My Favorite Theorem

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Episode 95 - Kyne Santos May 21, 2025

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Episode 95 - Kyne Santos Kevin Knudson & Evelyn Lamb Download 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 theore…

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Episode 94 - Jeremy Alm Feb 06, 2025

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Episode 94 - Jeremy Alm Kevin Knudson & Evelyn Lamb Download 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 i…

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Episode 93 - Robin Wilson Dec 02, 2024

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Episode 93 - Robin Wilson Kevin Knudson & Evelyn Lamb Download 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…

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Episode 92 - Kate Stange Jun 10, 2024

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Episode 92 - Kate Stange Kevin Knudson & Evelyn Lamb Download 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…

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Episode 91 - Karen Saxe Apr 03, 2024

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Episode 91 - Karen Saxe Kevin Knudson & Evelyn Lamb Download 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…

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Episode 90 - Corrine Yap Jan 23, 2024

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Episode 90 - Corrine Yap Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm your host Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City. And this is your other host. Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. How's it going? EL: All right. Yeah, I was I was trying to think about what to say. And I was like, well, the most exciting thing in my life right now is that our city is starting a pilot program of food waste, like a specific food waste bin. KK: Okay. EL: But then I realized I also did an 80 mile bike ride last Saturday, and that's the first time I've biked that far. And that might be slightly more exciting than compost. KK: Are you working up the centuries? Are you are you heading for? EL: We’ll see. I felt pretty fine after 80. I also don't feel like I wanted to do 20 more miles. So we'll see. Someday, maybe KK: The last century I did was, wow, it was 2003. It was 20 years ago. This is one called the six gap century in Georgia. And it goes over six mountain passes in the mountains of North Georgia, one of which has, like, a 15% grade, which is quite steep. It took me about eight hours. And then I hung up my bike and didn't ride it for like three months. EL: Well, I mean, it would probably take me at least eight hours to do a flat century. KK: Yeah, but back in my youth I could do a flat century in about five, but not anymore. Not anymore. So let's keep this banter going because I — so Ben Orlin, a former guest on our podcast, I saw Math with Bad Drawings today had various golden ratios. One of which was the golden ratio of hot fudge to ice cream in a hot fudge sundae, which he argues is one to one, but that's way too much fudge. EL: That is so much fudge! KK: But the podcast, like, substance to banter golden ratio, he claims is like two to one. So like a third of this should just be like us, you know, just shooting it. EL: Saying nothing. KK: Yeah. EL: Well, I must admit, that's why I listen to fewer podcasts that maybe I would want to because I have a low banter tolerance. Which brings us to our guest today. Yeah, so we are very happy to welcome Corrine Yap today. Would you like to tell us a little bit about yourself? Corrine Yap: Yes. So I am currently a visiting assistant professor at the Georgia Institute of Technology, Georgia Tech, in the math department. I'm also a postdoc affiliated with the Algorithms and Randomness Center. But I just got my PhD in the spring from Rutgers University. KK: Congratulations! CY: Thank you! I do a lot of, like, probabilistic combinatorics, and stuff around that. So that's sort of my main research work. I also do some performing and some playwriting as well. I actually just got back from a performance yesterday Worcester Polytechnic Institute in Massachusetts. EL: Oh, wow. CY: So very busy this time. EL: Yeah. That’s actually one of the reasons that I've been wanting to invite you for a while. And I was like, well, I should wait until I've seen one of her shows. And then it just has not aligned to work out. Because I know you've done them at the Joint Meetings and things, and the times that I have been there and you have been there, it’s just not been a good time. So it's like, well, I'm not going to put this off forever. So even though I have not yet seen one of your shows, I'm very glad that that we could invite you and have you here and yeah, well, can you talk a little bit about the kinds of theater that you do, or kinds of — I don't know if it's mostly theater or more, like other? I don't know, speaking performances? CY: Yeah. So it really started when I was a lot younger. And also in college, I primarily studied both mathematics and theatre, with no sort of vision as to what that would turn into in terms of a job or career or anything. I just really enjoyed doing both of them. And as an undergraduate I thought I was mainly interested in acting, but I started studying playwriting while at Sarah Lawrence College in Westchester, New York. And I started writing this play, which is the play that I continue to perform. It's called Uniform Convergence. And it's a one-woman play that's about math. It tells the story of Sofia Kovalevskaya, who is a historical Russian mathematician. She was born in 1850. And it tells a little bit about her life and how she faced a lot of obstacles to be successful as one of the first few women in academia. But it also has a portion that is sort of inspired by my experiences being Asian American, and also being a woman pursuing mathematics. And the setting is that of a real analysis classroom, a lecture where the character Professor…. EL: Hence, uniform convergence. CY: Yeah, and she is lecturing to her students. So at one point, they do reach the point of the class where they do uniform convergence as a topic. So, you know, in the past, I did a lot more — like, in college, I did, you know, the auditioning for plays and being involved in rehearsals, and all this sort of stuff. But since going to graduate school, and now having an actual job, this one play is sort of the main way that I keep my ties to doing theater and the theater world. KK: Very cool. EL: Yeah. Well, that's cool. I didn't realize that Kovalevskaya was the subject of this. I actually just read Alice Munro's short story, Too Much Happiness, which is based on her life. And actually was not my favorite short story in the collection that it’s in, but it, you know, she is such a compelling figure and another woman who was interested in math, you know, at a time when it was a lot harder for a woman to have an academic career in any field, and was interested in literature. Wrote, I think both memoirs and fiction? CY: She also wrote a play. EL: Oh, wow. CY: Yeah. But it wasn't about math. But yeah, she was very much also in both of these worlds in, you know, sort of a more artistic, creative mindset as well as a mathematical one. EL: Yeah. Fascinating person. So yeah, that's really interesting. And hopefully someday I'll get to see it. CY: Yeah, I'm still performing. I didn't think I necessarily would be. But it's been since 2017. I've been performing it at different college campuses, and sometimes at conferences at different parts of the country. And I still get invited places. So as long as that keeps happening, I'll keep going. EL: Yeah, when I was still in academia and doing a postdoc, I did, you know, I'd started doing writing. And sometimes I would get invited to do both like a research seminar talk and a public engagement kind of talk. And so that that might be in your future as well. CY: Yeah, maybe. KK: Yeah. Broader impacts. EL: Wearing both hats on one trip. CY: Yeah, I actually, I forgot I am doing that. I think this is the first time I'm doing it. At Duke in October, when one day I'll be giving a seminar talk, and then the next day, I'll be performing the play. EL: Yeah, cool. Well, we invited you on here to talk about your plays, but also to talk about your favorite theorem. So what have you chosen? CY: