little theorem within a bigger theorem

It’s these quite sturdy—you know, you don't need to worry about ripping the pieces as you put them together—but you can create these things that look really intricate, and you can create positive curvature, or flat things, or negative curvature in all these different conformations. So the Gilbert-Varshamov bounds says that there's a sequence of codes with R at least 1−Hq(∂). EL: Yeah. SC: So the only thing I think that's worth mentioning is I do some Royal Institution maths master classes, where we go out and we take our favorite bit of math, and we go and take it to students who are between the ages of about 14 to 17. That's sort of the point, right? AS: Yeah. Yeah. I’m not saying it's entirely the same, but people have been made to feel like their voice isn't good enough. And when the opportunity presented itself to take a position at the Institute of American Indian Arts here in Santa Fe, it's a tribal college serving indigenous communities from all over the all over the nation and North America, I wanted to take that. So it's only about a third, let's say, as tall. And then, hey, let's look at curves with many points. SS: I could try too. It might might never have been popular enough to get to the new format. EL: We will help you. YHL: So my name is Yoon Ha Lee. And maybe just to make a picture, let's take the interval from 0 to 1. There’s just this nice, well-behaved statement. SS: Well that’s nice. EL: Oh, cool. I've read half of them, maybe, but I'm going to read them. KK: Did you make it from scratch? It's really, really well done. Although that was going to be my other favorite theorem. So he goes to the US and to Canada, and he writes letters all around the world, he shares it with all his friends. And if you look at Bosse, okay, to draw Desargues’ diagram, you need 10 points: the three on the first triangle, the three on the second triangle, the sun, that gives you seven, and then the three along the axis. Or it intrigues them so much that you want to learn more. EL: Yeah. KK: What’s that Japanese word, sort of the joy of having unread books? AW: Yes. Galois is one of the few people who actually has an interesting death date, whose death is historically significant, and there's a Twitter account that tweets based on on these little biographical snippets, and does it for his birthday rather than his death day and then says, like, “Galois made fundamental contributions to Galois theory.” So this was my response to that account, those tweeting from these biographical snippets saying there's there's more to history than just when people died and what theory named after them they contributed to, and tried to do something a bit more creative with that. EH: Well, it depends on which machine. KK: All the way with Gauss-Bonnet! CC: Two people have chosen the Brouwer fixed point theorem. In fact, descriptive statistics is one of the smallest parts of statistics, and one of the least powerful. It has a start point, it has an end point. He taught real analysis every semester, so I had to take the class with him every time and at some point, I cracked his code. Harry Turtledove is one author who, he likes to have the story where aliens invade during World War II and then the Nazis and the allies have to have to team up against the aliens kind of stories there. And when it really is differentiable, you get the correct differentiable answer, and when it's not, then you get another answer that's still mathematically meaningful. But it's still just cute. Because I mean, you you're a frequent collaborator with my husband, Jon Chaika. But we're watching a little more TV these days, and that might be a good one for us to go look at. The proof had eluded Grothendieck. But yeah. And stop talking about us; start talking to us. It's like, you break it into enough discrete steps where each thing seems okay. So I'm interested in seeing how we can take this lesson and how we can think about how we can be more critical about the ways we think about mathematics itself. So what would be the advantage of choosing the singular values over the eigenvalues, for example? The reason that I thought it would be great to have you on is that one of the things is a podcast called Mathematically Uncensored. TDB: Yeah, that's right. Ben, why don't you introduce yourself? This there's the three bars are evenly spaced, but in perspective, they're not. Is that related to the kind of math you like to think about? LK: Yeah, maybe a lot of these end up there. Really, this is a really challenging time for everybody. KK: Because the fundamental theorem, at least for one variable, is almost a trivial corollary of the mean value theorem. Okay, you were all young once, and you remember—oh, we’re all young, all the time, sorry—but divisibility by 9. KK: But