My Favorite Theorem

Latest Episodes:

Episode 65 - Howard Masur Apr 08, 2021

Show notes

Episode 65 - Howard Masur Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Hello, and welcome to My Favorite Theorem, a math podcast where there's no quiz at the end. I remember we did that tagline, like, I don't know, probably two years ago or something. And I forgot that I wanted to keep doing it. But I did it today. I'm Evelyn Lamb, one of your hosts. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host. Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. I forgot that tagline, too, and it's a pretty good one. Let's, let’s—look. EL: We’ll see if we remember later. KK: After our last recording session, we agreed we needed we needed a real tagline. So yeah. We're recording this on February 18, which means that Texas is largely without power and frozen. EL: Yeah. KK: And it's 82 degrees in Florida today. EL: Oh wow. Yeah. Most of my family is in Texas, and it is not great. KK: Do they have power? Or? No? EL: Most of them do. All of them do sometimes. KK: Right. Actually, now I think water is getting to be a problem now. Right? EL: Yeah. I haven't heard about any problems with that from my family. But yeah, it's not great. I hope that it warms up there soon and everything can come back online. But yeah, today, we're very happy to be talking with Howard Masur, who is in a place that is very used to being cold and snowy. So yeah, Howard, do you want to introduce yourself? Tell us a little bit about yourself? Howard Masur: Okay, thank you. First of all, thank you very much for inviting me to do this. I’ve been very excited thinking about about it. Yes, I'm on the math faculty at the University of Chicago. And I've, I guess, been working in mathematics for quite a long time and still enjoy it a great deal. It’s a major part, a very big part of my life. And your invitation to talk about my favorite theorem led me to, you know, think about what that would be and why I chose what I did. And and it made me think that, yes, what I really like the most in mathematics, or one of the things, is mathematics that connects different fields of mathematics. And maybe unexpectedly connects different fields. And I personally, have worked on and off in complex analysis and geometry and dynamical systems, another field. And I love the part of mathematics that sort of connects them. EL: Well that's perfect. Because I mean, you you're a frequent collaborator with my husband, Jon Chaika. But also with my advisor, Mike Wolf, who, you know, isn't quite in the same area of math generally. So yeah, you have worked in a lot of a lot of different fields that I feel like your name pops up, you know, in a very wide range of things related to geometry, analysis, dynamics, but yeah, you’ve got your finger in a lot of pots. KK: Right. Well, okay, so what is it? What's your favorite theorem? HM: Okay. It's called the Riemann mapping theorem. KK: Yes. HM: So, let me let me give a little bit of background. The first thing, it involves subsets of the plane which are called simply connected. And this is a notion from topology. And let me just say I looked at one of your podcasts and someone else talked about the Jordan curve theorem, where if you have a simple curve in the plane — it could be very, very complicated — a simple closed curve, then it has an inside and an outside, then the inside is simply connected. And a way of thinking about what simply connected means is heuristically it doesn't have any, it has no holes. But as also has been pointed out, they can be very complicated, Jordan curves. Certainly they can be simple looking like a circle. The inside of a circle is simply connected, the inside of a rectangle. But on the other hand, the Jordan curve can be very complicated like a snowflake, a Koch — I never remember how to pronounce that; is it “coke” snowflake? KK: Let’s go with Koch [pronounced “coke”]. HM: Pardon me? KK: Let’s go with that. HM: Okay. And so that's very complicated. It's the boundary — the curve is a fractal. So already simply connected domains can be very complicated, but they don't even have to be just the inside of a Jordan curve. You could take the plane itself, there’s a very simple example. You could take all the positive real numbers, include zero, and take it away from the complex plane. So the plane minus the positive real axis and also subtract the origin, that’s simply connected, it doesn't have any holes. And it's not the inside of a curve. You could also, on the other hand, here's something that isn't simply connected: you could remove the interval [0,1], including zero and one from your plane, just that interval on the real axis. And that is not simply connected because the complement, or the plain minus that, has a hole, which is that interval [0,1], it can be thought of as a hole. So that's the notion of simply connected. I don't know whether I should say more. I mean, that's what I thought to say about what simply connected means.KK: That’s great. Yeah, yeah, that's a good explanation. HM: Okay, and so that's a topological notion. And then the other thing that goes into this theorem is a notion from geometry, well, actually a notion from geometry and a notion from complex analysis. But let me let take a basic notion from geometry, which is called conformal. And the idea is that if you suppose you have two domains in the plane, and you have a transformation from one to the other, you say it's conformal if it's angle-preserving. So that means that if in the first domain, you have a pair of arcs — or maybe you prefer to think of them as straight lines, but it's better to think of a couple of arcs — that meet at a point, and then you apply the transformation, and you get a pair of arcs that meet in the image under the transformation. And you could measure the angle that you started with between the pair of arcs and the angle of the images of the pairs of arcs, and if the angles are equal at every point for every pair of arcs at those points, then you say the transformation is conformal, angle-preserving. Now, in some ways, the nicest — so let me give some examples that are and are not. The nicest transformations, certainly of the plane, are linear transformations. KK: Sure. HM: Given by two by two matrices, and they turn out not to be typically conformal. There are some that are, for example, a rotation about the origin is conformal. You know, if you have two lines and you rotate them, the angle they make after rotation is the same as the angle they started with. If you — this isn't strictly a linear transformation, it’s called affine — if you take a translation of the plane, if you take every point and you add the same vector, think of them as vectors, that's angle-preserving, that's a conformal transformation. Here's another one that's back to linear. If you take, for example, every point, which has, say, coordinates (x,y), and you multiply x by 2 and y by 2, so you multiply the coordinates by the same number, 2. That's called a scaling. And that's angle-preserving. One can sort of check that out. What that transformation does is, for example, it takes a square with one vertex at the origin, a unit square, and then another vertex on the x-axis at the point (1,0) and another at the point (0,1), last point at (1,1), and it takes a unit square to a two by two square, and that's angle preserving. But that's it — well, and the composition — but typical linear transformations are not angle-preserving. So, for example, if you took (x,y) and the transformation took (x, y) to (2x, y/2), so it multiplies in the x direction by 2 and multiplies in the y direction by a half, it takes a unit square into a rectangle, and that's not angle-preserving. It preserves the right angles, but it doesn't preserve other angles. EL: Yeah, you can imagine the diagonal is, you know, [demonstrates with arm gestures that are very helpful to podcast listeners]. HM: The diagonal is closer to the x-axis, so the diagonal which made an angle of 45 degrees will be moved with the x-axis. The x axis goes to itself, and the image of the diagonal is moved closer to the x axis. Yeah, exactly. So there aren’t maybe, there aren't so many linear transformations of the plane to itself, and so let me tell you what the theorem is, and this is a beautiful, beautiful theorem, I think, and it was really a cornerstone of, in the 19th century, of the beginnings of complex analysis. Oh yes, I’m sorry. Before I do that, let me also connect conformal, as I had mentioned, to complex analysis. One also can think of the euclidean plane as the complex plane, where (x,y) becomes x+iy, becomes a complex number z, and then conformal, another way of saying it, is that the map, the transformation from some region in the plane to some other region in the plane has a complex derivative. It’s what you call complex analytic. It has a derivative and the derivative is not zero. Again I looked at your podcast. Someone talked about the Cauchy-Riemann equations, and that's exactly what complex analytic means is that the Cauchy-Riemann equations hold. Where where w is u+iv and z is x+iy, then it's complex analytic if ux=vy and −uy=vx. That’s the Cauchy-Riemann equations, and that's from complex analysis. It has the names Cauchy and Riemann, who where in some sense the founders of complex analysis. And that's equivalent to conformal, so even there just in this, there's already kind of an amazing theorem that relates — I think obviously you had somebody on your podcast maybe talk about this — that relates complex analysis to geometry, conformal meaning angle-preserving and complex analytic meaning, let's say, the Cauchy-Riemann equations hold. KK: Right. HM: Okay, so the theorem is that if I take any simply connected set’s domain in the complex plane, other than the complex plane itself, okay? And I take the unit disc — so that's inside the circle of radius one, so that's simply connected — I can find a conformal transformation from the unit disc to this simply connected domain, and maybe thinking about the inverse, it's a conformal transformation from that (maybe crazy) simply connected domain to the unit disc, and so that's the Riemann mapping theorem EL: Yeah, and it's just amazing. I mean I think there's part of me that still doesn't believe that it's true. I've actually just, I don't know when it was, maybe a month or two ago, I think I was brushing my teeth or something and just thinking, why hasn't someone pick the Riemann mapping theorem yet for My Favorite Theorem? HM: Okay, all right. KK: It's a really mind-blowing theorem. So when I teach the undergraduate complex analysis course that we have, I don't get to it until the very end. HM: Yeah. KK: And it's kind of hard. You can't even really prove it especially at that level, but students just look at me like, there's no way this is true. This just can’t be true. So it's really remarkable that anything — I mean, you're right. I mean, these simply connected domains can be bizarre. But they're conformally equivalent to the unit desk. That's just blows my mind still. Yeah, EL: Yeah. It's just hard to imagine, like, this fractal snowflake, you know, how can you straighten that out enough to just be like a circle? HM: Let me contrast it — and this also goes back kind of to the founding mathematicians of the subject. If I take what's called an annulus, let's say I take the circle of radius 1. And I take the circle of radius R, where R is bigger than 1. And I take the region between them. So the region between two concentric circles, that's not simply connected because it has a hole, namely, the inside of the unit circle is the hole. And so if I take one of radius, the inner is radius 1, the outer radius is R, and I take another one, inner radius 1 and outer radius R’. And let's say R’ is not equal to R. So it's a different outer radius. They are not conformally equivalent, even though they are very simple boundaries, their circles. So there was something very, very special about simply connected. And that's also kind of what makes the theorem amazing. And then the fact that it doesn't work for something not simply connected started a whole field of mathematics that has been going on for close to 200 years. EL: And so was this kind of a love at first sight theorem for you the first time you saw it? HM: You know, I guess I'm not 100% sure. I was in college a little while ago, and I don't don't think I had complex analysis in college. And so I may not have run into it then. But certainly, as a first-year graduate student at University of Minnesota, and my professor, who then became my thesis advisor within a year, you know, for my PhD advisor, that was somehow his field. And so I certainly learned it as a graduate student. And that led me — again, I can't exactly say it led me to what I do — but, you know, it certainly had a big influence, and things that I do sort of have grown out of this whole history of this, from the from from the Riemann mapping theorem. KK: So, is this one of those theorems is actually named correctly? Did Riemann actually prove it? HM. I don't know, I'm not a historian. You know, I mean, I could ask. For that matter, are the Cauchy-Riemann equations named after the right people? Yeah. I mean, I know the modern proof that one sees in books on the Riemann mapping theorem is not due to Riemann. It’s I think, early 20th century. EL: Is it Poincaré maybe? HM: You know, my mind is going blank here for a second. EL: It’s someone. HM: I don't know. I'm not a historian, and I did not look it up to say “Does Riemann really deserve credit?” KK: But wait, I looked at Wikipedia. I’m cheating. The first rigorous proof of the theorem was given by William Fogg Osgood in 1900. HM: Oh, okay. Okay. Yeah. KK: So apparently Riemann, this is in his thesis, actually. But there were some issues, it depended on the Dirichlet principle. And Hilbert sort of fixed it enough that it was okay. But Osgood is credited with the first rigorous proof. HM: Well, isn’t it also somehow the case again, that mathematicians 200 years ago did not quite have the rigor that we have now? KK: That’s true. Cauchy sort of put limits on the right footing more or less, but I think it still took a little while to get it cleaned up, right? So are there any really interesting applications of this theorem that you like? Or is it just beauty for its own sake? HM: Gosh, you know, I'm not sure. I think beauty for its own sake, I mean, but also to my mind, it opened up a whole branch of mathematics where you study, well, for example, you study surfaces. Or maybe it's the difference between topologists and geometers. A topologist says that a doughnut, famously, a doughnut is the same as a coffee cup with a, you know, with a handle and so forth. And geometers say, well, we could put different ways of measuring angles, different metrics on a torus that are not conformally equivalent, that there's no transformation from one to the other that preserves angles. KK: Right. HM: And this Riemann mapping theorem says, No, you can't do that for simply connected. They are conformally the same. But as soon as we move to topologically more complicated things like a torus or even these annuli, or surfaces with more holes, genus, then there are different ways of putting metrics and measuring angles and so forth. And so it opened up, and again, this actually also has Riemann’s name to it, it’s the Riemann moduli space, is the st…