Yeah, so I've chosen Mantel’s theorem as my favorite theorem. So this is a theorem that is in the area called extremal combinatorics. And I'll explain what that means. But the statement of the theorem is pretty straightforward. It says that if you have a graph, which I’m a combinatorialist, so for me graphs mean, collections of vertices with edges connecting pairs of vertices. If you have a graph on N vertices, then the maximum number of edges you can have without forming any triangles — so just three edges and three vertices connected to each other — the maximum number of edges you can have with no triangles is N squared over four with appropriate floor. KK: Yeah, sure. CY: And this seems like, okay, this is this is just a statement, maximum number of edges. What's so cool about that? You actually, we actually also know where the N squared over four comes from. It’s, the extremal example is the complete bipartite graph on parts of size N over two. So what that means is, you split your vertices up into two sets, each of size half the total universe. And all of your edges go between the two parts. So from one part to the other, not inside the vertices of the parts. So complete means you have all the possible edges crossing between the parts, and then bipartite because you have the two parts of the vertices, and that has N squared over four edges. And it has no triangles in it. KK: Not even any cycles. CY: Yes. Yeah, no odd cycles. Yeah. KK: Okay, all right. CY: Yeah. So, one reason I really liked this is because when I first learned it, I didn't really think much of it, I learned it in an undergraduate class in combinatorics. And there are, like, three, maybe four proofs that we learned that were all pretty short and straightforward. One of the most basic proofs is just via induction on the number of vertices, and there's nothing, there's no really heavy machinery that's needed at all. And I didn't think much of it. And I didn't have any context as to like, why do we care about this sort of thing. But every year, I learn more and more things that make me appreciate this theory, more and more, because it really was the foundation for this whole field that we call extremal combinatorics, which is really centered on these questions of, like, what are the maxima and minima of certain things that we want to count when we put certain constraints on the problem? So this is an example we want the maximum number of edges. And our constraint is we have no triangles. And you can, there are a lot of different directions you can go with this sort of theorem. One of the most sort of classical foundational ones is just to replace triangle with a different type of graph. Like you could say, Okay, if I want the maximum number of edges with no cycle of length four, or cycle of length 10, right, what can I say? Or if I want the maximum number of edges with no complete graph of size five, where complete means you know, the vertices, you have every possible edge between every pair of vertices. And this type of problem, sort of replacing triangle with other things. It's called a Turán type problem, because there's Turán's theorem that generalizes mantle's theorem to complete graphs of higher orders. And we basically know the answer of what the extremal number is, and the extremal constructions for almost every graph, except for when you consider a bipartite graph as your, instead of triangles. EL: As the thing you're trying to avoid? CY: Exactly. And there's a reason for this, there's a theorem where it basically fails, or it's trivial in the case that your forbidden graph is bipartite. And so there's been a lot of study, it's still a very active area of research. And what people are doing is sort of taking different flavors of this Turán type problem that sort of started with Mantel’s theorem. And my first paper in graduate school was on a topological version of this theorem, where we were looking at these higher-dimensional structures called hyper-graphs, which you can think of as a higher dimensional version of a graph, and looking at a more geometric or topological viewpoint on these hyper-graphs by making them into simplicial, abstract simplicial complexes. So we don't have to go into the details of that. But I found it, you know, when I did that project, I found it very cool that that we could take this seemingly purely combinatorial, graph theoretic statement about just counting edges, and somehow turn it into something that requires a little bit more of a geometric or topological point of view, which is not something I had spent much time with before. And so that's sort of one direction at the beginning of my grad school career, where I felt like I had suddenly a much greater appreciation for this theorem. And on the other end, where I am now, it's also connecting very heavily to the research direction that I'm currently pursuing, which is in statistical physics, which is for me an entirely unexpected application of this sort of thing. But if you think about it, this sort of characterization of the extremal structure saying, okay, we can achieve the maximum with a complete bipartite graph, you can view this as sort of a ground state, if you will, if you want to think of the vertices as like particles in some sort of distribution, and you can take a probabilistic point of view on these sorts of counting problems. For example, it turns out that the triangle-free graphs and the bipartite graphs, if you think of these two collections, triangle-free graphs and bipartite graphs on N vertices, they're very closely related to one another. In fact, almost all triangle-free graphs are bipartite. This is a theorem by Erdős, Kleitman, and Rothschild. So you can sort of ask how far does that behavior persist if you add more constraints to your problem? And you can think about it as in a probabilistic sense of thinking, well, what if I have a probability distribution on my triangle-free graphs? And I have a probability distribution on my bipartite graphs? How are those distributions related to one another? And what is the counting statement, say, in terms of the probability distributions when we when we consider a randomness point of view on these things. And the sort of magical thing is that when you go to a probabilistic point of view, there are very natural ways that you can put it into a statistical physics context, where in statistical physics, you are thinking inherently about probability distributions on certain particles, on particles in space, or different configurations of particles in space, where there's maybe a physics motivation underlying the distribution you define. But ultimately, you can distill it down into something that is, that is simply triangle-free graphs, or different discrete structures. So one thing that I'm really interested in right now is just exploring more of this somewhat mysterious, but somewhat really amazing connection between questions that arise in graph theory and combinatorics that, you know, for a long time, we have just thought of in that context, in the graph theoretic context, and how, looking at them from a more statistical physics perspective, can help us gain new insight into how to tackle these problems. KK: Yeah, and hopefully, it'll go in the other direction. I mean, I think we have this idea that because we learn calculus, and we think about physics being based on calculus, but inherently, right, the universe has to be kind of discrete, so you can't divide stuff forever. So I mean, it sort of makes sense that the underlying business, when you get down to it, might have to involve some kind of graph theory questions. EL: Yeah, that is remarkable that there's this connection. So this is maybe a naive question about what you're talking about doing. Like probability distributions on graphs, are you saying things like, the likelihood that that two vertices have an edge between them? Or are we talking about some other kind of probability distribution? CY: Yeah, so there, I purposely didn't include too many details, just because there are a lot of a lot of actually interesting and all valid ways that you can think about imposing probability, you know, into this world into these problems, these extremal combinatorics problems. So one flavor is what you said, we can think of what's called the random graph model. The most common one is the Erdős–Rényi random graph model, where you simply have your N vertices, and for…

