed. And I haven't looked at these proofs, at least not in a long time. But then later in that century, Weierstrass is credited with giving a proof that, well, first, he proves that a solution does in fact exist. And he did use the calculus of variations to get this proof. So that's, that's sort of the story of the, of the theorem. EL: Yeah, this actually — you know, we say the Greeks knew this, but I kind of wonder if this is one of those things that humans would kind of intuitively know, even if they're not in a framework where they have language about proving mathematical theorems, even if that's not an aspect of, of their culture, but it seems like you're trying to get into the mentality of like, what is really intuitive or innate about mathematics for humans? And I wonder if that, you know, we kind of would understand, well, if I took a square or something, I could sort of bow it out a little bit, and get a little more area with the same string. KS: Actually, I mean, one reason I love this theorem is you can give string to kids, and I used to do this, like in elementary schools, and tell them make the biggest shape. And you have to tell them what closed is, no, you have to describe that the string has to come back to where it started. And they all come up with a circle. And this is, you know, second, third grade kids. So it is really intuitive. Yeah. So what it's meant by the Greeks knew this theorem is not 100 percent clear. KK: Because they didn’t even use pi, right? KS: And then actually, Evelyn to what you just said, you know, there's something that's quite interesting to me, which is that, you know, if you think about, you know, shapes of constant width, you know what I'm talking about? EL: Yeah. KS: So, if you take the fixed perimeter, there's an infinite number of these, the circle’s the largest one and those Reuleaux, I think that's how you say his name, those triangles are the ones of smallest area. EL: Okay. KS: And you were just kind of alluding to that, like take a triangle and go puff out the sides, or something. KK: And you can push in. KS: Yeah. Right. And you can do it for any regular polygon. EL: Yeah. Well, British money has a couple of these that are I think heptagons, Reuleaux heptagons? Are they all called Reuleaux? Or just the triangles? I don't know. KS: No, but you’re right about that, they do. And so it's kind of funny, I saw something that was talking about these points, like, what possessed them to make those points? And if you have a machine that has a hole size, and you know, it could fit a circle, it has a diameter, right, but it can also obviously fit one of these other shapes. Yeah. So that works. And I think you're right. It's a heptagon, heptagonal version of those. EL: Yeah. The first time I went to the UK, this was, I think, the most exciting things on my trip to me, was these coins. Like, who thought to make these? And I actually, I remember, I wrote a blog post about it and discovered that it was a little hard to figure out if I had the rights to use a picture because all the images of these coins are like, technically property of the Crown. KS: That’s funny. EL: Abolish the monarchy, man. KK: Her Majesty relented in the end? EL: Yeah, so strange. I was like, well, I'm not gonna beg the queen for the right to post this on my math blog. So I don't remember what happened with that. Hopefully, I'm not opening myself to takedown. KS: I think you’re probably okay. EL: Hopefully the statute of limitations has run out on that. Anyway. KK: I recently came across, I was going through an old notebook, and I found — I don't know why I tucked it in there — from the late 90s. I had one of these 10 Deutsche Mark notes that had Carl Gauss on it. EL: Oh, nice. KK: And so I put it on Instagram. And I'm now starting to worry. Wait a minute. Will the German government come after me? Although it's not really legal tender anymore. EL: Yeah, the pre-2000, whenever they went to the Euro, government. KK: It was pre-Euro. Yeah, I think I’m safe too. EL: But anyway, getting back to the math, Karen. So, has this been a favorite of yours for a long time? I guess to me, this is one that I don't think the first time I saw it, I would have been super impressed by it. So what was your experience? What's your history with this theorem? KS: Right. So like, why did I decide I liked it? Because yeah, it's sort of like, okay, I mean, it's appealing, because everybody can understand it, it’s very intuitive. It's got this, the proof has this interesting history. But why I like it is because you probably know that I'm pretty engaged with congressional redistricting. And when they do measures of compactness of districts, this is the theorem that kind of motivates all their measures. KK: The Polsby-Popper metric, right? KS: Yes, exactly. And so you take the Polsby-Popper measure, which was come up in 1991. So like, different states, should I say something about redistricting? KK: Sure, yeah. KS: So yeah, I mean, just like the very brief thing is every 10 years, we have to do the census. This is mandated in our Constitution, for the purposes of reapportionment of the House of Representative seats to the state so then after the census is done the seats, which we now have 435 of them, they're doled out to the states. And how that's done is a whole nother you know, interesting math problem, more interesting, probably. But then once the states get their number of seats, like how many in Florida? KK: We’re up to 27? [Editor’s note: It’s actually 28.] KS: So let's pretend there's 27 for a minute. KK: I think that’s right. [Ron Howard voice: It wasn’t.] KS: Okay. Then, you know, the Florida Legislature, probably, I don't know who does it in Florida, but somebody. KK: Let’s not talk about that. KS: Yeah, let’s not talk about that. Whoever’s in charge has to carve up the state geographically into 27 districts, one for each representative, and how they do that geographic carving up is extremely complicated. And to answer the question, “Has