Full show notes at the publisher

Episode 64 - Pamela Harris and Aris Winger Mar 11, 2021

Show notes

Episode 64 - Pamela Harris and Aris Winger Kevin Knudson & Evelyn Lamb Download Kevin Knudson: Welcome to My Favorite Theorem, a math podcast. We need a better tagline, but I'm not going to come up with one today. I'm Kevin Knudson, professor of mathematics at the University of Florida. Here is your other host. Evelyn Lamb: Hi, I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City. And I think that our guests might be able to help us with that tagline. But we'll get to that in a moment because I have to share with you a big kitchen win I had recently. KK: Okay. EL: Which is that that I successfully worked with phyllo dough! It was really exciting. I made these little pie pocket things with a potato and olive filling. It was so good. And the phyllo dough didn't make me want to tear out my hair. It was just like, best day ever. KK: Did you make it from scratch? EL: No, I mean, I bought frozen phyllo dough. KK: Okay, all right. EL: Yeah, yeah, I’m not at that level. KK: I’ve never worked with that stuff. Although my son and I made made gyoza last month, which, again, you know, that that's a lot of work to because you start folding up these dumplings, and you know. They’re fantastic. It's much better. So, yeah, enough. Now I'm getting hungry. Okay. It's mid afternoon. It's not time for supper yet. So today we have we have a twofer today. This is this is going to be great, great fun. It's like a battle royale going here. This will be so much fun. So today we are joined by Pamela Harris and Aris winger. And why don't you guys introduce yourself? Let's start with Pamela. Pamela Harris: Hi, everyone. I like how we're on Zoom, and so I get to wave. But that’s really only to the people on the call. So for those listening, imagine that I waved at you. So I am super excited to be here with you all today. I'm an associate professor of mathematics at Williams College. And I have gotten the pleasure to work with Dr. Aris Winger on a variety of projects, but I'll let him introduce himself too. Aris Winger: Hey everybody, I’m Aris Winger. I'm assistant professor at Georgia Gwinnett College. I've been here for a few years now. Yeah, no, we, Pamela and I have been all over the place together. I've been the honored one, to just be her sidekick on a lot of things. PH: Ha, ha, stop that! EL: So we're very excited to have you here. So you've worked on several things together. The reason that I thought it would be great to have you on is that one of the things is a podcast called Mathematically Uncensored. And it's a really nice podcast. And I think it has a fantastic tagline. I was telling Aris earlier that it just made me very jealous. So we've we've never quite gotten, like, this snappy tagline. So tell us what your podcast tagline is. And a little bit about the podcast. PH: Maybe I can do the tagline. So our tagline is “Where our talk is real and complex, but never discrete.” AW: Yeah, that's right. That is the tagline. And yeah, it's a good one. And sometimes I have to come back to it time and again to remember, so that we live up to that during the podcast. We're taping the podcast later today, actually. And so it should be out on Wednesday. So yeah, the show is about really creating a space for people of color in the mathematical sciences and in mathematics in general, I think. And so one of the ways—I think for us the only way that can happen—is we have to start having hard conversations. Right. And so a realization that comfort and staying on the surface level of our discussions doesn't allow for us to have the true visibility that all people in mathematics should have. And so for too long, we've been talking surface-level and saying, “Oh, we have diversity issues. Oh, we should work harder on inclusion.” No, actually, people are suffering. No, actually, here's our opinion. And stop talking about us; start talking to us. So it really is a space where we're just like, you know what, screw it. Let us say what we think needs to be said. Listen to us. Listen to people who look like us. And yeah it’s hard. It's hard to do the podcast sometimes because when you go deeper and start to talk about harder topics, then there are risks that come with that. Pamela and I, week after week, say, “Oh, I don't know if I really should have said that.” But ,you know, it's what needs to be said, because we're not doing it just for us. We're doing it to model what what needs to happen from everybody in this discipline, to really say the things that need to be said. KK: Have you gotten negative feedback? I hope not. AW: Yeah, that’s a good question. So I mean, I think that the emails we've gotten are have been great and supportive. But I think, so for me, I'm expecting no one to say — I’m expecting the usual game as it is, right, that people aren't going to say anything, but of course there's going to be backlash when you start saying things that go against white privilege and go against the current power structures. You know, I'm expecting to be fired this year. PH: Yeah, those are the conversations that we have constantly — that we’re having on the podcast are things that Aris and I are having conversations about privately. And so part of what's been really eye-opening for me in terms of doing a podcast is that I forget people are listening. There are times Aris and I are having just a conversation, and I forget we're recording. And I say things that I normally would censor. If I were in a mixed crowd, if I were in a department meeting, if I were at a committee meeting for, you know, X organization. And I think it's not so much that we would receive an email that says, “Hey, you shouldn't have said X, Y, and Z,” it’s that we are actually getting targeted. For example, I was just virtually visiting Purdue University giving a talk about a book that Aris and I wrote, supporting students of color. And accidentally, the link got shared to the wrong people. And all of a sudden, I'm getting Zoom-bombed at a conversation. That's targeted, right? So those are the kinds of things that we are experiencing as people of color, and we have to have conversations about how are we ensuring that this isn't the experience when you bring a Black or brown mathematician to talk virtually at your colloquium. And if we're not talking about that, then no one is talking about that, because people are trying to hide their dirty laundry. Purdue University is not putting out an email to their alumni saying, “By the way, we invited Pamela Harris to show up and talk about how we best support students of color. And then we got Zoom-bombed, and somebody was writing the N-word and saying f BLM.” Right? Like, that's not happening. AW: Yes. Wait, they didn't say anything about it? PH: Well, they're actively doing things about it. But you know they're not putting out the message. AW: Right. So then it gets sanitized, right? So a traumatic attack gets sanitized to be something else. There are two things about the podcast that Pamela and I, and the Center for Minorities in the Mathematical Sciences, really are trying to work with is making sure that we call out these things, but then not to center it, right, because the the podcast itself is supposed to be about our experiences. But a lot of ways there's a significant part of our experiences that is tied to having to continuously fight against this type of oppression against us. EL: Yeah. And I think it's really important to have that. And it's so important that it decenters— I think I was listening to an episode recently where you talked about the white gaze and what you have to deal with all the time in trying to present things to a majority white audience. And I think it's really important for us white people to listen to this and realize that not everything is about and for us. And I mean, there are so many things in life where this is true: movies, TV shows, books and stuff. And yeah, I think it's great that that your voices are there and having these conversations, and I think that people should listen to your podcast. AW: I appreciate that. Yeah. Because it requires a deep interrogation, a self-interrogation by white people to really deal with the feelings. Let me just step back and give the usual disclaimer. Everybody's nice. Everybody's good. Nobody's mean. Nobody is a bad person. Let me just say all that to get that out of the way, right? But what we're talking about is that when I say something on the show, when Pamela says something on the show and you get this feeling like “Wow, that doesn't feel good to me,” then you need to take some time and figure out why it is that you're feeling this way. And it's tied to your privilege, something that you need to interrogate, and it will make you a better person and for everybody. KK: I don't know. I can't wrap my head around people, like, Zoom-bombing. This is nothing that would ever come to my mind. “You know what, I'm going to go Zoom-bomb this person.” I just… EL: Well, I mean, it’s just a bad way to spend your time, but not everyone has the same time priorities. AW: Well, no. So I think that's a great question. And let me just say that it that's how deep and pervasive it is in people, right, that people grow up and have this experience of being raised by other people who have ingrained within them that it is fundamentally, and in some sense, it just burns their soul to have somebody who does not look like them, have someone who is “lesser” than them take the center stage, be deemed the expert. And so again, I'm not calling these people bad. But there is something within some of us that says — and it’s called white supremacy, by the way — that we all have, that we all have to fight, that is so ingrained in some people that they feel compelled to do it. And so they, again, no one's going to fix that for them. And the person who did this to Pamela has it in spades, right? And so when we say that, so I think too often we make it an intellectual exercise, right? We say that it just makes no sense. Right? It doesn't make any sense because white supremacy makes absolutely no sense. But it is a thing. And it's there. And that's what it is, right? So I've been working a lot on calling, naming things so that we don't get confused, because as long as we don't name it, then it just gets to be out there. Like, “Oh, I don't understand.” We understand this exactly. It's called white supremacy. And we need to fight it in our discipline, and across the board. PH: And it doesn't always just show its face via Zoom-bombing with the N-word in the chat, right? It shows up with who you invite to your podcast. It shows up with who's winning awards from our big national organizations. It shows up with who gets tenure, who even lands into a tenure track position, who even gets to go into graduate school, who actually majors as a mathematician, who actually goes to college, who actually graduates high school, who actually gets told that they're a mathematician. Right? So this is showing its ugly head in very visual ways that we all feel a huge sense of, “Oh, no, this is terrible. I'm sorry, this happened to you.” But the truth is that white supremacy is in everything within the mathematical sciences. And so you know, we got to pull it at its root, my friends. At its root! AW: Yes. PH: So this was just one way in which it showed itself, but I want to make it clear that it is pervasive. KK: Sure. Right. EL: So what I love about hosting this podcast is that we get to know both people and their math and their relationship to their math. And so we're gonna pivot a little bit now, maybe pivot a lot now, and say, Okay, what are your favorite theorems? And, yeah, I don't know who wants to go first. But, yeah, what's your favorite theorem? KK: Yeah, let’s hear it. PH: I’ll do it. I’ll go first. I always like hearing Aris talk. So I'm just like “Aris, go,” right? But no, I’m going to take the lead today. Alright, so I wanted to tell you about this theorem called Zeckendorf’s theorem. I don't know if you know about it. KK: I do not. PH: And it goes like this. So start with the Fibonacci numbers without the repeated 1. So 1, 2, and then start adding the previous two, so 3-5-8, and so on. Alright. So if you start with that sequence, his theorem says the following, if you give me any positive integer N, I can write it uniquely as a sum of non-consecutive Fibonacci numbers. AW: Oh, wow! EL: Uniquely? PH: Yes. And this is why you need to get rid of the 1, 1. Because otherwise you have a choice. But yeah. So it's hard to do off the top of my head, because I'm not someone who, like, holds numbers. But say, for example, we wanted to do 20. Maybe we wanted to write the number 20 as a sum of Fibonacci numbers that are not consecutive. So what would you do? You would find the largest Fibonacci number that fits inside of 20. So in this case, it would be 13. AW: Yeah. PH: 13 fits in there. Okay, so we subtract 13. We're left with 7. Repeat the pattern. KK: Ah, five and two. KK: Five and two! They're non consecutive. KK: Okay. PH: Yeah. AW: Wow! PH: Three is in between them, and eight is in between the others. And so you can do this uniquely. And so this is using what's known as the greedy algorithm because you just do that process that I just said, and it terminates because you started with a finite number. KK: Sure. PH: And so the the proof, of course, there's the, you know, “Can you do it?” but then “Can you do it uniquely?” So the thing that you would do there is assume that you have two different ways of writing it, each of which uses non-consecutive, and then you would argue that they end up being exactly the same thing. So that, in fact, they use the same number of Fibonacci numbers and that those numbers are actually the exact same. KK: Sure, okay. EL: Yeah. Like I'm trying to figure out — and I don't, I also am not super great at working with numbers in my head just on the fly. But yeah, I'm trying to figure out, like, what would have gone wrong if I had picked eight instead of 13 to start with, or something? And I feel like that will help me understand, but I probably need to go sit quietly by myself and think about it. Because there’s a little pressure. PH: Yeah, it's a little subtle. And it might be that you don't get big enough, you end up having to repeat something. EL: Yeah, I feel like there's not enough left below eight to get me there without being consecutive. PH: Yeah. Right. AW: Right. Because you’ve got to get 12. Yeah, yes. Yeah. KK: Yeah, it makes sense, right? Like, I guess, you know, if you pick the largest one less than your number, then it's more than halfway there. That's sort of the point, right? So that's how you prove it terminates, but also the the non-uniqueness, the non-uniqueness seems like the hard part to me somehow, but also the non-consecutive. Wait a minute, I don't know, which is. AW: Well it sounds hard, period. KK: Yeah. I like this theorem. This is good. What attracts you about this theorem? What gets you there? PH: So I, in part of my dissertation, I found a new place where the Fibonacci numbers showed up. And so once you find Fibonacci numbers somewhere new, I was like, what else is known about these beautiful numbers? And so this was one of those results that I found, you know, just kind of looking at the literature. And then I later on started doing some research generalizing this theorem. So meaning, in what other ways could you…