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Episode 89 - Allison Henrich Nov 12, 2023

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Episode 89 - Allison Henrich Kevin Knudson & Evelyn Lamb Download 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 I'm joined as always by your other and let's be honest, better, host. Evelyn Lamb: I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City. And tomorrow is my 40th birthday. So everything I do today is the last time I do it in my 30s. So, like, having my last mug of tea in my 30s, taking out the compost for the last time in my 30s, going for a bike ride for the last time in my 30s. So I'm, I'm kind of enjoying that. KK: Well, congratulations. Let's not talk about how long ago I passed that landmark. I will say there's a switch that goes off when you turn 40. So riding your bike will be more difficult tomorrow, I assure you. EL: Well I’d better get one in then. KK: Any big plans? EL: I’m actually going to the Janelle Monae concert. She's in town on my birthday. I'm sure that's a causal relationship there. KK: It must be. EL: So yeah, I'm excited about that. KK: Okay, so fun fact, my Janelle Monae number is, is two. So I have a half brother. Very long story. I have a half brother, who also has a brother by — his mother has two children with — my dad was one of them. And then another man was the other one. So this other one, his name is Rico. He was a backup dancer for Janelle Monae. EL: Wow. So yeah, brush with celebrity there. KK: I mean, of course I've never met Janelle Monae, but you can — actually if you look him up, so there's a style of dance, sort of Memphis Jook, it’s called. Dr. Rico. He's something else. Amazing dancer. EL: Wow. Interesting life. KK: That's right. That's right. So anyway, enough about us. We have guests on this show. So today we're pleased to welcome Allison Henrich. Allison, introduce yourself, please. Allison Henrich: Hi. Yes. I'm Allison Henrich. Happy birthday. I'm so excited for you. EL: Yes, you get to be on my last My Favorite Theorem of my thirties! AH: Yes, awesome! I feel so special. So I'm Allison Henrich. I'm a professor at Seattle University, and I'm also currently the editor of MAA Focus, which is the news magazine of the Mathematical Association of America. KK: I have one on my desk. AH: Woo-hoo! Is it one of mine? EL: Yeah, and when we were chatting before we started recording, you made the mistake of mentioning that you've done some improv comedy. Is that something you do regularly? AH: So I wasn't an improv artist. This is such a cool event. This science grad student at the University of Washington started this type of improv comedy where they have two scientists give short five-minute talks. And then this improv comedy troupe does a performance that's loosely based on things that they heard in the science talk. And so I gave a talk about some basic knot theory ideas, and it was so funny. I wish everyone could have the experience of an improv comedy troupe doing a whole set about your like research or your job. Yeah, it was so amazing. EL: Cool. But also, it sounds a little stressful. A little bit. Yeah. AH: Yeah. You want to not be too boring. And you gotta, like — it's really interesting. The other speaker tried to work in things for them to make jokes about, and they totally didn't take the bait. And they found like more interesting things to make jokes about, but I definitely tried to work in some things that would help them riff off of my talk, and it worked pretty well. Like just referring to knots with quirky names and making jokes about knot theorists and whatnot. KK: Sure. What-knot. Hahaha. AH: There are a lot of good knotty puns. KK: Sure. Okay, so this podcast does have a theme. And the question is, what's your favorite theorem? AH: Yes! This is a hard question. KK: Of course. AH: I’ve decided to tell you about my second favorite theorem. Should I admit that? KK: Sure. EL: I’m sorry, that’s a different podcast, My Second Favorite Theorem. It has two slightly worse hosts. AH: It’s the cheap knockoff. KK: No, it's the sequel, once we get rid of this one, we're gonna move on. AH: Just, we're all out of mathematicians, we’ve got to go through them again. So my, let's call it my favorite theorem. KK: Sure. AH: My favorite theorem is the region crossing change theorem. So I have to tell you a bunch of stuff before I can explain what this theorem is. KK: Sure. But it must be about knots. AH: It is about knots. So, you know, knots we represent, typically, with two-dimensional pictures called knot diagrams, where you have ways of representing when a strand is going over and when a strand is going under at a crossing. And so every type of knot that there is has infinitely many diagrams you can draw of it. But no matter how you draw a diagram of whatever your favorite knot is, it can always be unknotted if you're allowed to do a special kind of move called a crossing change. So if you have your favorite knot diagram, and you're allowed to switch the over and under strands on whichever crossings you want, you can always turn that knot diagram into the diagram of an unknot, which is like a trivial knot that'll fall apart if you unravel it a little bit. EL: Basically just a circle, right? AH: Yeah, a circle. I mean, all knots are circles, so I have trouble. Like, a geometric circle. EL: A boring circle. AH: Yeah, a boring circle. EL: And so this theorem, does it come with like, a number of how many of these crossing changes? AH: Ah, so this is not my favorite theorem. This is a theorem that's going to help us understand my favorite theorem. EL: Okay. AH: So this theorem has a really interesting proof that Colin Adams calls “proof by roller coaster.” So the the theorem that says you can unknot any not diagram by changing crossings, you can accomplish unknotting using a certain algorithm where you choose a starting point to travel around a knot, and you decide that every time you encounter a crossing for the first time, you're going to go over it. So the fact is that you're kind of like always traveling downwards. And then when you get to the very end, you take a little elevator back up to where you started. So this will always create an unknot. So it's not that surprising that this is true, that if you're allowed to change whatever crossings you want, you can unknot things. What is surprising is my favorite theorem, which is that region crossing changes can unknot any knot diagram. So let me tell you what a region crossing change is. So you have your knot diagram in the plane. A lot of us kind of imagine that this plane is on a big sphere. So can we picture not diagram on a ball? Is that okay? KK: Sure. Make it a big enough ball, and it looks like a knot diagram. AH: Exactly. Yup. So we've got a knot diagram on a ball, and the knot diagram basically separates the surface of the ball into different regions, right? So this amazing theorem uses this operation called a region crossing change. And what a region crossing change is, is you choose a region in the diagram, and you change every crossing along the boundary of that region. So in my head, I'm picturing kind of like a triangular region in the diagram. And if I do a region crossing change on that region, I'm going to change all three crossings that are kind of around that region. So this is the amazing result: every not diagram can be unknotted by region crossing changes. So you no longer, seemingly, have control over individual crossings, you can only change groups of crossings at a time. KK: Okay. EL: But you can still do it. AH: Yes, you can still do it. KK: Right. That seems less likely. The other one, you told us and I thought, Well, yeah, I can kind of see, before