this been gerrymandered?” there are certain measures of what's called compactness, and this is like a whole nother thing I could talk for hours on. And compactness sort of measures the lack of convexity, sort of, so like, are there long skinny arms going out? And this is where obviously, like a podcast is, is not the best. But in any case, you know, are there long skinny arms going out, or does the thing look like a circle? So the Polsby-Popper measure tells you how close to a circle, or a disk because it's filled in, but in any case, your district is. Well, that's kind of weird, because if you think about tiling any state with circles, it’s just not going to happen. EL: Right. KS: Yeah. So just to sort of fetishize circles is bizarre. But I guess, like, what are your other options? Well, there are lots of other options. But the Polsby-Popper is the most common. There's a handful of states that require specific compactness measures in their process, and many other states that require compactness, but they don't specify the actual measure. In any case, the Polsby-Popper is the most common. And the other common measure is called the Reock measure, and that also fetishizes circles. It's a similar type thing. So with the Polsby-Popper, it's kind of interesting, because they they first published it in a law journal in 1991, in this context for redistricting, but it has actually been mentioned, as far back as the late ‘20s. And can I read you a funny a funny opening line? EL: Yeah, sure. KS: So the it first appeared, as far as I know, in a 1927 paper in the Journal of Paleontology. Okay. And how's this for the start of a paper? In quotes: “How round is a rock? This is a question that the geologist is often forced to ask himself.” Okay. EL: Nice. KS: So that's a great opening sentence. And then it kind of carries on: “when he wishes to consider the amount of erosion that a stone has received.” And then the paper is actually about measuring the roundness of grains of sand. EL: Oh, cool. KS: So there's a lot to say here that the paper is filled with hilarious hand drawings, you know, but also, of course, that geologists seem to be male is another observation. EL: Yeah, well, and the grammar rules of the time. KS: Yeah, exactly. But even just this past January, I ran into a paper that was published, and uses this to measure the aggressiveness. It's in, like, a cancer journal. I can't remember which one. And I wrote it down, but of course, what do you know, I can't see it. Anyways — oh, Cancer Medicine is the name of the journal — and it used the Polsby-Popper measure to measure aggressiveness of tumor growth. So you know, it has a life. EL: That's so so interesting. When you said Journal of Paleontology, I was just like, how is that going to come up in paleontology? But what do you say? Yeah, how round is a rock? It's like, yeah, you do need to measure that. I actually, just the other day watched this interesting video about sand grains and like, certain beaches, or, and certain dunes have different acoustical properties. Due to, like, if they've got a lot of the same sized sand grains and if they pack really well, or if they don't, sometimes there can be the squeaking effect, like when you walk on it, or in a dune, like when there's wind, there can be these like deep, deep resonances, like almost a thunder sound that happens. KS: Oh, that is interesting. EL: And this this video went and looked under the microscope at the sand on these different beaches, and kind of showed how some of them packed together better or worse, and some of them are more uniform. So they might secretly be using that metric. KK: They might. KS: That’s fascinating. I mean, I heard I've heard that squeaky sound on beaches. EL: I never have I'm not a huge beach person. So I guess, yeah, but I'm curious about going to one of these beaches someday now. KS: Yeah. And when you said that I was thinking of the packing, like how they pack, but that would have to do with their shape, and their size. Well, I don't know. KK: So this is a sphere packing question now. And it's yes. EL: Or a “how sphere-y is your sphere”-packing question. KS: How spherey is your sphere? EL: Not quite as catchy. KK: Right. So the other part of this podcast is we like to ask our guests to pair their theorem with something, so what pairs well with the isoperimetric inequality? KS: So naturally, you know, a mathematician would ask, are there analogs in higher dimensions? Right? And then back to how spherey is your sphere, so I play tennis quite a bit. So I'm going to pair it with tennis. EL: Excellent. KS: The shape of the ball abides by the theorem. KK: Yes. Right. KS: And works for so many reasons. EL: Yeah. Well, and you are not the the first My Favorite Theorem guest to pick tennis, actually. KK: That’s right. Yeah. EL: Yeah, we've had Dr. Curto. KK: Carina. EL: Yeah. Carina Curto, paired paired hers with tennis. It was it was about linear algebra. That's right. Yeah. Yeah. Hers was about how this thing kind of goes back and forth. When you're doing this thing in linear algebra. So you picked different aspects of tennis to pair with your theorem. KK: Yep. Do you play much do you, you play, you play a lot? KS: I play — it’s embarrassing to put on a very well listened-to podcast — that I do play a lot, because I don't know how good I am. KK: That doesn’t matter. KS: But I play a couple times a week. KK: I used to play quite a bit. So as a teenager, certainly. And then in my 30s I played a lot. I played a little league tennis. This is when I lived in Mississippi. And actually, my team won the state championship two years running at our level. KS: Oh, wow. KK: But I'm not any good. This was like, you know, I'm like a 3.5. Like, you know, just a very intermediate sort of player. KS: Yeah, that's what I am. KK: Yeah, my shoulder won't take it anymore. KS: I still, I feel lucky. Because physically, I can do it. Right now. I'm in a 40+ league, and that's good. But next season, whatever you call it, or next