Full show notes at the publisher

Episode 63 - Lily Khadjavi Feb 11, 2021

Show notes

Episode 63 - Lily Khadjavi Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Hello, and welcome to My Favorite Theorem, the podcast from 2021. I don't know why I said that, just, it's a math podcast, and it is currently being taped in 2021. I'm your host Evelyn Lamb. I'm a freelance math 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. No, look, it's important to say it's 2021 because 2020 lasted for about six years. It was—I couldn't wait for 2020 to be over. I don't think 2021 is much better yet. It's January 5. I'll leave our listeners to figure out what's going on right now that might be disturbing. And, and yeah, but anyway, no, happy new year. And I had a very nice holiday. My son has been home for nine months now. He's going to go back to school finally next month to finish up his senior year in college. And I did nothing for a week. I mean, like when I say nothing, I mean nothing. Just get up, watch some TV, like we’re watching old reruns of Frasier, like this is the nothing levels I saw. It was fantastic. EL: Very nice. KK: How about you guys? Did you have a nice holiday? EL: Um, I had a bad bike accident right before Christmas. So I had some enforced rest. But I'm mostly better now. I have gotten on my bike a couple times, and nothing terrible has happened. So still a little more anxious than usual on the bike. We were taking a ride yesterday and I could tell I was just like, not angry, but just, you know, nervous and worried. And it's just like, Okay, I'm just at the scene of the trauma, which is my bike seat, and getting over it. But I hope I will continue to not fall off my bike and keep going. KK: That’s the only thing to do. Back in my competitive cycling days when I was a postdoc, I had some pretty nasty crashes. But yeah, you just get back on. What else are you going to do? So anyway, enough of that. Let’s talk math. EL: Yes. And today, we are very happy to welcome Lily Khadjavi to the show. Hi, will you introduce yourself and tell us a little bit about yourself? Lily Khadjavi: Hi. Oh, thanks, Evelyn. It's so great to be here. I'm Lily Khadjavi, as you said. I'm a professor of mathematics at Loyola Marymount University, which is in Los Angeles, California. I'm a number theorist by training, but I'd say that I'm lucky to have taken some other mathematical journeys, especially since graduate school, and I don't know, for example, this past year, maybe my biggest excitement is I was lucky to be appointed to a state board in California. So by the Attorney General, Xavier Becerra, to be appointed to an advisory board looking at policing and law enforcement and the issue of profiling. And so that's an issue that's very important to me. And it was an unexpected mathematical journey. EL: Yeah. LK: If you’d asked me 20 years ago, what would I be up to, I might not have thought of that. And I've taken many a bike spill in my day, so I could feel some nice affinity being here today. You’ve just got to get back on and be careful, of course. EL: Yeah. And that that must be an especially important issue in LA, because I know the LAPD has been the subject of some, I guess, investigations and inquiries into their practices and things like that. LK: That's exactly right. And over the years, it was under a consent decree, so an agreement between the US Department of Justice and the City of Los Angeles, with many aspects monitoring police practice. And actually, some of that included data collection efforts looking at traffic stops. And that, combined with teaching a statistics course, is what really gave me a window more into policing practice, into problems that where I wanted authentic engagement for my students with the real world and took me on, maybe I'll say unexpected journeys to law conferences and elsewhere, as I started to learn more about the issues, the ways that as mathematicians, we can bring tools to bear on on these social questions too. EL: Yeah, very cool. KK: Yeah. EL: So what is your favorite theorem? And I know that's an unfair question, but I will ask it anyway. And then, you know, you can run away with it. LK: Yeah. I know this podcast is not visual, but I'm already kind of smiling in a terrified way because I found this question so difficult, really an impossible task, because I thought it's like asking me when my favorite song—I don't know, do you have a favorite song? EL: That is hard to say. If you asked me, I would start listing things. I would not, probably, be able to tell you one thing. LK: What do you think, Kevin? KK: I, uh, Taxman? LK. Okay, I thought you would name the opening the music for the podcast as a favorite too. EL: Oh, yeah. LK: You know, shout out to that. KK: I do like that. But now, you know, maybe What Is Life by George Harrison? Single? LK: Oh, yeah. Okay, well, maybe I'll count that as listing, which is what Evelyn started to do. Because it's very, difficult. KK: It is. LK: You know, I was really wrestling with this. And it got me kind of thinking about why do we like certain theorems. I think I pivoted to what Evelyn said. I started wanting to make lists. And of course, it's fun to talk about things that are new to everyone. And, you know, it's been a remarkable podcast, and lots of people have staked out, I mean, they've grabbed those beautiful favorite theorems. But I started thinking, could you have a taxonomy? I really saw a taxonomy of theorems. Not by discipline. So not a topological statement or an analytic proof, but by how mathematicians feel about them, or the aesthetic of them. And so my first you know, category had to be sort of the great workhorses, like those theorems that get so much done, but they also they never cease to amaze you. And I mean, it’s hard not to point right away to the fundamental theorem of calculus, and I think maybe in your very first episode. That's right, that might be what? KK: Yeah, Amie Wilkinson. EL: Yes, Amie Wilkinson just came in and snatched that one. Although as everyone knows, we do double theorems, you know, we don't have a rule that you can't use the same theorem again. LK: No, because that's one we use again and again and again. You know, even this past semester, I was teaching multivariable calculus. And you know, we have this march through line integrals, double, triple integrals, and we build, of course, to Green’s theorem, Stokes’ theorem, the divergence theorem. So these main theorems in calculus that the machinery is heavy enough for the students that even if I'm trying to put them in a context where, “Oh, this is really all the fundamental theorem of calculus,” I think that gets obscured obscured for students first trying to get their head around these theorems. Even though you relate them, you say, Oh, but they've got the boundary of this—maybe endpoints of a curve or some other surface boundary, and you're relating it as the relationship between differentiation integration, and it's so it's beautiful stuff. But I think I'm not convinced my students thought of it as the same theorem, even if I tried to emphasize this perspective. But still, they, all of us can be blown away by how powerful the theorem is in all of its incarnations. And so that's a great workhorse. So we don't have to talk at length about that one. It's been here before, but you know, you just have to tip your hat to that one. But I was wondering, are there other great workhorses something you put in that in that category? KK: So I argue—I mean, so you mentioned the fundamental theorem—the workhorse there is actually the mean value theorem. LK: Hmm. KK: Because the fundamental theorem, at least for one variable, is almost a trivial corollary of the mean value theorem. And I didn't appreciate that until I taught that sort of undergraduate analysis course for the first time. And I said, “Wait a minute.” And then I sort of came up with this joke, I'm actually going to write a book. It's like a “Where's Waldo” style thing: Where's the mean value theorem? Because in every proof, it seemed like, Well, wait a minute, by the mean value theorem, I can pull this point out. Or there is one, I don't know where it is, but it's in there somewhere. So I really like that one. LK: That’s a really great perspective. I also will say that I did not happen on that feeling until teaching analysis for the first time, of course, versus, you know, for seeing these theorems or learning about them, and even learning them in analysis, not just using them in calculus. Know, that reminds me that it wasn't till grad school, maybe taking a differentiable manifolds class, and that's not really my area. But seeing, Oh, you can define a wedge product, you can define these things in a certain way. Oh, they really are literally all the same theorem. But I like this perspective, maybe that would have been a way to convince my students a little bit more, to kind of point to the mean value theorem, because it would put them on more familiar turf too. I really like that. Yeah. Are there other workhorses? EL: So the first one that came to my mind was classification of surfaces, in topology, of like, you know, the fact that you can do that—I feel like I it's like so internalized to me now. And yeah, I don't know, that for some reason that came to mind, but it's been a long time since I did research and was keeping up with, you know, proving things. So yeah, it’s—but yeah, I think I would say that anyway. KK: Yeah. And I would sort of think anything with fundamental in its name right. LK: Yeah, I was thinking that. KK: So the fundamental theorem of arithmetic, okay, so that you can factor integers as products of primes, or the fundamental theorem of algebra, that every polynomial with complex coefficients has a root. But then more obscure things like the fundamental theorem of algebraic K-theory. You guys know that one? LK: That one, I'm afraid does not trip off my tongue. KK: All it is, is it's a little bit weird. It just says that the K-theory of if you have a ring, maybe it needs to be regular, that if you look at the K-theory of the ring, and the K-theory of a polynomial ring in one variable over it, they're the same. And the topological idea of that is that, you know, it's a contractibility argument somehow. And so it's fundamental in that way. LK: These are great workhorses. Yeah. And also, Evelyn, you mentioned the classification, like these results are just so fundamental. So in whether they have fundamental in the name or not, they are. EL: Like, naming it fundamental, it's almost like cheating that point. Or, maybe not cheating, maybe stealing everyone else's thunder. It’s like, “No, I already told you that this is the fundamental theorem of this.” LK: My poor students, whenever I want them to conjure up the name and think of something that way, I make the same corny joke. I'm like, “It's time to put the fun back into…” and they’re like, “Ugh, now she's saying fundamental again.” So yeah, I was thinking, too, that in different fields, we reach back, even as we're doing different things in our own work, back to those disciplines that we were sort of steeped in. And I think for topologists, there are so many great theorems to reach to. KK: Sure. LK: But I was thinking even like the central limit theorem in statistics and probability, so this idea that you could have any kind of probability distribution—start with any distribution at all—but then when you start to look at samples, when the samples are large enough, that the mean is approximated by a normal distribution. That somehow never ceases to amaze me in the way that the fundamental theorem of calculus, too. Like, “Oh, this is a really beautiful result!” But it's also a workhorse. There are so many questions in statistics and probability that you can get at by gleaning information from the standard normal distribution. So maybe I’d put that into a workhorse category. KK: Sure. EL: Actually, Heine-Borel theorem, maybe could be kind of a workhorse, although I'm sort of waiting for for you to say that it's actually the mean value theorem too. KK: No, it's just, it's just that, you know, compact sets are closed and bounded. That's it. Right? EL: Yeah. Yeah, actually, yeah, that, once again, is such a workhorse that it's often the definition that people learn of compactness. LK: That’s right.EL: Like the first time they see it. Or, like such an important theorem that it it almost becomes a definition. Actually the Pythagorean theorem, in that case, is almost a definition. KK: Sure. EL: Slash how to measure distance in the Euclidean plane. LK: Yeah, that's a good example. So maybe now we have so many workhorses, well, another category I was thinking of — it's beautiful stuff. I was thinking of those theorems where the subtlety of the situation kind of sneaks up on you. So maybe you hear the statement, and you kind of even think, “Oh yeah, I believe that,” like the Jordan curve theorem, I think you had a guest speak about this, too. So this, you know, idea of a simple closed curve. So you just draw it in the plane, there's an inside, and it divides the plane into an inside and outside. And I kind of really remember—I can't tell you what day of the week it was—but I remember the first time this came up in a class, and I thought, “Yeah.” But then we started thinking about how would you go about proving something like this, or even just being shown, someone drawing, a wild enough crazy curve, where suddenly you can't just eyeball it and immediately see what's inside and what's outside. So I don't know what this category or set of theorems should be, but the subtlety sneaks up on you even though statement seems reasonable. EL: “I can't believe I have to prove this.” Maybe that’s slightly different. Well, what I mean is like, I can't believe this is a—It seems so intuitive that understanding that there is something to prove is a challenge, in addition to then proving it. LK: Yeah. And maybe you can't even prove it—Well, how about the four color theorem? So this map coloring theorem, this idea that the four colors suffice, so if you have states or counties or whatever regions, you want to make your map of, that if they share a common edge boundary, then use different colors, that four colors is enough. I don’t know, has a human being ever proven that? My understanding is that it took computing power. KK: It’s been verified. EL: I think they’ve reduced the number of cases, also, that have to be done from the initial proof, but I still think it's not a human-producible proof. KK: That’s right. But I think Tom Hales actually verified the proof using one of these proving software things. So I mean, yeah, but that was controversial. LK: That brings up a neat question about what constitutes proof in this day and age. I've seen interesting talks about statements where, or journals where something's given as this: “Okay, here's a theorem. And here's the paper that's been refereed.” And then later, oh, here's something that contradicts it. And people are left in a sort of limbo. Well, that's another discussion, things unproven, un-theorems, I don't know. Well, anyway, in this category, that's going to help the subtlety of the situation sneaks up on you. If I start coloring maps, testing things out, after a while, I’d say, “Oh, there's a lot to this.” But the statement itself has an elegant simplicity. KK: Well, it…