we even saw the proof, I could sort of imagine, well, yeah, you just lift them up basically. AH: Exactly. You lift it up, and then if it gets stuck, you know, change that crossing. But you can only change groups of crossings with the region crossing change. But amazingly, it's still an unknotting operation. So that just blew my mind when I heard that. KK: Okay, so now I have questions. So, more than one, right? You can't expect to be able to just do one of these, right? AH: Right. I mean, so if you have a region that just has one crossing on it, it's like a super boring region, because it's just a little loop. KK: Yep. AH: And that's actually called a reducible crossing. KK: Sure. AH: If you just have a little loop, it doesn't matter which way, which is going over and which is under. KK: No, but I guess I meant, so you know, you've got one region, right? AH: Yeah. KK: So there might be multiple regions, you might have to change many of these, right? AH: Yes, yes. KK: What if two are adjacent, then you do one flip on one and one flip on the other, then you're undoing some of the flips from the other. AH: Exactly. KK: Is that why it works, maybe? AH: That is why it works. So it’s a really cool proof. It's actually a proof by induction, which is so cool, that you can have like a proof on knot diagrams that's a proof by induction. But it's by induction on the number of reducible crossings. So the number of these crossings that you could sort of flip out of the diagram. They're not really necessary for the knotedness of the knot. But the base case is the most interesting part of the proof, where you have a knot diagram that has no reducible crossings. So no extraneous little loops or flips on it. But it's very constructive, and it uses things like checkerboard colorings, and it uses splices, or smoothings, which is where you take a crossing and you turn it into — like, you basically get rid of the crossing by cutting it and reattaching ends so that it's just — I’ve got this picture in my head, how do I say it? What's the best way to say that? So you have a crossing, and you want to get rid of it by cutting it and reattaching ends so that there's no crossing anymore. Does that make sense? KK: Well, it’s sort of like a braid, right? I mean, so you imagine sort of a braid cross, you just clip the string above and below and then you just reattach, then you don't have it, right? Is that what you’re doing? AH: Okay, yeah, what you just said totally makes sense because I could see your fingers. KK: This would be a better video podcast, I suppose. AH: I know. Yeah, at least for topology, or geometry. But the proof basically creates a checkerboard coloring that tells you how to find a collection of regions where you can basically control which crossing you're going to change. So I can change just one crossing, by carefully selecting a group of regions where exactly one, or exactly three of the regions involved in that crossing are going to get changed, but every other crossing in the diagram is next to either zero, two, or four regions that are being changed. So if it gets changed, it'll get changed back and look like it like it started. KK: Right. Okay. All right. EL: So I have not thought about knot theory, probably since we talked with, like, Laura Taalman on this podcast years ago. It's not something I think about a whole lot. And so I was not expecting this induction to be on the number of reducible crossings because they're so silly, you can just undo it, and then your diagram doesn't even have it anymore. So, yeah, why not the number of crossings or the number of regions or something? KK: Yeah. AH: So the reason reducible crossings are annoying for region crossing changes is because at a reducible crossing — you know, at any crossing, if you zero in on it, it looks like there are four different regions involved in the crossing, but with a reducible crossing, two of those four regions are actually the same region. KK: Right. AH: So it can look locally like you're changing two regions, so that you know, the crossing shouldn't flip. But you're really changing one, so the crossing does flip. So that's why reducible crossings are the annoying thing that you need to carefully control. KK: Okay. All right. AH: Yeah. And so once you get into the inductive step, you basically want to take a reducible crossing, change it so that you have two pieces, one has one fewer reducible crossings, and you know how to deal with that. And then one is a totally reduced diagram of a knot. EL: Yeah. But the base case is the hard part, it sounds like. AH: Yes, yes, yes. Totally. EL: Interesting. AH: Yeah. So yeah, so one of the reasons I love this is because I love unknotting. In general, I find unknotting questions really interesting. And I highly recommend everyone go listen to Laura Taalman's My Favorite Theorem podcast because she talks a lot about unknotting problems. But also the woman who proved this result is named Ayaka Shimizu. She’s a Japanese mathematician, probably my age, maybe a little bit younger, maybe she's about to have her 40th birthday or something, I don't know. But she is one of the coolest mathematicians I've ever met. She's definitely the cutest mathematician, and her talks are so cute that you're like, oh my gosh, I'm watching such a cute talk! And then you realize, oh my God, this result that she just proved is really amazing! So she's just super, super cool. I love her so much, and I think it's amazing that she proved this result that, you know, the Japanese math community wondered about for a long time, but no one came up with a proof before her. And she must have, maybe she was even a grad student at the time, or she was definitely a very young mathematician when she proved this result. So I love it. KK: So here's a question: why would you want to allow such operations? I mean, because physically, changing the crossing, I mean, that would be great when your shoes are knotted, right? Like, you could just go Oh, snap, that's unknotted. Right. Is there a practical reason? And by practical, it could be including things like, it doesn't change the knot invariants or something? Or I don't know, it must if you get to the uknot, but I mean, is it… or is it just fun? AH: Well, so the other thing — yeah, it's just fun. The other thing you need to know about me is that I study games that you can play on knot diagrams. KK: Okay. AH: And this result enabled this Lights Out-type game, they actually have a website. You can search for this game called Region Select. It's a really fun solitaire game that's a lot like Lights Out if you've heard of that game. And basically, the fact that the region crossing change is an unknotting operation basically means that any lights out game that you can think of or any region select game you can think of is playable, so you can have a knot diagram. Basically, at each crossing, instead of a crossing, you have a light. So it looks a lot like a graph, actually. You have a light and the lights are, some of them are on and some of them are off, and you need to select regions to try and turn all of them on or turn all of them off. And it's a really fun solitaire game that comes from this. KK: Okay. AH: But I’ve actually use the region crossing change to invent one of the many games that I've studied. It's called the region unknotting game. And basically, I'm super interested in these types of two-player games, where you start with a knot diagram, or maybe the shadow of a knot diagram. And you have two players doing something to the diagram, and one player wants to create the unknot and the other player wants to create something knotted. And so we have many games of this variety we've invented. One is the knott…