Audrey Hepburn should be breakfast somewhere right? That's sounds like a really neat. And yesterday was June 30. And you go “Wait, what, what was that?” “Oh, well, don't worry. So there's this question, there's this kind of obvious pool of candidates, of outstanding early- to mid-career mathematicians, including people like Oscar Zariski and André Weil, and Schwartz's eventual co-medalist, Atle Selberg. KK: Oh. MB: And then there's a long subtitle. There's no other value that when you cube it, gives you zero. KK: Yeah, that's pretty cool. It shows up with who's winning awards from our big national organizations. Right. EL: An observation. Just saying that, you know, if you were bored some time and wanted to sink a bunch of time into that. What's new? Yeah. EL: Yes. And the scaling factor, which is often denoted λ, is called the eigenvalue associated to that eigenvector. EL: Yes, actually. KK: Okay. EL: Yeah, well, do you want to say a website or Twitter account or anything where people can find you online? KK: I mean, I was teaching topology this this semester, and I was proving Poincaré duality, which is a similar sort of thing, and it's highly non-trivial. Now I am an associate professor of math at Bates College in Maine. And then it turns out that that's not true at all, that Brownian motion—so you look at pollen dancing around on the surface of some water, and it's jumping around in these really crazy aggressive ways. But it involves cohomology. AS: Yeah. And so then it's a question of taste in math. Is there anything else you would like to share? Quite small! And so when you unravel all of this, it's almost like Mad Libs, you have, like, this Mad Lib template. And, of course, people like me, who only were aware of the Fields Medal once they started grad school in math—I wasn't particularly aware of anything before that—Think of it as the very best mathematicians under 40 because it has sort of morphed into that over the intervening decades. And I'm just super thankful that people purchase the book, and we're supporting our future work. And so it opened up, and again, this actually also has Riemann’s name to it, it’s the Riemann moduli space, is the study of all metrics on a space. KK: Yeah, that's one thing, right. Anyway, so I like that. And Hilbert sort of fixed it enough that it was okay. 73 degrees there? Yeah, definitely. AW: Oh right, yes! Is that a difficult theorem to prove? AW: Absolutely. And this will be a timely one, at least for the US, because it will be airing—I mean, I guess the past two years basically have been part of the 2020 presidential season—but really in the thick of it. Is it sort of a combination of stuff? Of course, twin primes, you don't need a whole arithmetic progression, you just need two of them. So you just moved to Athens, correct? But at this moment, they are unusable. EL: Yeah. Well, thank you so much for joining us. And we were talking before we started taping that I'm not actually remembering for sure whether we've used the words eigenvector and eigenvalue yet on the podcast, which, I feel like we must have because we've done so many episodes, but yeah, can we maybe just say what those are for anyone who isn't familiar? Jean Leray, a French mathematician who worked on spectral sequences as a prisoner of warOlivier Messiaen's Quartet for the End of Time, composed when he was a prisoner of warA paper generalizing the Zeckendorf theorem by Harris and coauthorsOur episode with Amie Wilkinson, who also chose the Fundamental Theorem of Calculus, making it 2 for 2 among mathematicians with the initials AW. Leila Schneps has done these amazing volumes about a topic called dessins d’enfants, or children's drawings, but I have to read a piece of this because he wrote something like “Oh, Belyi announced this very result.” So this idea, he says actually, “Deligne when consulted found it crazy indeed, but without having a counterexample at hand. So if you are willing to ignore some finite number of points, then no rational polynomial is ever more than four-to-one. Because to me that wasn't the first reaction. And I hope—I mean by the time this is—we have a bit of a backlog in our past episodes, and so who even knows what's going to be happening when this is airing. And it was this disconcerting direction, as Grothendieck said, that actually took less somehow. So he was looking at maps from curves to the projective line. PH: So I was trying to think about my favorite food, and when it was the epitome of perfection, and I came up with, okay, so if we're going to pair it with something to drink, I was like, I want to think about happy moments. So that top eigenvalue, that one with the largest absolute value, is really controlling the long-term behavior of your dynamic process. So you get this sort of control. And I don't know, it