Full show notes at the publisher

Episode 62 - Tai-Danae Bradley Jan 15, 2021

Show notes

Episode 62 - Tai-Danae Bradley Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Welcome to my favorite theorem, a math podcast. I'm Evelyn Lamb, one of your hosts. And here's your other host. Kevin: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. It's been a while. I haven't seen your smiling face in a while. EL: Yeah. I've started experimenting more with home haircuts. I don't know if you can see. KK: I can. It's a little a little longer on top. EL: Yeah. KK: And it's more of more of a high and tight thing going here. This is Yeah. All right. It looks good. EL: Yeah, it's been kind of fun. And, like, depending on how long ago between washing it, it has different properties. So it's very, it's like materials science over here, too. So a lot of fun. KK: Well, you probably can't tell, but I've gone from a goatee to a plague beard. And also, I've let my hair grow a good bit longer. I mean, now that I'm in my 50s, there's less of it than there used to be. But I am letting it grow longer, you know, because it's winter, right? EL: Oh yeah. Your Florida winter. It's probably like, what? 73 degrees there? KK: It is 66 today. It's chilly. EL: Oh, wow. Yeah, gosh! Well, today we are very happy to invite Tai-Danae Bradley to the podcast. Hi, Tai-Danae. Will you tell us a little bit about yourself? Tai-Danae Bradley: Yeah. Hi, Evelyn. Hi, Kevin. Thank you so much for having me here. So I am currently a postdoc at X. People may be more familiar with his former name, Google X. Prior to that, I recently finished my PhD at the CUNY Graduate Center earlier this year. And I also enjoy writing about math on a website called math3ma. EL: Yes, and the E of that is a 3 if you're trying to spell it. TDB: Yeah, m-a-t-h-3-m-a. That's right. I pronounce it mathema. Some people say math-three-ma, but you know. EL: Yeah, I kind of like saying math-three-ma my head. So, I guess, not to not to sound rude. But what does X want with a category theorist? TDB: Oh, that's a great question. So yeah, first, I might say for all of the real category theorists listening, I may humbly not refer to myself as a category theorist. I'm more of, like, an avid fan of category theory. KK: But you wrote a book! TDB: Yeah, I did. I did. No, I really enjoy category theory, I guess I'll say. So at X, I work on a team of folks who are using ideas from—now this may sound left field—but they're using ideas from physics to tackle problems in machine learning. And when I was in graduate school at CUNY, my research was using ideas in mathematics, including category theory, to sort of tackle similar problems. And so you can see how those could kind of go hand in hand. And so now that I'm at X, I'm really just kind of continuing the same research interest I had, but, you know, in this new environment. EL: Okay, cool. KK: Very cool. EL: Yeah, mostly, we've had academics on the podcast. We’ve had a few people who work in other industries, but it's nice to see what's out there, like, even a very abstract field can get you an applied job somewhere. TDB: Yeah, that's right. EL: Yeah, well, of course, we did invite you here to talk about your job. But we also invited you here to ask what your favorite theorem is. TDB: Okay. Thank you for this question. I'm so excited to talk about this. But I will say, I tend to be very enthusiastic about lots of ideas in mathematics at lots of different times. And so my favorite theorem or result usually depends on the hour of the day. Like, whatever I’m reading at the time, like, this is so awesome! But today, I thought it'd be really fun to talk about the singular value decomposition in linear algebra. KK: Awesome! TDB: Yeah. So I will say, when I was an undergrad, I did not learn about SVD. So I think my undergrad class stopped just before that. And so I had to wait to learn about all of its wonders. So for people who are listening, maybe I could just say it's a fundamental result that says the following, simply put. Any matrix whatsoever can be written as a product of three matrices. And these three matrices have nice properties. Two of them, the ones on the left and the right, are unitary matrices, or orthogonal if your matrix is real. And then the middle matrix is a diagonal matrix. And the terminology is if you look at the columns of the two unitary matrices, these are called the singular vectors of your original matrix. And then the entries of the diagonal matrix, those are called the singular values of that matrix. So unlike something like an eigen decomposition, you don't have to make any assumptions about the matrix you started with. It doesn't have to have some special properties for this to work. It's just a blanket statement. Any matrix can be factored in this way. EL: Yeah, and I, as we were saying, before we started recording, I also did not actually encounter this in any classes. KK: Nor did I. EL: And yeah, it’s something I've heard of, but not never really looked into because I didn't ever do linear algebra, you know, as part of my thesis or something like that. But yeah, okay, so it seems a little surprising that there aren't any extra restrictions on what kind of matrices can do this. So why is that? I don't know if that question is too far from left field. TDB: Maybe that's one of the, you know, many amazing things about SVD is that you don't have to make any assumptions. So number one, in mathematics, we usually say multiplying things is pretty easy, but factorizing is hard. Like, it's hard to factor something. But here in linear algebra, it's like, oh, things are really nice. You just have this matrix, and you get a factorization. That's pretty amazing. I think, maybe to connect why is that—to connect this with maybe something that's more familiar, we could ask, what are those singular vectors? Where do they come from? Or, you know, what's the proof sketch of this? EL: Yeah. TDB: And essentially, what you do is you take your matrix, you multiply it by its transpose. And that thing is going to be this nice real symmetric matrix, and that has eigenvectors. And so the eigenvectors of that matrix are actually the singular vectors of your original one. Now, depending on like, if you multiply them the transpose of the matrix on the left or right, that will determine whether, you know, you get the left or right singular vectors. So, you might think that SVD is, like, second best: “Oh, not every matrix is square, so, we can't talk about eigenvectors, oh, I guess singular vectors will have to do.” But actually, it's like picking up on this nice spectral decomposition theorem that we like. And I think when one looks out into the mathematical/scientific/engineering landscape, and you see SVD sort of popping up all over the place, it's pretty ubiquitous. And so that sort of suggests it’s not a second-class citizen. It's really a first-class result. EL: Yeah. Well, that's funny, because I did, when I was reading it, I was like, “Oh, I guess this is a nice consolation prize for not being an invertible square matrix, is that you can do this thing.” But you're telling me that that was—that’s not a good attitude to have about this? TDB: Well, yeah, I think SVD, I wouldn't think of it as a consolation prize, I think it is quite something really fundamental. You know, if you were to invite linear algebra onto this podcast and ask linear algebra, what its favorite theorem is, just based on the ubiquity and prevalence of SVD in nature, I'd probably bet linear algebra would say singular value decomposition. EL: Yeah, can can we get them next? KK: Can we get linear algebra on? We’ll see. Okay, so I don't know if this question has—it must have an answer. So say your matrix is square in the first place. So you could talk about the eigenvalues, and you do this, I assume the singular values are different from the eigenvalues. So what would be the advantage of choosing the singular values over the eigenvalues, for example? TDB: So I think if your matrix is square, and symmetric, or Hermitian, then the eigenvectors correspond to the singular vectors. KK: Okay, that makes sense. TDB: But, that's a good question, Kevin. And I don't have a good answer that I could confidently go on record with. KK: That’s cool. Sorry. I threw a curveball. TDB: That’s a great question. KK: Because then singular values are important. The way I've always sort of heard it was that they sort of act like eigenvalues in the sense that you can line them up and that the biggest one matters the most. TDB: Exactly, exactly. Right. And in fact, I mean, that sort of goes back to the proof that we were talking about. I was saying, oh, the singular vectors are the eigenvectors of this matrix multiplied by its transpose. And the singular vectors turn out to be the square roots of the eigenvalues of that square matrix that you got. So they're definitely related. KK: Okay. All right. Very cool. So what drew you to this theorem? Why this theorem in particular? TDB: Yeah, why this theorem? So this kind of goes back to what we were talking about earlier. I really like this theorem because it's very parallel to a construction in category theory. KK: Yes. TDB: Maybe people find that very surprising. We're talking about SVD. And all of a sudden, here's this category theory, curveball. EL: Yeah, because I really do feel like linear algebra almost feels like some of the most tangible math., and category theory, to me, feels like some of the least tangible. KK: So wait, wait, are you going to tell us this is the Yoneda lemma for linear algebra? TDB: No. Although that was going to be my other favorite theorem. Okay, so I'm excited to share this with you. I think this is a really nice story. So I'm going to try my best because it can get heavy, but I'm going to try to keep it really light. But I might omit details, but you know, people can maybe look further into this. So to make the connection, and to keep things relatively understandable, let's forget for a second that I even mentioned category theory. So let’s empty our brains of linear algebra and category theory. I just want to think about sets for a second. So let me just give a really simple, simple construction. Suppose we have two sets. Let's say they're finite, for simplicity. And I'll call them a set X and a set Y. And suppose I have a relation between these two sets, so a subset of the cartesian product. And just for simplicity, or fun, let’s think of the elements of the set X as objects. So maybe animals: cat, dog, fish, turtle, blah, blah. And let's also think of elements in the set Y as features or attributes, like, “has four legs,” “is furry,” “eats bugs,” blah, blah, blah. Okay. Now, given any relation—any subset of a Cartesian product of sets—you can always ask the following simple question. Suppose I have a subset of objects. You can ask, “Hey, what are all the features that are common to all of those objects in my subset?” So you can imagine in your subset, you have an object, that object corresponds to a set of features, only the ones possessed by that object. And now just take the intersection over all objects in your subset? That's a totally natural question you could ask. And you can also imagine going in the other direction, and asking you the same question. Suppose you have a subset of features. And you want to know, “Hey, what are all of the objects that share all of those features in that subset I started with?” A totally natural question you could ask anytime you have a relation. Now, this leads to a really interesting construction. Namely, if someone were to give me any subset of objects and any subset of features, you could ask, “Does this pair satisfy the property that these two sets are the answers to those two questions that I asked?” Like, I had my set of objects and, Oh, is this set of features that you gave me only the ones corresponding to this set of objects and vice versa? Pairs of subsets for which the answer is yes, that satisfy that property, they have a special name. They're called formal concepts. So you can imagine like, oh, the concept of, you know, “house pet” is like the set of all {rabbits, cats, dogs}, and, like, the features that they share is “furry,” “sits in your lap,” blah, blah, blah. So this is not a definition I made up, you know, you can go on Wikipedia and look at formal concept analysis. This is part of that. Or you can usually find this in books on lattice theory and order theory. So formal concepts are these nice things you get from a relation between two sets. Now, what in the world does this have to do with linear algebra or category theory, blah, blah, blah? So here's the connection. Probably you can see it already. Anytime you have a relation, that’s basically a matrix. It's a matrix whose entries are 0 and 1. You can imagine a matrix where the rows are indexed by objects and the columns are indexed by your features. And there's a 1 and the little x little y entry if that object has that feature and 0 otherwise. KK: Sure. TDB: And it turns out that these formal concepts that you get are very much like the eigenvectors of that 0-1 matrix multiplied by its transpose. AKA, they're like the singular vectors of your relation. So I'm saying it turns out—so I'm kind of asking you to believe me, and I'm not giving you any reason to see why that should be true—But it's sort of, when you put pen to paper paper and you work out all of the details, you can sort of see this. But I say it's like because if you just do the naive thing, and think of your, your 0-1 matrix as a linear map, like as a linear transformation, you could say, okay, you know, should I view this as a matrix over the reals? Or maybe I want to think of 0 and 1 as you know, the finite field with two elements. But if you try to work out the linear algebra and say, oh, formal concepts are eigenvectors, it doesn't work. And you can sort of see why that is. we started the conversation with sets, not vector spaces. So this formal concept story is not a story about linear algebra, i.e., the conversation is not occurring in the world of linear algebra. And so if you have mappings—you know, from sets of objects to sets of features—the kind of structure you want that to preserve is not linearity, because we started with sets. So we weren't talking about linear algebra. So what is it? It turns out it's a different structure. Maybe for the sake of time, it's not really important what it is, or if you ask me, I'll be happy to tell you. But just knowing there's another kind of structure that you'd like this map to preserve, and under that right sort of context, when you're in the right context, you really do see, oh, wow, these formal concepts are really like eigenvectors or singular vectors in this new context. Now, anytime you have a recipe, or a template, or a context, but you can just sort of substitute out the ingredients for something else, I mean, there's a bet that category theory is involved. And indeed, that's the case. So it turns out that this mapping, this sort of dual mapping from objects to features, and then going back features to objects, that, it turns out, is an example of adjunction in category theory. So there's a way to view sets as categories. And there's a way to view mappings between them as functors. And an adjunction in category theory is like a linear map and its adjoint, or like a matrix and its transpose. So in category theory, an adjunction is — let me say it this way, in linear algebr…