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Episode 88 - Tom Edgar Oct 09, 2023

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Episode 88 - Tom Edgar Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm your host 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. It's Friday. Hooray! EL: Yeah, yeah. KK: Long Weekend. Yeah. EL: It’s the start of a new month. Everything — anything is possible. KK: Right. EL: Including a great conversation with our guest. KK: Yeah. I think it will be good. It's been an okay day so far. EL: Great. KK: The hurricane notwithstanding. EL: Yeah. KK: But yeah, that went by. But yeah, Hurricane Idalia really did some serious damage. And it’s, yeah, it's rough. EL: Yeah, and there was recently the tropical storm on the other side of the country that actually kind of affected our weather, and today, I am hoping that the gale of wind outside my window isn't too much, too hear-able on the audio. KK: I don't hear it, so it must be okay. Yeah. EL: Great. Well, anyway, we are here today to talk with Tom Edgar about his favorite theorem. So Tom, would you like to introduce yourself? Tom Edgar: Yeah, sure. Hi. Thanks for having me. It's fun to be here. I love your podcast, as you both know, but now everybody knows I love your podcast. I'm Tom Edgar. I'm a professor of mathematics at a small, comprehensive university in Tacoma, Washington called Pacific Lutheran University, just south of Seattle, about 35 minutes, maybe. Depending on traffic, like an hour and a half. I'm also currently the editor of Math Horizons, which is the undergraduate-level periodical from the Mathematics Association of America. And spend a lot of my time on those two things right there and just getting ready to go back to teaching here starting next week. KK: Oh, you guys start after Labor Day. Okay, good for you. EL: Oh, yeah. That is nice. Yes. And I think we've worked together a little bit on various Math Horizons things. TE: Yeah, both of you have. So I mean, Kevin's on my editorial board, and he's written a couple of things. And then, Evelyn, I met you I think it in person at ICERM back forever ago. And I remember you were nice enough to do a piece about your awesome calendar, which I still have. I actually have a second copy now because I just have two now. EL: Excellent. Yeah. Well, I would recommend getting one for every room. TE: It doesn't hurt: one for the office, one at home. EL: I’m not biased at all. TE: No, one for your for your classrooms for your students. It's a great idea. KK: Right. And it's universal. It's not year-specific. So reminder to all of our listeners, go to the AMS bookstore where they seem to be having a sale all the time, right? EL: Yeah. Can’t afford not to! That's right. Anyway, Tom, now that you've so kindly plugged my calendar for me, what is your favorite theorem? TE: And just that wasn't planned either. Right? That was just, you know, it's a nice thing that you've done. It's really cool. Yeah, so my favorite theorem is a hard thing. Because I've been listening your podcast for a number of years, and I was like, hey, if I ever get a chance, I wonder what I would talk about. And I had one that I was going to talk about, but I I've changed recently. There have been some projects that I've done in the past few years that kind of have changed my viewpoint. And so the theorem that I want to talk about is a pretty elementary theorem, in some sense. Most mathematicians will have seen it, a lot of, any math-adjacent people will have seen it. And it's the formula for the sum of the first N positive integers. So if you were to add up, say one plus two plus three plus four plus five, right, you can do this addition problem. My son, who's eight, can do this addition problem. But is there a quick way to get to the answer? And so the result is that if you add up one plus two plus three plus four plus five, you can actually get that in sort of fewer computations by multiplying five by six and dividing by two. And so the general formula is, if you were to add up the first N positive integers, pick your favorite number to stop at, N, then the theorem says that that sum should be N times N plus one divided by two. So the number that you stop at, multiplied by the next number, and then take half of that. So I really love this theorem for a variety of reasons. KK: So there’s the apocryphal, probably apocryphal, story about Gauss, right? TE: Yeah, for sure. So I definitely enjoy this aspect of it because most people think, oh, there is this story. So the story is, I'm not even going to tell the story because I've read — Brian Hayes has an article where he tries to get to the bottom of this actual story and where it came from, but the general idea is that, you know, some teacher of Gauss gave this as an exercise, to find this sum and expecting it to take a long time and Gauss produces the answer almost instantaneously. I like talking about this because a number of people have changed that story over the years. And so it gets more dramatic, or things like that, or a lot of people think that this is Gauss’s sum formula, that Gauss was the very first person to come up with this, like in the 1800s, like, nobody knew that, you know, this was it. But this has certainly been known — you know, one of my favorite proofs is the picture proof where you imagine the sum of the first N integers is sort of almost like a staircase diagram, one box at the top, two boxes below that, three boxes below that, and so on. And you take two copies of this staircase diagram, rotate one 180 degrees, and stick them together, and you have an N by N +1 rectangle. And Martin Gardner attributes this to the ancient Greeks, right? So presumably, people been drawing this in sands, and all sorts of things, for as long as people been thinking about counting, right? EL: I must admit, I do — like, that story always bugs me because people, I don't know, people will use it as evidence of like this amazing genius. And I'm sorry, if this is, I don't know if I sound like I’m bragging or something. But like, I figured this out when I was in school, and I'm not a Gauss, by any stretch. KK: Don’t sell yourself short. EL: And it's like, you sit around playing with numbers a little bit, then, you know, you can figure this out, it's figure-out-able, which I think is good for people to know, rather than think, Oh, you have to be, you know, some native genius to be able to figure something like that out. TE: Yeah, for sure. And, and I think, like, I don't know if you've read Brian Hayes’s article on it or not. EL: I think so. TE: Yeah. He brings up the point that maybe the reason people like it is because it's sort of, like, the student having this victory over the the mean classroom teacher. And somehow we just love this idea, not necessarily the genius myth, but this idea that like, oh, the the student won, or something like this. But yeah, but it's fun to talk about too. And just that always opens up the conversation with people about all the misattribution that we have in mathematics, right? Theorems named for people that maybe don't even have anything to do with that theorem, for one reason or another. KK: So let's talk proofs. So you mentioned the one that Martin Gardner did with the picture. Okay. What's your favorite proof? Do you have one? TE: Yeah. I mean, that one's pretty amazing, if you ask me. You know, I mean, another reason I like this is that this is sort of, if not the, it's probably the standard first induction proof that any undergraduate sees, right? So you learn about induction, and then you prove this formula by induction. I dislike that proof in one sense, and I love that proof in the other, right? So it's nice from learning induction. On