just changed me. Was that before or after Schwartz? Below are some links you might enjoy as you listen to the episode.Arrow's impossibility theoremCardinal voting, an alternative to voting systems that are based on ranking the optionsOur episode with Henry Fowler, who was at the time on the faculty of Diné College and is now at Navajo Technical UniversityOur episode with Moon Duchin, who studies gerrymandering, among other thingsBelin Tsinnajinnie on Twitter. SS: The book, as I say, was hard because I tried to combine history and applications and big ideas without really showing the math. EL: You have accurately stereotyped me. When you think, okay, arithmetic, geometry, those are totally different fields, people study them in totally different ways, but in fact, it turns out that the geometry of a curve can tell you information about the arithmetic. But if I look at how much I turned, I didn't go through 360 degrees. And yet, it's surprisingly useful. EL: Well, I mean, it’s just a bad way to spend your time, but not everyone has the same time priorities. I still find it amazing that we're still going. Winger's profile on Mathematically Gifted and Black, Mathematically Uncensored, the podcast they cohostMinoritymath.org, the Center for Minorities in the Mathematical Science, a website with information and resources for people of color in mathematicsAsked and Answered: Dialogues On Advocating For Students of Color in Mathematics, their bookZeckendorf's theorem and a biography of Edouard Zeckendorf. And I personally, have worked on and off in complex analysis and geometry and dynamical systems, another field. So the book is called, I think, Codes and Curves. Evelyn Lamb: Hi, I'm Evelyn Lamb. I have these correlations, maybe I organize them into a matrix matrix, I have data and organize it into a matrix. Do you have any suggestions? And I'll call them a set X and a set Y. You mentioned the Pythagorean theorem, so this idea that, that you could find numbers where the sum of two squares is itself a square, like three squared plus four squared equals five squared, but what if you had cubes instead, could you find a cubed plus b cubed equals c cubed, or any a to the n plus b to the n equals c to the n. And, you know, that's a statement that, although the machinery of number theory that's developed to ultimately prove this is so technical, and involves elliptic curves and modularity, all kinds of neat stuff, but that the statement was very simple. Like, like the ham sandwich theorem. Because it requires a deep interrogation, a self-interrogation by white people to really deal with the feelings. It's a probabilistic proof. I’m Kevin Knudson, professor of mathematics at the University of Florida. My faculty have actually been great. It's called distributing!” Right? So it's only possible for one prime to show up there, namely 3. KK: That’s right. BT: So I'll say more about why I'm kind of drawn to this theorem. They relate that to how sometimes third party candidates can be a spoiler for an election even though overall, it looks like a plurality of voters might prefer one candidate over another. And so the number of rooms and the number of people are the same—the word is cardinality because you don't want to say number because you can't count that. Ironically, most of what I was interested in doing when I did my undergraduate degree was abstract algebra. CC: Yeah? So 1, 2, and then start adding the previous two, so 3-5-8, and so on. So I see over your shoulder a little bit of a Sierpinski triangle. And the idea is that if you suppose you have two domains in the plane, and you have a transformation from one to the other, you say it's conformal if it's angle-preserving. Curto’s website A short video of Curto talking about how her background in math and physics is useful in neuroscience and a longer interview in Quanta Magazine An article version of Curto’s talk a few years ago about topology and the neural code Curto ended the episode with a plug for linear algebra as a whole. Yeah. So I will say, when I was an undergrad, I did not learn about SVD. And in fact, I mean, that sort of goes back to the proof that we were talking about. So that's really cool. But yeah, before that, I started my own undergraduate research program together with colleagues in Hawaii, at the University of Hawaii at Hilo. EL: Okay. EL: Oh yeah. SS: Yeah, it may not be the best Cubist example. I've never met one. I don't know. CC: So I have many, but the one I chose for today is the Perron-Frobenius theorem. KK: Okay. So even if I knew nothing about the surface other than this particular loop, I would then know that the surface inside must be mostly positively curved, like a sphere. And then actually, I went ahead and printed out ahead of time, these quotes of Grothendieck, who was so struck when