Full show notes at the publisher

Episode 61 - Yoon Ha Lee Dec 10, 2020

Show notes

Episode 61 - Yoon Ha Lee Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Hello, and welcome to My Favorite Theorem, coming at you from the double hurricane part of 2020 today. I mean, I'm not near the Gulf Coast so it's it's not quite as relevant for my life, but that is the portion of the year we are in right now. I am one of your hosts, Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And here's your other host. Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. It's just hot here. But you know, there have been, like, fire tornadoes, right, in California? This is all very on-brand for 2020. This year can’t end soon enough. EL: Yeah, we say that. I feel like I've said that at the end of many previous years, and then it's not great. Yoon Ha Lee: As a science fiction writer, I have to say never assume it's the worst. It can always get worse. EL: Yes. KK: Right, right, right. EL: Yes. And that is our guest, Yoon Ha Lee. So yeah, would you like to introduce yourself, tell us a little bit about yourself, and maybe talk about your writing a little bit, how you got to writing from the degrees that you have in math. YHL: So my name is Yoon Ha Lee. I'm from Houston, and I'm a science fiction and fantasy writer. I actually went to Cornell to get a degree in history, and then I realized that history majors starve on the street. So I switched to math, so that I could have an income and ended up not becoming a mathematician. My best-known books are probably the Machineries of Empire trilogy, which is Ninefox Gambit, Raven Stratagem and Revenant Gun. It's space opera, lots of ships blowing up everywhere. And then a kid's book, Dragon Pearl, which is out from Disney Hyperion in the Rick Riordan Presents series. And that one is also a space opera, because ships blowing up is just fun. EL: Yeah, well, and that's funny. I think I just put together—I had seen the Rick Riordan publishing imprint before, and I just started reading Percy Jackson the other day. And so it's like, oh, that's who that guy is. KK: And I think I might be the only one among us who is old enough to have seen the biggest space opera, Star Wars, in the theater in its first release. YHL: Yeah, my parents let me see it on the television when I was six years old, and I was terrified at the point where Luke gets his hand cut off. KK: That’s Empire. YHL: I think the second one? I forget which movie it was, but he gets his hand cut off and I had nightmares for weeks. And I'm like, Mom and Dad, Why? Why? Why did you think this was an appropriate movie for a six-year-old? And then I got all the storybooks and I wanted the lightsaber and everything, so I guess it worked out. KK: Of course, yeah. Well, my movie story—we’re getting off track, but it's it's a good movie story. So when I was six years old, in 1975, my parents thought it would be a good idea to take me to the drive-in to see Jaws. And I had nightmares for months that there was a shark living under my bed, a huge shart that was going to get me. EL: My husband was born I think right around the time one of them was released. I don't remember which one now. But we were talking with one of his colleagues one time and figured out that on the day he was born, that colleague was going to see that movie, like, the day it came out. KK: I’m going to guess it was I'm gonna guess it was Jedi. I don't know exactly how old you guys are. But that's that's my guess. EL: That sounds right. Yeah, I'm not a big star wars person. But yeah, I guess I've always not been sure, like, “space opera.” The term is something that I feel like I know it when I see it. But I don't really know, like, how to describe it. Is it just—do you feel like a categorization of space opera is, like, ships blowing up? YHL: Ships blowing up, generally bigger, larger-than-life characters, larger-than-life stakes, big galactic civilization types of things. It's basically the Star Wars genre. EL: Yeah KK: It works. EL: yeah. And the Machinery of Empire—the reason that I invited you on here is because I just read Ninefox Gambit a few weeks ago and just thought, you know, this person sure uses a lot of math terms for a novel! So mathematicians might be especially interested in reading this one, it has shenanigans with calendar systems that are based on math and arithmetic and stuff. So yeah, that was fun. So you, in addition to getting a bachelor's degree in math, you got a master's in math education, right? YHL: Yes, at Stanford. And I ended up not using it for very long. I was a teacher for, like, half a year before I left the profession. EL: Okay, and was it just that your writing was taking off and you wanted to do that more? Were there other reasons? YHL: A kid came along. That was the big reason. Yeah. EL: Oh. Yeah. That definitely can take a lot of time. KK: Ah yeah, just a little bit. EL: Well, that's great. So what is your favorite theorem? YHL: My favorite theorem is Cantor's diagonalization proof. And I discovered it actually in high school as a footnote in Roger Penrose’s The Emperor's New Mind. It was really just sort of a sidelight to the extremely complicated and hard-to-follow argument that he was making in that book on the nature of consciousness and quantum physics, which, as a high schooler, you know, it basically went over my head. But I was sitting there staring at this footnote and going “I don't understand this at all.” He said in the footnote that Cantor had proven that the real numbers, the set of real numbers, has a cardinality greater than the set of natural numbers. And of course, I was a high schooler. I hadn't had a lot of math background. So my understanding of these concepts was very, very shaky. But he said if you make a list of, you know—pretend that you have a list of all the real numbers and you put them, you know, 1, 2, 3, 4, you put them in correspondence with the natural numbers, and then you go down diagonally, first digit of the first number, second digit of the second number, third digit of the third number, and so on. And then you shift it by one. So if the numeral in that place is two, it becomes three, if it's nine, it becomes zero, and so on. So you can construct a number that is not on the list, even though your premise is that you have everything on the list. And I think this was the first time that I really understood what a proof by contradiction was. My math teachers had attempted very hard to get this concept into my head. And it just did not go through until I read that proof and meditated upon it. And it's funny, because I spent most of my life as a kid thinking that I hated math. And yet there I was in the library reading books about math, so I guess I didn't hate it as much as I thought I did. EL: Yeah, I was thinking a high schooler reading that Penrose book is definitely—yeah, you had some natural curiosity about math, it sounds like. KK: Yeah, I'm sort of sort of surprised that your high school teachers were trying to teach you proofs by contradiction. That's kind of interesting. I don't remember seeing any of that until I got to university. YHL: I don't know that they got into depth about it. But this was at Seoul Foreign School, which was a private, international school in South Korea. And they tried to make the curriculum more advanced, with mixed results. KK: Sure. It’s worth a shot. EL: Yeah, and this, this is really one of those Greatest Hits. Like if you're putting together the like, record that you're going to send out or something, like, Math’s Greatest Hits with would include this diagonalization argument. It's so appealing. And we've had another guest select that too, Adriana Salerno a few months ago and yeah, just people. I think a lot of people who eventually do become mathematicians, this is one of those first moments where they feel like they really understand some some pretty high-concept math kind of stuff. So did you see this this proof later in school? YHL: No. Ironically, most of what I was interested in doing when I did my undergraduate degree was abstract algebra. So I didn't even take a set theory course at all. But I knew it was sort of out there in the water, and I don't know, one of the things I loved about math and that led me to switch my major to math was the idea that there were these beautiful ideas and these beautiful arguments, and just sort of the elegance of it, which was very different from history, where—I love history, and I love all the battles and things, like the defenestration of Prague and all the exciting things happening. But you can't really prove things in history. Like you can't go back and run the siege of Stalingrad again, and see what happens differently. KK: Maybe we could though, right? We have the computing power now. Maybe we could do that. This sounds like your next novel, right? So simulation of Stalingrad, and this time, the Nazis win or something? I don't know. YHL: Oh no. I mean, science fiction writers totally do that. There's this whole strand of alternate history, science fiction or fantasy. Harry Turtledove is one author who, he likes to have the story where aliens invade during World War II and then the Nazis and the allies have to have to team up against the aliens kind of stories there. There is a set there is a readership for these things. Sure. EL: So you use a lot of math concepts in your writing, your fiction writing. So have you ever tried to work in diagonalization, or this kind of idea, into any of your stories? YHL: This one? No. I mean, occasionally, I remember writing a story in college, actually, called Counting the Shapes. And it was just everything in the kitchen sink, because I was taking point-set topology, and so I used it as a metaphor for a kind of magic that worked that way, and other ideas, like, I don't know, I had recently read James Gleick’s Chaos. So I was really interested in chaos theory and fractals. And I don't know that I was super systematic about it, and I sort of suspect that a real mathematician would look at it and poke holes. You know, I'm using this as a magic system, not as rigorous math, more as a metaphor, I guess, or flavor. KK: Oh, but I mean, writers do that all the time, right? So I, I taught math and lit class with a friend of mine in languages a few years ago. And, you know, Borges, for example, you know, this sort of stuff is all over his work, these ideas of infinity and, and it's even embedded in Kafka and all this stuff, and it can be a wonderful way to to get your readers to think about something from a point of view they might not have thought of before. YHL: Well, the interesting thing about Ninefox Gambit and the math terminology that I used for flavor is that 20 publishers turned the book down because they said it had too much math. And I my joke about this is that they saw the word diagonalization in the linear algebra matrix context, and they didn't know what that meant, and they ran away from it. Which was extremely discouraging when my agent at the time, Jennifer Jackson, and I were going out on submission with this book. And it's like, it's basically a space opera adventure where people blow each other up. You don't have to worry about the occasional math term. It's just there as flavor for the magic system. But a lot of people—I’m sure you have encountered the fact that a lot of people in the US have math phobia, and this really does affect the readership as well. KK: Really? EL: Yeah, that’s funny, because in some way, I mean, you definitely use the the math language to give a certain flavor to the system that this universe is in, but you could sub it out for, like, any Star Trek term, YHL: Exactly. EL: t’s just like, oh, yeah, you could put tricorders and dilithium crystals, or, you know, anything in to serve that that because you know, you're it's not a math textbook, no one's learning linear algebra from reading Ninefox Gambit. YHL: No, exactly. I actually, when I was originally writing the book, like the rough draft, I had my abstract algebra textbooks out and ready to go. And I was going to construct sort of a game engine, a combat engine of how these battles were going to work in an abstract algebra sense. And my husband who, he's not afraid of math, he's actually a gravitational astrophysicist, and he's arguably better at math than I am. But he sat me down and said, “Yoon Ha, you can't do this. You're not going to have any readers because science fiction readers who want to read about big spaceships blowing each other up do not want to have to wade through a math textbook to get to the action.” And I mean, it turned out that he was absolutely correct. So I ended up not doing that and just using it as, you know, “the force,” except with math flavor. KK: Linear algebra is the force. All right! EL: That’s so interesting. I noticed on your website that you have a section for games. So do you also like to design games? YHL: I do design games. And by design games, I mean tiny little interactive, interactive fiction text adventures or really small tabletop RPGs in the indy sense. You know, three page games for five people, no GM, that kind of thing. So I do enjoy doing that. And it is related to math, I think, but it's certainly not something that we learn to do in any of our math classes. EL: Yeah, well, I mean, personally, I think it would be very cool. Have you have you written up this potential game, the abstract algebra game thing into an actual game? Or was that kind of abandoned on the editing floor while you were putting the book together? YHL: It got abandoned on the editing floor. Also because it would have been a tremendous time suck. And, you know, it would have been a fun idea. But if I wasn't going to use it in a book, and it certainly wasn't going to be used in like a computer game or some something like that, there just didn't seem to be enough incentive to go ahead and do it. EL: Yeah, probably the market of math mathematicians who read sci fi is, you know, not a tiny market but maybe not quite the demographic you're looking for. But I'm just imagining, like, hauling out the Sylow theorems to, like, explode someone’s battle cruiser or something. Just saying that, you know, if you were bored some time and wanted to sink a bunch of time into that. YHL: if somebody else wrote it, I would definitely buy it and read it, I have to say. KK: All right. The challenge is out there, everybody. Everybody should get on this. EL: Yeah, very cool. Yep. KK: So another thing we do on this podcast is we ask our guests to pair their theorem with something. So what pairs well, with Cantor's diagonalization argument? YHL: Waffles. KK: Waffles? Oh, well, yeah. YHL: Because sort of that grid shape. I know, this is super visual. But the waffles I'm thinking of, my husband did his postdoc at Caltech, so we lived in Pasadena. And when we were there, there was this delightful Colombian hotdog place. And they also made the best waffles with berries and fruit and syrup and whipped cream. And those are the waffles I think of when I think of the diagonal slash proof. KK: Right. And so the grid is actually fairly small. Is it one of those waffle makers? YHL: Yeah. KK: Yeah. Okay, so I have a Belgian waffle maker, and it's fine. It makes four at a time, but those holes are pretty big. Right? I'm thinking of, like, the small, Eggo style, right? You can put a lot of digits. EL: You could also, like, I guess, maybe a berry is too big to fit in them, but I'm just thinking you can put different things in a…