the other hand, it's like, man, it's induction. I didn't get anything out of that. Whereas that picture proof from the ancient Greeks, right, just tells you exactly what what to do, right? EL: Yeah. And I'm trying to remember is there a book or something called, like Proofs without Words or something like that? And it's a great proof without words, because it doesn't take a whole lot of scaffolding to show this picture and the numbers and to see exactly what's going on. TE: For sure. Yeah, yeah. So Roger Nelson has three compendia now, like Proofs without Words, right? So this is three books, maybe almost a total of 600 pages of diagram proofs. And that one is in the first edition. And it's definitely — I mean, there's a couple iconic proofs without words, and I would put it as one of the top four iconic proofs without words. There's the Pythagorean theorem with a couple, and a couple of other ones that go along with it. But that's that. But my favorite proof actually — well, so, back in, like 2019, right at the end of 2019. Right, the beginning 2020 Before the before, sort of all the craziness, a mathematician named Enrique Treviño, who's a professor at Lake Forest College in Chicago, he was posting some things on Twitter about different proofs of this theorem and I knew a couple and I sent it to him, he's like, Hey, we should write these all up. So we got together and wrote these all up. And so we have a compendium that's online of 35 proofs so far, of the of the fact. And we finished that just before — I think it was end of January 2020, we sort of finished it. We've been working on it here and there ever since. But one that came out of there that's my favorite — and it's hard to describe, so I'll see what I can do — but it's also a picture proof. But instead of taking two triangular diagrams, so two staircase diagrams, you take eight staircase diagrams. The same kind of picture, instead of two and you just glue them together and you get a rectangle, you take eight. So again, the visual here should be sort of a right triangular stack of squares, N squares on the bottom, one square on the top, and then it's right oriented. And when you put eight of these together, you get a perfect square, except there's this one missing cell in the middle. And so it tells you that eight times this, this number, which these are called the triangular numbers, because they fit into these triangular arrays. So eight times the Nth triangular number is basically the Nth odd square. So (2N+1) squared, except missing one, missing one cell, so minus one. And this proof to me, it's much more complicated, in some sense. Like, why don't you just use the real picture proof, the easy one with two? But this one indicates that there are a lot of other things going on. So you can use this proof essentially, to prove that odd squares are congruent to one mod eight and these kinds of things right here. I mean, it sort of falls right out of that. And then this was key to Gauss’s — what's it called? — three triangle theorem, which says that every positive integer can be written as the sum of three triangular numbers. And so this fact plays a role. This visual proof plays a role there. KK: Okay. EL: Oh, nice. KK: Very cool. EL: Yeah, I'll have to draw that out later. I'm not quite sure I believe you, but I'll take your word for it for now. TE: You’re going to have to draw it out, for sure. I was like, Oh, should I? Kevin asked my favorite. I wasn't going to necessarily going to talk about that one, but for some reason, I liked that one because it opened my eyes to a lot of other things going on in math as well. So it just has a connection, you know, thinking about what are called figurate numbers. So these are numbers that can be arranged in certain geometric patterns. So the triangular numbers, the squares, these are familiar ones to us, but there are just so many cool mathematical ideas that somehow I never picked up as an undergraduate or a graduate student about these, like Euler’s pentagonal number theorem, or Fermat’s polygonal number theorem, just amazing facts out there that I just never would have come across. EL: Yeah, well, I guess that one is kind of an overpowered proof for that particular formula. But like you said, yeah, it kind of opens the door to a few different things, a sledge hammer for a mosquito. KK: I like that. TE: Those are some of my favorites of the ones that that Enrique and I compiled. One of the ones that sort of blew my mind that we came across was this idea that you can use Euler’s polyhedral formula for planar graphs, right? So the the planar graph version, you can use this and it proves the sum of the integers formula if you just find the right graph, and that's like a sledgehammer! KK: Oh, nice. TE: But it’s a beautiful, really powerful theorem for topologists. I think both of you somehow are topologists or topology-adjacent. Am I wrong about Evelyn? Not you? EL: Yeah. Oh, yeah. Why not? KK: No, it's true, Evelyn. TE: The fact that you can use you know, this Euler’s polyhedral theorem, which I know has been featured on your podcast before, and maybe even recently, you know, to me was really powerful, like, oh, you're using something really strong. But it's also a way that you can introduce people to a cool idea with this relatively simple fact, elementary fact that they might be encountering as early undergraduate-level mathematicians, or even earlier than that. KK: Very cool. All right, so I know visual proofs are kind of your thing. So have you animated this one? I know you like to animate these things. I see them on Twitter occasionally. TE: Yeah, so I spend my time animating. For the past year and a half, two years this, this arose out of the pandemic, right, we all went online, and some of us were teaching online and kind of upset with maybe some of the digital content that we could produce. And so I spent some time trying to figure out how to how to do some animations. But yeah, so this one I animated, I animated 12 of them, so a dozen of the proofs from Enrique and I, that we compiled I animated a dozen of them last year. This was part of, I submitted as part of Three Blue One Brown, Grant Sanderson, runs this summer of math exposition stuff. So I submitted that last year as my video, the idea being that you should think deeply about simple things because you can encounter a lot of things along the way. And this is not my quote, this is a quote from Ken — the person who started the Ross program, and I'm forgetting the Ross program, I'm forgetting the founder. His last name is Ross but I can't necessarily remember the first name. Okay. So yeah, so I have animated some of them. And I believe I've animated, I think I've animated Euler’s polyhedral theorem, Pick’s theorem, the classic visual proof, there's combinatorial proofs. So there's like, a double counting proof. And then there's one that uses bijective proof. So just some really cool ones out there to see and explore. KK: On YouTube? They’re on YouTube, right? TE: Yeah, that’s on YouTube. Yeah. Mathematics Visual Proofs is the name of the YouTube channel at this point. Who knows? It changes if you have to change it, right? KK: Well, we'll link to it. We'll find it. TE: Okay. I appreciate that. Thank you. All right. Cool. EL: Yeah. And so you said maybe this isn't the theorem you would have picked, if we had asked you, you know, three years ago or something. So how, how did this theorem get — Was it this project with Enrique that got you interested in it? TE: Yeah, I mean, I've always loved the theorem, but sort of seeing all of all of the available proofs and the ways th…

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Episode 87 - Tatiana Toro Sep 07, 2023