this theorem was announced or proven because he'd been thinking along these lines, but was surprised at the simplicity of their proof. So for people who are listening, maybe I could just say it's a fundamental result that says the following, simply put. AC: So the first one, the one from 2011, is called Viewpoints with a subtitle “mathematical perspective, and fractal geometry in art,” and that's suitable for, like, a first-year seminar in math and art. Two of them, the ones on the left and the right, are unitary matrices, or orthogonal if your matrix is real. But what does X want with a category theorist? One time we were just flying by the coast of Greenland. And one of the things that I think, you know, you think the rationals are dense but they really shouldn't be? EL: I will be visting it as soon as we get off this phone call. EL: Yeah. We need a better tagline, but I'm not going to come up with one today. So this intersects also with algebraic geometry. EH: Well, I think I've already mentioned lettuce. But if some other domain fool should try to get Miss Y’s affection, So yeah, I mean, my daughter recently — I didn't realize this. And it turns out that — so now Belyi’s theorem — he realized that in the situation where you're branched over at most three points, so in the picture, we had over 0 and also infinity. JW: Yeah. BO: Yeah, yeah. I think it’s, like, blueberry and watermelon? Before we get to the episode, I wanted to let you know about a very special live virtual My Favorite Theorem taping. Yeah. Is Desargues’ theorem related to that theorem? I'm a math and science writer in Salt Lake City, Utah. And that's an easy proof to do. EL: Yeah. So when I was six years old, in 1975, my parents thought it would be a good idea to take me to the drive-in to see Jaws. He told us about several favorite theorems of the moment before zeroing in on one of his first mathematical discoveries: a way to determine whether a number is divisible by 7. But you're right. Thank you for having me. In fact, one of the applications of the Gauss-Bonnet theorem in nature is how do you create a surface that sort of fits onto itself and fills a lot of space—or doesn't fill that much space but gives you a very high surface area to volume ratio. So you have the first number in decimal. Those are the zeros. And oh, by the way, that perspective is called category theory. Barely flavored. I think he probably would have said that—I really do think that he in particular would have said that to any student. I don't know if that question is too far from left field. I’ve been very excited thinking about about it. Well, here's to linear algebra. That definitely can take a lot of time. Judy Walker: Hello. DL: But if you have an arithmetic progression, so a bunch of numbers which differ by all the same amount, and they're not all divisible by some single number, then Dirichlet’s theorem tells you that there are infinitely many primes in that sequence. Yeah. Yeah, the question of drawing other curves is really interesting because of how you do or don't define curves in projective geometry. KK: Well, you know, about as pleasant as it sounds. And since you want to write all positive integers, well, you’ve got to start with 1 somewhere. CC: So it's actually really old. So I run a sleepaway camp at Union College, in which students learn proof-based mathematics for the first time. I kind of discovered it randomly and started learning a lot about it and became fascinated. I mean, part of it is part of the batter cooks longer than the rest of it. Yes!” Some people say math-three-ma, but you know. Yeah. KK: Yeah. But I really wanted to put this in the category of something that that bridges different areas, because this picture I was describing earlier really was just a graph with three edges and four vertices. KK: Yeah, on Instagram, I follow a lot of bird watching accounts. And so I think this is a theorem that you can really play with and see in the world. I didn't do it just for the sake of promoting diversity in the math major, but it's sort of unintentionally has done that. But we still got it because it was cheap. But, yeah. My people are not indoors. So I went to a conference in Colombia, we visited Tayrona which is a beach in Colombia. EL: But yeah, I remember seeing you do a poem at this this joint math poetry thing and then kept seeing you at various things and then we met, you know, a few years ago when I was at Eau Claire, I guess, we actually met in person then. EL: It’s someone. CC: Right, exactly. Often people don't want that set answer. 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Look in a sort of standard geometry, you know, not this one you saying. When people think it 's space opera is, last year ’ s injective and,...

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