season, I guess, I'm in an 18+ League, and I've done this before. It means the other players are allowed to be as young as 18. It’s a little humbling, even if we can serve, you know, we have the technical skills, like they’re, you know, like the shots you use in the 40s, like, lobbing is not a good strategy in 18+ because they can run. KK: Back when I was in my 30s and played, I played a lot of singles still, and I could still do it. But when I would come up against the 20-year-olds, it'd be a lot harder. But then I also learned, I used to play a lot of doubles with with these guys in their 70s. And they destroyed me every time. They were just —because they knew where to be. They had such skill and good instincts for where the ball was going to be. It was humbling in that way. KS: Yeah, it's it's fun. And I prefer playing doubles these days. It's just more fun and different strategy. KK: Yeah, and less court to cover. That helps. KS: Less court to cover. And it’s more social. It's a lot of fun. KK: Yeah, so you haven't succumbed to pickleball, have you? KS: I played once, on my 60th birthday. Because no one would play tennis with me. And I got invited to a pickleball thing. And I was like, Okay, we're gonna do it. And, you know, it was fun, but I haven't really. It’s a challenge in Minnesota playing pickleball because it's so windy and the balls are so light, and it’s like whiffle ball. KK: That’s what they are, basically. KS: The ball kind of blows around all over the place. So yeah, I haven't I succumbed to doing that. In DC I'm lucky to have enough people to play tennis with. There's a lot of them. KK: Cool. All right. EL: Yeah. Great pairing. KK: Yeah, yeah. So we also give our guests a chance to plug anything they're working on. You sort of already did that. I mean, you're doing all the work. Anything else you want to pitch? KS: I mean, back to what I do, one reason I love this new job is I get to go in and and make connections to any Congressperson. You know, they have their own interests motivated by their own history, their own life, their own constituents. And this can be — there are obvious things we think about, like people, congressional members who are interested in their electric grid, or ocean modeling for the Hawaii delegation. But it's fun. And it's a fun challenge to think of things. So there's one newish member who was a truck driver before he was elected to Congress. And, we went in and their office was like, we can't make a connection to math. And we started talking about logistics, you know, truck routing. And it was great. It turned into a great conversation where they hadn't really thought about that. So this is what I really love about my job, trying to connect math to anything they’ve got. What they’re interested in, I'm gonna I'm gonna try to connect math, and there are very few issues that that can't be connected. EL: Yeah, well I actually have a question, something that our listeners might be interested in is like if a mathematician is listening to this, and wonders, how can I get more connected to what's happening? How can I understand what math and science, you know, representatives do on the hill? Is there a newsletter or a website or something that you have that they could look at? And, you know, maybe find ways to get more involved? Or at least more informed? KS: Yeah, definitely. So first of all, I used to write a blog, but I don't do that anymore for the AMS. The AMS Government Relations page — so my office is the Office of Government Relations. And I believe if you search, AMS government relations, you'll get to my webpage, you know, the one that I call mine, and you'll see a lot of different things there. There are ways to get engaged. We offer felt three fellowships. Two are for graduate students, one is for a person with a PhD in mathematics to come and to come here physically and do things. One is a boot camp for graduate students, a three-day graduate boot camp to come learn about legislative policy. And then the the biggest one is a year long fellowship and working in Congress. I do hill visits with people. And you know, I'm pretty willing to bring almost any mathematician to the hill, and that can be virtual these days. So we have volunteer members through our committee work who fly in and do these hill visits. We did this last Wednesday, we had about 25 AFS, volunteers fly in, and that was a fantastic day. But I can do them virtually. I've done them with big groups of grad students from departments, and people can email me if they want. And I think you guys have my email. EL: Yeah. Thanks. KS: So those are the big ways. And then for AMS members who are a little more advanced in their careers, you can volunteer for AMS committees. And there's the Committee on Science Policy, which really focuses on this one. And then I'm also in charge of the Human Rights Committee for the AMS, which can be of interest to a lot of people. KK: Sure. EL: For sure. KK: Lots going on there. KS: Yeah, lots going on. KK: Well, Karen, this is terrific. Thanks so much for taking time out of your day, and thanks for joining us. KS: Thank you. [outro] In this episode, we enjoyed talking with Karen Saxe about her work as the director of the American Mathematical Society's Office of Government Relations and her favorite theorem, the isoperimetric theorem. Below are a few links you might find relevant as you listen:Saxe's website and the homepage of the AMS Office of Government RelationsA survey of the history of the isoperimetric problem by Richard Tapia The 1995 proof by Peter LaxEvelyn's blog post about 50 pence coins and other British objects of constant widthThe Polsby-Popper test to measure gerrymanderingA public lecture by mathematician Moon Duchin about mathematics and redistrictingThe 1927 Journal of Paleontology article that first uses the Polsby-Popper metric (though not with that name)An Atomic Frontier video about squeaky sandOur episode with fellow tennis-enjoyer Carina CurtoThe 10 Deutsche Mark note