Full show notes at the publisher

Episode 60 - Michael Barany Nov 12, 2020

Show notes

Episode 60 - Michael Barany Kevin Knudson & Evelyn Lamb Download Kevin Knudson: Welcome to My Favorite Theorem, a math podcast for your quarantine life. I'm Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host. Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer in beautiful Salt Lake City, Utah. KK: Yeah. EL: How are you, Kevin? KK: I'm okay. I had my—speaking of quarantines, I had my COVID swab test this morning. EL: How was it? KK: Well, you know, about as pleasant as it sounds. But yeah, I'm sure you've been to the pool and gotten water up your nose. That's what it feels like. EL: Yeah. KK: And then it's over. And it's no big deal. I should have the results within 48 hours. It’s part of the university's move to get everybody back to campus, although I don't expect to go back to the office in any serious way before August. But this is late May now for our listeners, who will probably be hearing this in December or something, right? EL: Yeah. Who even knows? Time has no meaning. KK: Hopefully this will all be irrelevant by the time our listeners hear this. [Editor’s note: lolsob.] We'll we'll have a vaccine and everything. It will be a brave new world and everything be fine. EL: It’ll be a memory of that weird time early in year. KK: That’s right. The before times. So anyway, today, we are pleased to welcome Michael Barany. Michael, why don’t you introduce yourself and let us know who you are and what's up. Michael Barany: Hi. So I'm a historian of mathematics. I'm super excited to be on this podcast. I feel like I've been listening long enough that the Gainesville percussionists must be in grad school by now. KK: No. One of them is my son, and he just finished his third year of college. MB: Okay, yeah. So older than he was anyway. EL: Yeah. MB: Yeah, so I’m a historian of mathematics. I'm based at the University of Edinburgh, where I'm in a kind of interdisciplinary social science of science and technology department. So I get to teach students from all over the university how to think about what science means when you step back and look at the people involved and how they relate to society, how ideas matter, how technology's changed the world, all that fun stuff that gets people to really rethink their place in the world and the kind of things they do with their science. KK: That’s very cool. EL: And I know some people who are historians of math will get a degree through a math department and some get it through a history department, I assume. And which are you? I always wonder what the benefits are of each approach. MB: Yeah, that's great. History of mathematics is a really strange field. It’s actually, as a field, a lot older than history of science as a field, and even older than history as a profession. EL: Huh. MB: So history of mathematics started as a branch of mathematics in the early modern period. So we're talking like the 1500s, 1600s. There are always debates about what you classify as this or that. And it started as a way of trying to understand how mathematical theories came about, how they naturally fit together. The idea was that if you understood how mathematical theories emerged, you could come up with better mathematical theories, and you could understand the sort of natural order of numbers and the universe and everything else that you want to understand with mathematics. And then more toward the 19th and the 20th century, there are all these different variations of history of mathematics that branched out of fields like history and philosophy, and philosophy of science and history of science. So my undergrad training was in mathematics. My PhD is from a history department, but from a history of science program in that department. But it's possible to get a PhD in history of mathematics from a mathematics department, it's possible to sort of straddle between different departments. And it makes it a really rich and interesting field. Mathematics education departments or groups sometimes give PhDs in history of mathematics. And they really use the history for different purposes. So if your goal is to make mathematics better, you're taking the perspective of someone doing it from a mathematics department. If your goal is to become a better educator, then you can use history for that in a math education context. I tend to do history as a way of understanding how things fit together in the past and trying to make sense of social values and social structures and ideologies and ideas and how those fit together. And that's the approach that that you come at from a history or history of science perspective. KK: Very cool. And How did you end up in Edinburgh of all places? MB: Well, so the academic job market is bad enough in mathematics, right, but in history of mathematics, in a good year, there may be two to three openings in history of science jobs in general. So that's the cynical answer. The more idealistic answer is Edinburgh has this really important place in the sociology of science. In the 1970 s and 80s especially, there was this group of kind of radical sociologists in at the University of Edinburgh who sat down. It was called the Edinburgh School of the sociology of scientific knowledge, which is known for this sort of extreme relativism and constructivism view of how politics and ideology shape scientific knowledge. And I did a master's degree in that department many years later, in 2009-2010, sort of getting my feet wet and starting to learn that discipline. And that approach has been really formative for me and my scholarship. And so it was an incredible stroke of luck that they just happened to have an opening in my field while I was on the market. And I was even even more lucky to have the chance to go there. KK: Wow, that's great. I’ve always wanted to go there. I've never been to Edinburgh, MB: It’s the most beautiful city in the world. KK: Yeah, it looks great. All right, well, being a historian of math, you must know a lot of theorems. So the question is, do you actually have a favorite one? And if so, what is it? MB: So my favorite theorem is more of a definition. But I guess the theorem is that the definition works. KK: Okay, great. EL: That works. MB: Which, actually—saying what it means for a definition to work is actually a really hard problem, both historically and mathematically. So it's interesting in that regard. Ao the definition is the definition of the derivative of a distribution. KK: Okay. MB: So distributions, as you’ll recall from, from analysis—I guess, grad analysis I is usually when you meet them. EL: Yeah, I think it wasn't until grad school for me at least. KK: I don't know if I've ever met them, really. MB: So distributions were invented in 1945, more or less. And in the early years, actually, people were saying you could teach this as a replacement for your basic calculus. So the idea was, this would be something that even beginning college students or even high school students would be learning. So it's interesting to see how they have people pitched that the level of a theory or the the relevant audience, and that's part of the story, too. But in earlier stages of one's calculus education, you learn that there are functions that are integrable but not continuous; continuous but not differentiable; differentiable but not continuously differentiable, and so on. And so a big problem is how do you know something's differentiable when you're studying a differential equation or trying to prove some theorem that involves derivatives. And distributions were the kind of magic wand that was invented in the middle of the 20th century to say that's not actually a problem. Basically, if you pretend everything's differentiable, then all the math works out. And when it really is differentiable, you get the correct differentiable answer, and when it's not, then you get another answer that's still mathematically meaningful. But it's sort of your magic passphrase to be able to ignore all of those problems. So a distribution is this replacement for a function. Where functions have these sort of different degrees of differentiability, distributions are always differentiable and they always have antiderivatives, just like functions do, but every distribution can be differentiated ad nauseam for whatever differential equation you want to do. And the way you do that is through this definition—my favorite definition/theorem—which is you use integration by parts. So that's a technique you use in calculus class, too, as a sort of trick for resolving complicated integrals. And distributions actually don't tend to look at the things that make the calculus problems challenging or interesting, depending on what kind of student you are, or what kind of teacher you are. So you set them up in a way where you don't have to worry about boundary conditions, you don't have to worry about what the antiderivative things are, because you're working with things where you already know what the antiderivative is. And the definition of distribution uses this fact from integration by parts that you essentially move the derivative from one function to another. So we don't have an exact way of saying functionally what the derivative of a distribution is. You can still say if you multiply it by a function that's super-smooth and over a bounded domain—so you don't have any boundary conditions to worry about, and so you always know how to differentiate that—if you multiply that by a distribution, and take the integral, then if you want to take the derivative of that distribution, integration by parts says you can instead throw in a minus sign and take the derivative of that smooth function instead. And so using that kind of trick, of moving the derivative onto something that is always differentiable, you can calculate the effect of differentiating a distribution without ever having to worry about, say, what the values of of that distribution are after you’ve taken the derivative, because distributions are often things that don't have sort of concrete values in the way that we expect functions to have. EL: And I hope this question isn't very silly. But when you think about integration by parts—you know, if you took calculus at some point and learned this, there's the UV, and then there's the minus the integral of something else. And so for this, we just choose a function that would be zero on the boundary, and that would get rid of that UV term. Is that right? MB: Exactly? Yeah. So the definition of distribution sets up this whole space of really nice smooth functions. All of them eventually go to zero, and because you're always integrating over the entire domain, and it's always zero when you go far enough out into the domain, those boundary terms with that UV in the beginning just completely disappear, and you're just left with the negative integral, and then with the derivative flopped over. EL: All right, great. So if anyone was worried about where their UV went, that's where it went. It was zero. Don't worry. Everything's okay. Yeah. Okay. So what is good about this? Or what do you like about this? MB: Yeah. So I think this is a really interesting definition from a lot of different perspectives. One thing that I've been trying to understand in my research about the history of mathematics is what it means for mathematics to become a global discipline in the 20th century, so to have people around the world working on the same mathematical theory and contributing to the same research program. And this definition is really helped me understand what that even means and how to understand and analyze that historically. So we think, well, you know, a mathematical theory or a mathematical idea is the same wherever you look at it, and whoever's doing it. As long as they can manipulate the definitions or prove the theorem, it shouldn't matter where they are. But if you look historically, at actual mathematicians doing actual mathematics, where they are makes a huge difference in terms of what methods they're comfortable with, how they understand concepts, how they explain things to each other, how they make sense of new techniques. I mean, learning a new mathematics technique is actually really hard in a lot of cases. And so the question is, how do you form enough of an understanding to be able to work with someone who you can't go and have a conversation with over tea the next day to sort of work out your problems? And the answer is, basically, you use things like this definition and take something you're really comfortable with—integration by parts—and give it a new meaning. And by taking old meanings and reconfiguring them and relating them to other meanings, you make it possible for everyone to have their own sorts of mathematical universes where they're building up theories, but to interact in a way where they can all sensibly talk to each other and develop new ideas and share new ideas. So that's one of the things that that's really exciting about that the definition to me. One of the other things is sort of how do you know what the significance of the definition is? I mean, a lot of people early on said, isn't this just like a pun? Isn't this just wordplay? Quite early on, when Schwartz was sharing this definition, and some people were getting really excited about it. Some people said, well, you know, it's a cool idea. But isn't this just basically integration by parts? What's new? What's interesting about this? And the history really shows this debate, almost, between people with different kinds of values and philosophies and goals for mathematics, for mathematics education, for the relationship between pure and applied mathematics, where they take different ideas of what's really going on with this definition. Is it something that's complex and difficult and profound and important in that way, or is it something that is utterly trivial and simple, and therefore really useful to people who may be, say, electrical engineers who are trying to work with the Heaviside calculus, and need some sort of magic way to make that all add up? And what made distributions and this definition really powerful is it could be these multiple things to multiple people. So you can have mathematicians in Poland, or in Manchester, or in or in Argentina come to these very, almost diametrically opposed views of what it is that's significant or challenging or easy about distributions, and they can all agree to talk to each other and agree that it's worth sharing their theories and inviting them to conferences, and reading their publications, and they can somehow all make a community out of these different understandings. KK: I’ve never thought about the sociological aspects in that way. That's really interesting. So the theorem that basically says that this definition is a good one. Is that a difficult theorem to prove? MB: So there are a lot of different parts. It’s not—I guess it doesn't even boil down to one statement. KK: Yeah, sure. Yeah, that makes sense. Yeah. MB: So there's the aspect that when you're dealing with a function, but dealing with it using the distributions definition, that anything you do is not going to ruin what's good about it being a function. So anything you do with a distribution, if you could have done it as though it were a regular function, you get the same answer. So that's one aspect of the theorem that sort of establishes this definition. Another aspect is that distributions ar…