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Episode 87 - Tatiana Toro Kevin Knudson & Evelyn Lamb Download Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I am one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida, and your other host is… Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, where we sadly are past our beautiful, not too hot spring and fully into summer. So we enjoyed it while it lasted. I didn't have to turn on any air conditioning until after the start of July. KK: I think we started air conditioning in March.EL: Slightly different.KK: Little different vibe down here in Florida, but that's where we are. So anyway, it's summertime here, which means that there are tumbleweeds rolling through my department and I'm answering a few emails a day and trying to work, trying to do math. And boy, sometimes it's hard, you know, but sometimes it isn't. So. Anyway, so today, though, we are — this is great — we are very pleased to welcome Tatiana Toto, who will introduce herself and let us know what she's all about.Tatiana Toro: Thank you very much for the invitation. I'm very glad to be here. And in fact, I'm very glad to see Evelyn's cloud that I had heard about in other podcasts. So I'm Tatiana Toro. I'm a mathematician at the University of Washington, where I have been a faculty member since 1996. And currently I am the director of the Simon's Lab for Mathematical Sciences Institute, formerly known as MSRI. And I'm in Berkeley, California, and summer hasn't arrived yet. KK: It never will.EL: Yeah, that’ll be November, right?KK: I had actually forgotten that the name of MSRI had changed to the Simon's business. That’ll take some getting used to. I think I mentioned before we started talking, I spent a semester there, way back in 2006, and my son came with me, and my wife did too, and he was seven at the time. And now he's an adult living in Vancouver. It's weird how things change. I love that building, though. And the panoramic view you have the bay, and you can watch the fog roll in through the gate at tea time. It’s just a really wonderful place. So congratulations. How long have you been director? Has it been a year yet?TT: It’s almost a year, a year August first.KK: Yeah. That's fantastic. What a terrific position. And I'm glad that you're willing to take it on. Do you split your time between Berkeley and Seattle? Or are you mostly in Berkeley these days?TT: I am mostly in Berkeley. My students are still in Seattle, so I see them mostly on Zoom. But once in a while on a Friday, in Seattle. KK: Oh, so you go there. You don't fly them down?TT: Some of them have come, actually one of them this year to the summer school.KK: All right. So what is this podcast about? Favorite theorems. And you told us yours ahead of time, but we'll let you share. What is your favorite theorem?TT: Okay, so my favorite theorem is the Pythagorean theorem, and I know that everybody's gonna say what on earth are you talking about?EL: No, I really, really love this choice. And, you know, I've said this on many other iterations of this podcast, but I love that, you know, we'll get things that span the gamut from Pythagoras theorem, or the infinitude of primes, or something like that, all the way up to something that you, you know, you need to have been researching for 20 years in some very ultra-specific field to even understand, and so, you know, it just like shows how math connects with us in different ways at different times in our lives, and how we can appreciate some maybe things that seem very simple about math, even when we have had math careers for for many years. So yeah, tell us about how did you end up settling on the Pythagoras theorem?TT: So, actually, it has played a very important role in my career. Like, when I describe it to my students, when I'm teaching a graduate class and I talk about the some of the theorems I'll describe in a minute, I tell them, you know, one of the key ideas in my thesis was the Pythagorean theorem. So let me explain. It appears in many other results in this area of geometric analysis. So for example — let me give you two examples. So what was my thesis about? You have a surface, a blob in space, and you're trying to — two dimensions in R3 — and you're trying to understand if you can find a parameterization, which means a good way to describe it in terms of the plane. So can you deform the plane in a nice way so that it covers the surface? And a nice way means that distances are not changed too much. So I had some specific conditions for this surface, and the answer, the key, is in the situation I was looking at, yes, you could do it. And when you go and deeply look at what makes this possible, it is the Pythagorean theorem because the basic point is that if you can control how distances are distorted, you can control how the whole shape is mapped from the plane. And at the time, it looked like a curiosity. You know, I graduated many years ago. At the time, a few years earlier, Peter Jones had solved the analyst’s traveling salesman problem, which I'm gonna — just in general terms, let's imagine you have a lot of points in a square, and you're trying to understand whether you can pass a curve of finite length to all of these points. You're going to tell me, “If they’re finite, of course you can.” But you want to do it in an efficient way, in a way that doesn't depend on the number of points. And so he had found the condition that told you if this condition is satisfied, then yes. And there's not an algorithm, that doesn't exist yet, that tells you what's the best curve, but there's a curve, and he tells you that the length is no more than something. And what's behind that is the fact that if you have a straight triangle that has sides, A and B, and the other one is B, A squared plus B squared equals C squared. And it really is understanding that. And there's another important thing, the fact that the square root also plays an important role in these, but really, really, if you ask me, “What are the tools you need in this area?” I'll tell you how the square root behaves in the Pythagorean theorem, and then a couple of good ideas and you're able to reconstruct the whole thing.KK: I’m now curious about this traveling salesman problem. So there's no algorithm though?TT: No, there's no algorithm. I used the word analyst’s traveling salesman problem because the analyst wants to know whether you can pass a curve of finite length. Maybe you can say you're not ambitious enough. You don't want the shortest possible curve. To build the shortest curve, there’s no algorithm. And the construction of Peter Jones builds a curve, but it's not necessarily the best one.KK: Sure. Yeah.TT: It doesn't tell you it tells you the length is no more than D. But it's not. Yeah, no.EL: Yeah. I'm trying to remember if, like, I think there probably are some algorithms or some, like results that say like, you can get within a certain percentage of something. But yeah, the algorithm for the actual fastest path doesn't exist yet. Which is, you know, it's one of those things, it's like, huh, that's kind of surprising that we don't have a way to do that yet. Just means that there's still work to be done. Still jobs out there for mathematicians.KK: Well, because the combinatorial on the graph theory one is, is NP complete, right? I mean, yeah. So that that are NP-hard, or whatever. NP-something. I've never been clear about the differences. But is this one known to be that too?TT: I believe. KK: Okay. All right. TT: But you can construct — you know, so this was what was interesting about the problem, the result of Peter Jones, is that — the result of Peter Jones, and I have to say, I was very ignorant of that result, which had just happened a few years prior to my thesis. I have to remind the young audience that at the time, there was no internet the same way, and there was no arXiv, and you know, there was no Zoom. And then Peter Jones had a couple of postdocs at Yale, Stephen Semmes and Guy David, who started working on this. And the truth is, may I tell story about my thesis? EL: Yeah. KK: Please do.TT: So my thesis came out of misunderstanding. I went to my advisor, and I showed that these surfaces that I was looking at, which were some that he had looked at, that there was this property about distances over the surfaces, like if an ant traveled on the surface between two points, you know, taking the shortest path, it was comparable to the Euclidean distance. And so I went to my advisor, Leon Simon, and I told him, you know, I've been able to do this about these surfaces. And then he told me, oh, then I guess they have about they admitted bilipschitz parameterization, which is this good description. So okay, so I went to the library, and I looked through every possible book that I could find, and I couldn't find that. So I went back two weeks later and asked if he’d mind giving me a reference for this results, and he said, oh, I don't have a reference. That must be true. KK: It must be true.TT: And that became my thesis problem. And then, oh, there were many iterations of attempts. And I could do specific cases, but I could not do the general case. And on May of my fourth year, finally, somebody gives a colloquium where he talks about good parameterizations. And he talks about things like what I was thinking. I was thrilled. I mean, I thought, oh, I'm going go read everything this guy has written and my answer will be there. And then I told my advisor afterwards, I think I'm going to go read this guy's work. And this guy was Stephen Semmes, and he comes from harmonic analysis. And my advisor says, no, stop reading, I don't want you reading anymore. You just prove that theorem and that’s it. I don't want you reading. But one good thing, you know, harmonic analysts use squares, rather than balls. That's the most useful comment my advisor had. EL: Huh!TT: And what's interesting is that Stephen Semmes was talking about a broader class of surfaces than mine. And for those, he was asking, “Do bilipschitz parameterizations exist?” And for those the answer still is not known. And if I had gone and read everything that he had written, I mean, he was the big shot, I was the student, I might not have gotten my result. And I remember when I told Stephen at some point in the fall, oh, you know, I proved this, his first question, his first reaction, was, “I don't believe you.” And he said, “How did you do this?” And I said, “Using the Pythagorean theorem.” And so that's why the Pythagorean Theorem really is very dear to my heart.EL: Yeah. So I imagine that you saw the Pythagorean Theorem many years before you were in grad school. Do you remember, did it make a big impact on you when you saw it in school for the first time? I don't know what what year that would have been, elementary or middle school or whatever it was?TT: So I remember, I think I remember when I saw it because I remember the book. I had a beautiful — I went through the French system. I'm Colombian, but I went through the French system, and in the French system at the time, they tracked us very early on. And so we had these beautiful math book that, you know, I still remember how it smelled, and it was in there. But I remember the book, not especially the theorem. I never thought much about it until I got to graduate school. I used it other times. EL: Right. I mean, I think maybe the beauty of that kind of thing isn't necessarily what you're looking at, when you're a kid and first seeing math. You’re more like, okay, how can I use this to do the problems on the homework or something like that? So you were tracked into math pretty early on? You knew very early on that you were interested in math? TT: Yeah.KK: It’s nice they let you just do math. I think in the US what happens, I think, is students who are good at math are told they should be engineers. As if they're kind of the same thing, and they're not.TT: But that, you see, now, you feel free to remove this if you want. That's what the boys were told. The girls — since math was roughly like philosophy, and I come from a South American country, it was okay.KK: That’s fascinating. Okay, interesting. EL: Yeah. Well, I mean, there's a lot of different, you know, philosophies about whether tracking that early, you know, kind of deciding on what direction you want to go that early, is good or not. You know, it works for some people and not others, definitely.TT: Absolutely. I think it worked for me very well. And it didn't work on any of my classmates who were in the same class. I mean, I thought everybody loved it the same way I did and had as much fun. And then, it's interesting. Later on, I've learned that that wasn't the case. And then some of them suffered through it, you know. But to me, it was great.KK: So this is a French system in Colombia? Okay, this is a bit — okay, let’s get there. How did that actually happen? Why were there French schools in Colombia?TT: Well, I'll explain why there were French schools in Colombia and how I got into a French school. So there's something that's called a cooperation agreement between France and developing countries, where they have schools. The primary reason to have them is so the kids of their diplomats can continue their studies, but then they also offer them to the general population at a very reasonable price. They are private schools, but they are not as expensive. They're a fraction, or they used to be a fraction, of what the other private schools were. And at the time, so Colombia for a long time was what was called a Sacred Heart country. And so the ties with the Catholic Church were very strong. And so in terms of education for the girls, it was most girls went to nun school. But I am not Catholic, and therefore I couldn't, that was not an option for me. And so we needed a coed school. I mean, my parents wanted a coed school. The girls schools were all nuns. They wanted a coed school, and we needed an affordable coed school, and public schools were not good, and still unfortunately are not good. That's how I landed in the French school. KK: Fascinating.EL: Wow. Okay. Yeah.KK: Our listeners are learning all kinds of stuff, right?EL: Yeah, yeah, we've wandered a little away. But luckily, we know, thanks to the Pythagorean theorem, that we can walk back in a certain amount of time. So yeah, the other things that we like to do on this podcast is have you pair your theorem with, you know, some food, beverage, sport, you know, whatever, delight in life you would like.TT: So I actually will pair it with walking. So I'm going to give myself the title of urban hiker. I do walk long distances around town and in cities on a regular basis. I mean, I walk about two hours a day, at least. And so I pair it with that, because most often when I walk, I'm actually doing exactly the opposite of the Pythagorean theorem. I want to go the longest possible way, not the shortest possible way. But once in a while, I take the diagonal. And now that I'm living here in Berkeley, there's a beautiful diagonal that I take. And so I think about that here often.EL: Yeah. Do you like the hills?KK: Yeah, I was about to say, do you actually hike all the way up to the building there? Because that is quite a hike.TT: Not when I'm coming to work. But sometimes on weekends I do. You know, I want to crease and it depends. It depends what I…