Full show notes at the publisher

Episode 59 - Daniel Litt Oct 08, 2020

Show notes

Episode 59 - Daniel Litt Kevin Knudson & Evelyn Lamb Download Kevin Knudson: Welcome to My Favorite Theorem, a math podcast and so much more. I'm Kevin Knudson, professor of mathematics at the University of Florida. Here is your other host. Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a math and science writer in Salt Lake City, Utah. I have left the county two times since this all happened. We don't have a car, so when I leave my home, it is either on feet or bicycle, which is your feet moving in a different way. But I have biked out of our county now into two different other counties. So it's very exciting. KK: Fantastic. Well, I do have a car. I bought gas yesterday for the first time since May 26, I think. And yesterday was June 30. EL: Yes. KK: And I've gotten two haircuts, but it looks like you've gotten none. EL: Yes. That’s correct. I’m probably the shaggiest. I've been in a while. My I normally this time of year is buzzcut city, which I do at home anyway. But I don't know. KK: I will say I’m letting it get a little longer actually. I know I said I got a haircut, but you know, Ellen likes it longer somehow. So here we go. This is where we are. My son's been home for three months, and we haven't killed each other. It's all right. EL: Great. Yeah, everything's doing as well as can be expected, I suppose. If you're listening to this in the future, and somehow, everything is under control by the time we publish this, which seems unlikely, we are recording this during the 2020 COVID-19 pandemic, right, which—I guess it still stays COVID-19 even though it's 2020 now, to represent the way time has not moved forward. KK: Right. Time has no meaning. And you know, Florida now is of course becoming a real hotspot, and cases are spiking. And I'm just staying home and, and I have four brands of gin, so I'm okay. EL: Yeah. Anyway! KK: Anyway, let's talk math. So we're pleased today to welcome Daniel Litt. Daniel, would you please introduce yourself? Daniel Litt: Hey, thank you so much. It's really nice to be here. I'm Daniel Litt. I'm an assistant professor at the University of Georgia in Athens, Georgia, likewise, a COVID-19 hotspot. I also have not gotten gas, but I think I've beat your record, Kevin. I haven't gotten gas since the pandemic began. KK: Wow. That’s pretty remarkable. DL: I’ve driven, maybe the farthest away I've driven from home is about a 15-minute drive, but those are few and far between. KK: Sure. DL: So yeah, I'm really excited to be here and talk about math with both of you. KK: Cool. All right. So I mean, this podcast is—actually, let’s talk about you first. So you just moved to Athens, correct? DL: I started a year ago. KK: A year ago, okay. But you just bought your house. DL: That’s right. Yeah. So I actually live in northeast Atlanta, because my wife works at the CDC, which is a pretty cool place to work right now. KK: Oh! EL: Oh wow. KK: All right. Is she an epidemiologist? DL: She does evaluation science, so at least part of what she was doing was seeing how the CDC’s interventions and deployers, how effective they were being help them to understand that. KK: Very cool. Well, now it would be an interesting time to work there. I'm sure it's always interesting, but especially now. Yeah. All right. Cool. All right. So this podcast is called my favorite theorem. And you've told us what it is, but we can't wait for you to tell our listeners. So what is your favorite theorem? DL: Yeah, so my favorite theorem is Dirichlet’s theorem on primes in arithmetic progressions. So maybe let me explain what that says. KK: Please do. EL: Yes, that would be great. DL: Yeah. So a prime number is a positive integer, like 1, 2, 3, 4, etc, which is only divisible by one and by itself. So 2 is a prime, 3 is a prime, 5 is a prime, 7, 11, etc. Twelve is not a prime because it's 3 times 4. And part of what Dirichlet’s theorem on primes in arithmetic progressions tries to answer, part of the question it answered, is how are primes distributed? So there is a general principle of mathematics that says that if you have a bunch of objects, they're usually distributed in as random a way as possible. And Dirichlet’s theorem is one way of capturing that for primes. So it says if you look at an arithmetic progressions—that’s, like 2, 5, 8, 11, 14, etc. So there I started at 2 and I increased by 3 every time. Another example would be 3, 6,9, 12, 15, etc—there I started at 3 and increased by 3 every time. So Dirichlet’s theorem says that if you have one of those arithmetic progressions, and it's possible for infinitely many primes to show up in it, then they do. So let me give you an example. So for 3, 6, 9, 12, etc, all of those numbers are divisible by 3. So it's only possible for one prime to show up there, namely 3. EL: Right. DL: But if you have an arithmetic progression, so a bunch of numbers which differ by all the same amount, and they're not all divisible by some single number, then Dirichlet’s theorem tells you that there are infinitely many primes in that sequence. So for example, in the sequence 2, 5, 8, 11, etc, there are infinitely many primes, 5 and 11 being the first two [editor’s note: the first primes after 2. But it’s just odd for an even number to be prime]. And it tells you something about the distribution of those primes, which maybe I won't get into, but just their bare existence is really an amazing theorem and incredible feat of mathematics. EL: So this theorem, I guess, for some of our listeners, and for me, it probably sort of reminds them in some ways of like twin primes or something, these other questions about distributions of primes. Of course, twin primes, you don't need a whole arithmetic progression, you just need two of them. That would be primes that are separated by two, which other than 2 and 3 is the smallest gap that primes can have. And, of course, twin primes is not solved yet. DL: Yeah, we don’t know that there are infinitely many. EL: Yeah, people think there are but you know, who knows? We might have found the last one already. I guess that's unlikely. But Dirichlet was proved a long time ago. So can you give me a sense for why this is a lot easier than twin primes? DL: Yeah, so part of the reason, I think, is that twin primes are much sparser than primes in any given arithmetic progression. So just to give you an example, if you have a bunch of numbers, one way of measuring how big they are is you could take the sum of 1 over those numbers. So for example, the sum of 1/n, where n ranges over all positive integers, diverges; that sum goes to infinity. And the same is actually true for the primes in any fixed arithmetic progression. So if you take all the primes in the sequence 2, 5, 8, 11, etc, and take the sum of one over them, that goes to infinity, since there's a lot of them. On the other hand, we know that if you do the same thing for twin primes, that sum converges to a finite number. And that number is pretty small, actually. We know, up to quite a lot of accuracy, what it looks like. And that already tells you that they're sort of hard to find. And if you have things that are hard to find, it's going to be harder to show that there are infinitely many of them. I mention this sum of reciprocals point of view because it's actually crucial to the way Dirichlet’s theorem is proven. So when you prove Dirichlet’s theorem, it's one of the these really amazing examples where you have a theorem that's about pure algebra. And you end up proving it using analysis. So in this case, the theory of Dirichlet L-functions. And understanding that sum of reciprocals is kind of key to understanding the analytic behavior of some of these L-functions, or at least it’s very closely related. KK: So I didn't know that result about the reciprocals of the twin primes converging. So even though we don't know that there are infinitely many, somehow… DL: Yeah, in fact, if there are finitely many then definitely that sum would converge, right? KK: Yeah, right. That’s—and we even know an estimate of what the answer is? Okay. That’s fascinating. DL: Yeah, and what you have to do to prove that is show that these primes are sufficiently sparse. And then and then you win. EL: So once again, I am super not a number theorist. So I'm just going to bumble my way in here. But to me, if I'm trying to show that something diverges, I show that it's sort of like 1/n, and if it converges, it's sort of like 1/n2 or, or worse, or better, or however, you want to morally rank these things. So I guess I could imagine it not being that hard to show that twin primes are sort of bounded by n2, or you're like bounded by 1/n2 squared, the reciprocals of that, would that be a way to do this? Or am I totally off? DL: It’s something like that. You want to show they're very spread out. Yeah, with primes, I do want to mention, so you mentioned like you want to say something like between 1/n or 1/n2. So primes are much, much rarer than integers, right? So it's really somewhere between those two. EL: Yeah. DL: So for example, understanding the growth rate of those numbers—the growth rate of the primes and the growth rate of the primes in a given arithmetic progression—is pretty hard. Like that's the prime number theorem, it’s one of the biggest accomplishments of 19th-century mathematics. KK: Right. Does that help you prove that, though? Maybe it does, right? Maybe not? DL: Yeah, so proving that the sum of the reciprocals of the primes diverges is much, much easier than the prime number theorem. And as you can prove that in, like, a page or page and a half or something. But it's very closely related to the key input of the prime number theorem, which is that the Riemann zeta function, the subject of the Riemann hypothesis, has a pole at s=1. KK: All right. Okay. So what's so compelling about this theorem for you? DL: Yeah, so what I love about it is that it's maybe one of the earliest places, aside from the prime number theorem itself, where you see some really deep interactions between algebra and complex analysis. So the tools you bring in are these Dirichlet L-functions, which are kind of generalizations of the Riemann zeta function. And they're really mysterious and awesome objects. But for me, what I find really exciting about it is that it's like the classic oldie. And people have been kind of remaking it over and over again for the last, like, century. So there's now tons of different versions of the Dirichlet theorem on primes in arithmetic progressions in all kinds of different settings. So here's an example. In geometry, you have a Riemannian manifold, which is kind of a manifold with a notion of distance on it. There's a version of Dirichlet’s theorem for loops in a Riemannian manifold, the first cases of which are maybe do that Peter Sarnak in his thesis. There are versions for over function fields. So I'm not going to be precise about what that means, but if you have some kind of geometric object that's kind of like the integers, you can understand it well and understand the behavior of primes and that kind of object, and how they behave in something analogous to an arithmetic progression. There's something called the Chebotarev density theorem, which tells you if you have a polynomial, and you take the remainder of that polynomial when you divide by a prime, how does its factorization behave as you vary the prime? So there's all kinds of versions of it, and it's a really exciting and cool sort of theme in mathematics. EL: So kind of getting back to the the more tangible number theory thing—which I guess it's kind of funny that we think of numbers as more tangible when they're sort of the first example of an incredibly abstract concept. But anyway, we'll pretend numbers are tangible. So how does this relate, I remember, and I don't even remember now, I must have been writing some article that related to this, but looking at your primes that are your 1 more than a multiple of 6 versus 1 less and looking at whether there are more or fewer of these. So these are two different arithmetic progressions. The one that's like, you know, 7, 13, let's see if I can add by 6, 19, this, that progression, versus the 5, 11, etc, progression. So is this related to looking at whether there are more of the ones that are one more one less or things like that? DL: For sure. EL: I feel like there are all these interesting results about these biases and the distributions. DL: Yeah, so people call this prime number races. EL: Yeah. DL: So what you might do is you might take two different arithmetic progressions and ask are there more prime numbers, like, less than a billion, say, in one of those progressions as opposed to the other? And there are actually pretty surprising properties of those races that I think are not totally well understood. So like even even this recent work of Kannan Soundararajan and Robert Lemke Oliver on this kind of thing. EL: Oh, yeah, that’s what I was writing about! DL: Which, yeah, shows some sort of surprising biases. And so that's the reason people think those are cool, is exactly this principle I mentioned before, this general principle of math that things should be as random as they can be. And there are maybe some ways in which our random models of the primes are not always totally accurate. And so understanding the ways in which they're inaccurate and how to fix that inaccuracy, like how to come up with a better model of the primes, is a really big part of modern number theory. EL: But I guess, the Dirichlet theorem is what you need before you start looking at any of these other things, is you need to know that you can even look at these sequences. DL: Right. Exactly. Yeah. I mean, how do you study the statistics of a sequence you don't know is infinite? Yeah. EL: Right. DL: One thing I’ll mentioned, one cool thing about it is it lets you—it’s not just an abstract existence result. Like, sometimes you just need a prime which is, like, 7 mod 23 to do some mathematical computation. Okay, and if it's 7 mod 23, then it's pretty easy to find one. You can take 7. But if you need a prime, that's a mod b, its remainder upon division by b is a, it's sort of hard to make one in general. And the fact that Dirichlet’s theorem gives them to you is actually really useful. So at least for a mathematician who cares about primes, it's something that just comes up a lot in daily life. KK: But it's not constructive, though. DL: Yeah, that's, that's right. It does kind of guarantee that there will be one less than some explicit constant, so in some sense, it's constructive, but it doesn’t, like, hand one to you. EL: But still, I guess a lot of the time, you probably don't actually need a particular one. You just kind of need to know that there is one. DL: Yeah. EL: And where did you first encounter this theorem? DL: I guess it was, I was probably reading Apostol’s number theory book when I was in college. But I think for me, I didn't really grok it until some other more modern version of it, like one of these remakes showed up for me in my own work. So I wanted to make a certain construction of algebraic curves. So that's some kind of geometric objects defined by some polynomial equations, which have some special properties. And it turned out that for me, the easiest way to do that was to use some version of Dirichlet’s theorem in some kind of geometric context. KK: Very cool. DL: So that was really exciting. KK: Yeah. Well, it's it's nice when, like you say, when the oldies…