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Episode 86 - Sarah Hart Jul 20, 2023

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Episode 86 - Sarah Hart Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. Or perhaps today we should say the maths podcast with no quiz at the end. My name is Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host. Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. It's Juneteenth. EL: It is, yeah. KK: And I'm all alone this week. My wife's out of town. And yesterday was Father's Day and I installed cabinets in the laundry room. This is how I spend my Father's Day, something we've been talking about doing since we bought the house. EL: That’s a dad thing to do. KK: 14 years later, I finally installed some cabinets in the laundry room. So it looks like you had a good time in France, judging from your Instagram feed. EL: Yes, yeah. And I'm freshly back, so I'm in that phase of jetlag where, like, you get up really early. And so it's 9am and I already went for a bike ride and did some baking and had a relaxing breakfast. At this point, I'm always like, “Why don't I do this all the time?” But eventually my natural circadian night owl rhythms will catch up with me. I'm enjoying enjoying my brief, brief morning person phase. KK: Yeah. Never been one, won’t ever be one as far as I can. EL: Yeah. Just keep moving west, and then you’ll be a morning person for as long as you can keep jetlag going. KK: That’s right. That's right. Yeah. EL: So yeah. Today we are very happy to have Sarah Hart on the show. Sarah, would you like to introduce yourself? And tell us a little bit about, you know, what you're all about? Sarah Hart: Ah, yes. So my name is Sarah Hart. I'm a mathematician based in in London in the United Kingdom. I'm a professor of mathematics, but my true passion is finding the links and seeing them between mathematics and other subjects, whether that's music or art or literature. And so I think there's fascinating observations to be made there, you know, the symmetries and patterns that we love as mathematicians are in all other creative subjects. And it's fun to spot them and spot the mathematics that's hiding in all of our favorite things. EL: Yeah. And of course, just a couple of months ago, you published a book about this. So will you tell us about it? KK: Yeah, SH: So this book, it's called Once Upon a Prime: The Wondrous Connections between Mathematics and Literature. And in the book, I explore everything from the hidden structures that are underneath various forms of poetry, to the ways that authors have used mathematical ideas in their writing to structure novels and other pieces of fiction and the ways that authors have used mathematical imagery and metaphor to enrich their writing, authors as diverse as you know, George Eliot, Leo Tolstoy, Marcel Proust, Kurt Vonnegut, you name it. And then I also look in the third section of the book at how mathematics itself and mathematicians are portrayed in fiction, because I think that's very, very interesting and shows us the ways in which those things at the time the books are written, how is the mathematics perceived? How has it made its way into popular culture? And how mathematicians are perceived as well, that tells us something fascinating, I think, about the place of mathematics in our culture. EL: Yeah, definitely. KK: We’re always portrayed as either mentally ill. Or just, like, absurd geniuses, you know, when really, you know, we're all pretty normal — most of us are pretty normal people, right? SH: Yeah. Well, we are, as everybody, there's a range. There's a range of ways to be human. And there's a range of ways to be a mathematician. But yeah, we're not all tragic geniuses, or kind of amoral beings of pure logic, or any of those things that you find in books. So yeah, and there are some sympathetic portrayals of mathematicians out there, and I know I talk about some of those, but yeah, it's very interesting how these these tropes, these stereotypes can creep in. EL: I must confess I'm about three quarters of the way through, I haven't quite finished that last section. But the first few sections that I've read, I've definitely — I keep adding books to my “Want to Read list,” so it’s a little dangerous. SH: Oh yeah, it should have a little warning, the book, saying “You will need a bigger bookcase.” Unfortunately, you know, you will want to go and read all of these books. And yeah, “Sorry, not sorry,” I think is the phrase. EL: Yes, definitely. I downloaded — so I don't need a bigger bookshelf because I put this one on my ereader — but I downloaded The Luminaries, which sounds like a really interesting book and excited to get to that, you know, in the neverending list of books that I'd like to read. KK: Right, we were talking about talking about our tsundoku business before [tsundoku is a Japanese word for accumulating books but not reading them]. So I actually I did, with a friend in the lit department, or in the language department, we taught a course on math and literature a few years ago. SH: That’s fantastic. KK: It was. It was so much fun. It's the best teaching experience I've ever had. But I was glad to read your book because we missed so much. Right? I mean, of course, we only had 15 weeks, you know, we and we talked about Woolf, like To the Lighthouse is kind of an interesting one. And yeah, I did finish the book. So sorry, Evelyn, I won. But no, it's it's actually, you know, it is spectacularly well written and, and I'm glad you're having success with it. Because it's — again, I like this idea, that you're sort of humanizing mathematicians and mathematics and showing people how it's everywhere. Isn't that part of your job? Aren’t you the Gresham professor, is that correct? SH: Yes, I’m the Gresham professor of geometry. So Gresham College is this really unique institution, actually. It was founded in 1597 in the will of Sir Thomas Gresham, who was a financier at the Court of Queen Elizabeth I in Tudor times. And in his will, he left provision for this college to be founded that would have seven professors, and their whole job was to give free lectures, at the time to the people of London. Of course now it's all livestreamed and it goes out and is available all over the internet. And anyone could go and it was just, you know, if you wanted to learn these subjects — and he thought there were seven most important subjects at the time that he said, I still say, geometry and mathematics more broadly, very important — but it was geometry, music, astronomy, law, rhetoric, physic, which is the old word for medicine, and I perhaps I’ve forgotten one. But yeah, these subjects, and so still today, this is what Gresham College does, free public lectures to anyone who wants to come. Now, you used to have to give them once in Latin and once in English. Now, you do not have to do it, thank goodness. KK: Yeah. Who would come? SH: I don’t know. Yeah, if I had to suddenly give my lectures in Latin, that might be slightly more of a challenge. My role there is to communicate mathematical ideas to anyone who wants to listen, so a general audience. And some of them will have mathematical training, but many will not. And they they're just kind of interested people who find things in general interesting, and mathematics is part of that. I love that idea, that mathematics is part of what a culturally interesting person might want to know about. And that is something that perhaps used to be more so than it is today. And I really would like mathematics to somehow be rehabilitated into what the cultural conversation involves, rather than it seems to be perhaps in a little bit, sometimes it's pigeon holed or put to one side, you have to be a geek to like mathematics. You have to be unusual. And it's really not true. It's not the case. EL: Yeah. Wow, that sounds like a dream job. I’m writing that down and putting it on my dream board? It's yeah. KK: I seem to remember, so I read the review of your book, I think by Jordan Ellenberg, who's also been on. SH: Yes. KK: It mentioned that the first person who held your chair invented long division. Is that right?SH: It’s true.KK: That's what used to get you a university job, is you invent long division. SH: Yeah. So that's, you know, what a lineage to be part of. I really feel honored and humbled to be in that role. And, actually, I'm the first woman to do this job in its 400 and whatever year history which, yeah, okay, you could say, yes, we might be a bit late with that one. But I feel it's a real privilege to do it. EL: Yeah. Well, that's wonderful. So we have invited you on this show to tell us what your favorite theorem is. So have at it. SH: Okay, so, my favorite theorem, I guess it's could be called a collection of theorems really, but the properties of the cycloid. So the cycloid is, it’s my favorite curve. And it's my favorite curve that probably unless you're a mathematician, you may not have heard of it. So people have heard of ellipses and circles and parabolas. And they've heard of shapes like triangles and things, but cycloids, people tend not to have heard of. And for me that's a surprise because they're so lovely. And the history of the study of the cycle of which, you know, we can we can talk about, is so fascinating and fun, and so many of the most famous mathematicians that people have heard of, like Isaac Newton, and Leibniz, and Mersenne, and Descartes and Galileo, and Pascal and Fermat, all of those people worked on the cycloid and were fascinated by it. And so there are these beautiful properties that it has, which we can bundle up into a theorem. And that would be my favorite thereom.EL: That’s great. And yeah, in case anyone listening to this doesn't know about the cycloid, it’s a cool curve. And it's actually, you know, it's a curve that a lot of people haven't seen as such, but it's one that does kind of arise sort of in everyday life, kind of. So yeah, do you want to describe what a cycloid is? SH: You can make a cycloid quite easily. It’s a fairly natural idea, I would say. Imagine a wheel rolling along the road. And now somewhere on the rim of the wheel, you paint the put a little blob of paint, or something like that, or if it's in the dark, you can put a little light. And then and then as the wheel rolls along, that blob of paint or little light will be following a particular path, as the wheel rolls. EL: Going up and down. SH: Kind of up and down. And eventually, sometimes it'll touch where the ground is. And then we'll go up and down again. And what you get is a series of arches, they look like arches. And that's what the cycloid is, normally you take one arch and call that the cycloid. KK: Right. SH: So this is quite a natural idea, what kind of shape will that be? And what is this arch shape? And the first thing you can say is, yeah, is it something I already know about? So early on in the study of this curve, which is first written down as a question, what is this shape? About 1500. Marin Mersenne, who is famous for Mersenne primes, among other things, so he thought maybe it's half an ellipse. And that's not too bad an approximation, but it isn't quite that. And so that's sort of question one. Is it something we already know? And it wasn’t. So then, people like Galileo started to ask, well, what do we what do we like to know about shapes and curves? So there are two questions really, at the time, they were called the quadrature question and the rectification. So quadrature is what's the what's the area? So if you make this arch, what's the area underneath this arch, between the arch and the road, I guess. That's question one. And the other one is the rectification: what's the length? So how long is this arch in terms of the circle that makes that makes the arch, the cycloid. And Galileo didn't know how to calculate either of those things. But he actually made, he physically made a cycloid. So he got a piece of sheet metal, and he rolled a circle along it, and he got the path. And then he cut it out and he weighed, he weighed the bit of metal that he had. EL: Oh wow! SH: To find an estimate for the area. Okay? So this is a real hands on thing. EL: Yeah, that’s commitment.SH: Because he did not know. So he physically made it and weighed it. And he got an answer that was around about three times the area of the of the circle that makes it, roughly speaking, and he said, Okay, if we all think, what’s a number that's roughly three, that's to do with circles, right? And so he wondered, could it be pi times the area of the circle? It isn't. It isn't pi times the area of the circle! Galileo never managed to work out exactly what it was. But this guy Roberval, Gilles de Roberval, did manage to work out what the area is. He didn't tell anyone how he'd done it because at this time in history, there were all these priority disputes, who sorted this thing first, who has done what first? People would sometimes go to the length of writing their solutions in code. So Thomas Hooke, who was another Gresham professor, when he worked out what we call Hooke’s law now, he wrote Hooke’s law down as an anagram in Latin, before he told anyone else. And then if anyone else came up with it, he could say, look, here's my anagram that I did earlier to prove that I thought of it first. So there were all these weird and wonderful things that people did at that time to establish priority. But Roberval, he had this incentive for not telling that he knew the area under a cycloid. And the incentive was this — it was not a good idea for them to do this — the job he had at the time, Roberval, was renewed every three years. And to get the job every three years, there were some questions that were set. And if you could answer those questions the best out of all the people who tried to do it, you could get that job for the next three years. But the person setting the question was the incumbent professor. So if you're the incumbent professor, you need to set questions that only you know the answer to, and then you get to keep your job. So for a few years, Roberval could say, you know, what's the area under this cycloid, and no one else knew. So he worked it out. And his proof was quite nice, but it wasn't published until 30 or 40 years after his death. But it actually — and this is the first lovely thing about the cycloid — the area, if you have a circle that's making this cycloid by rolling along road, the area underneath one of these arches is exactly not pi times, exactly three times the area of the generating circle. So a lovely whole number, simple relationship between the arts. EL: What are the odds? It’s almost miraculous. SH: Fantastic. So here's another equally miraculous thing that kind of adds to the first one. Then people try to work out what's the length of this cycloid? And the person who managed to solve that was, in fact, Christopher Wren. So he's well known as an architect, and he designed St. Paul’s, the wonderful dome of St. Paul's in London, and many other churches in London. But he was also a mathematician among many other things. So he solved the rectification problem, what's the length, and if the circle that makes this, the cycloid has diameter d. So we know that the circumference of that circle, the length around the circle would be pi times d. Well, another beautiful whole number relationship, the length of the cycloid arch is exac…

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