Full show notes at the publisher

Episode 58 - Susan D'Agostino Sep 10, 2020

Show notes

Episode 58 - Susan D'Agostino Kevin Knudson & Evelyn Lamb Download Kevin Knudson: Welcome to My Favorite Theorem, a podcast about math and so much more. I'm one of your hosts, Kevin Knudson, professor of mathematics at University of Florida. And here is your other host. Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a math and science writer in Salt Lake City, Utah. So how are you, Kevin? KK: I’m fine. It's it's stay at home time. You know, my wife and son are here and we're sheltered against the coronavirus, and we've not really had any fights or anything. It's been okay. EL: That’s great! KK: Yeah, we're pretty good at ignoring each other. So that's pretty good. How about you guys? EL: Yeah, an essential skill. Oh, things are good. I was just texting with a friend today about how to do an Easter egg hunt for a cat. So I think everyone is staying, you know, really mentally alert right now. KK: Yeah. EL: She’s thinking about putting bonito flakes in the little eggs and putting them out in the yard. KK: That’s a brilliant idea. I mean, we were walking the dog earlier, and I was lamenting how I just sort of feel like I'm drifting and not doing anything. But then, you know, I've cooked a lot, and I'm still working. It's just sort of weird. You know, it's just very. EL: Yeah, time has no meaning. KK: Yeah, it's it's been March for weeks, at least. I saw something on Twitter, Somebody said, “How is tomorrow finally March 30,000th?” EL: Yeah. KK: That’s exactly what it feels like. Anyway, today, we are pleased to welcome Susan D'Agostino to our show. Susan, why don't you introduce yourself? Susan D’Agostino: Hi. Thanks so much for having me. I really appreciate being here. I’m a great fan of your show. So yeah, I'm Susan D’Agostino. I'm a writer and a mathematician. I have a forthcoming book, How to Free Your Inner Mathematician, which is coming out from Oxford University Press. Actually, it was just released in the UK last week and the US release will be in late May. And otherwise, I write for publications like Quanta, Scientific American, Financial Times, and others. And I'm currently working on an MA in science writing at Johns Hopkins University. KK: Yeah, that's pretty cool. In fact, I pre-ordered your book. During the Joint Meetings, I think you tweeted out a discount code. So I took advantage of that. SD: Yes. And actually, that discount code is still in effect, and it's on my website, which I'll mention later. EL: Great. So you said you're at Hopkins, but you actually live in New Hampshire? SD: Exactly. Yes. I'm just pursuing the program part-time, and it's a low-residency program. So I’m a full-time writer, and then just one class a semester. It creates community, and it's a great way to meet other mathematicians and scientists who are interested in writing about the subject for the general public. EL: Nice. I went to Maine for the first time when I was living in Providence last semester and drove through New Hampshire, which I don't think is actually my first time in New Hampshire, but might have been. We did stop at one of the liquor stores there off the highway, which seems like a big thing in New Hampshire because I guess they don't have sales tax. SD: No sales tax, no income tax, “Live Free or Die.” Yeah, and you probably test right around where I live because I live in New Hampshire has a very short seacoast, about 18 miles, depending on how you measure it. We live right on the seacoast. EL: Oh yeah, we did pass right there. Wonderful. Yeah, the coast is very beautiful out there. SD: I love it. Absolutely love it. I'm feeling very lucky because there's lots of room to oo outside these days. So, yeah, just taking walks every day. EL: Wonderful. KK: So you used to be a math professor, correct? SD: Yes. KK: And you just decided that wasn't for you anymore? SD: Yeah, well, you know, life is short. There's a lot to do. And I love teaching. I had tenure and everything. And I did it for a decade. And then I thought, “You know, if I don't write the books I have in mind soon, then maybe they won't get done.” I've got my first one out already, only two years into this career pivot to writing, and I’m working on my next one. And I always had in mind, in fact, I have a PhD, but I also have an MFA. So I have a terminal degrees both in math and writing. And I always had one foot in the math world and one foot in the writing world, and I realized I didn't want to only live in one. So this is my effort to live fully in both worlds. KK: That’s awesome. EL: Yeah. Nice. So the big question we have now of course, is what is your favorite theorem? SD: Okay, great. My favorite theorem is the Jordan curve theorem. KK: Nice. SD: Yeah. It’s a statement about simple closed curves in a 2-d space. So before I talk about what the Jordan curve theorem is, let's just make sure we're abundantly clear about what a simple closed curve is. EL: Yes. SD: So, a curve—you can think about it as just a line you might draw on a piece of paper. It has a start point, it has an end point. It could be straight, it could be bent, it could be wiggly, it could intersect itself or not. The starting point and the end point may be different or not. And because this is audio, I thought maybe we could think about capital letters in a very simple font like Helvetica, or Arial. So for example, the capital letter O is a is a curve. When you draw it, it has a start point and an end point that are the same. The capital letter C is also a curve. That one has a different starting and end point, but that's okay. It satisfies our definition. Capital letter P also. That one intersects itself in the middle, but it's still it's a curve. Okay, so a simple curve is a curve that doesn't intersect itself along the way. It may or may not have the same starting and end point, but it won't intersect itself along the way. So capital letter O and capital letter C are both simple. But for example, the capital letter B is not simple, because if you were to start at the bottom, go up in a vertical line, draw that first upper loop and then the second upper loop, between the first and second upper bubbles of the B, you will hit that initial vertical line that you drew. So it's not simple because it touches itself along the way. And a closed curve is a curve that starts and ends at the same point. So the letter O is closed, but the letter C is not because that one starts in one place ends in another. KK: Right. SD: Moving forward as we talk about the Jordan curve theorem, let's just keep in mind two great examples of simple closed curves: the letter O, and even the capital letter D. It's fine that that D has some angles, in the bottom left and upper left. So corners are fine, but it needs to start and end in the same place and doesn't intersect itself other than where it starts and ends. Okay, so the Jordan curves theorem tells us that every simple closed curve in the plane separates the plane into an inside and an outside. So a plane, you might just think of as a piece of paper, you know, an 8 1/2 by 11 piece of paper, let's draw the letter O on it. And when you draw that letter O, you are separating that piece of paper, the surface, into a region that you might call inside the letter O and another region that you might call outside the letter O. And the second part of the Jordan curve theorem tells you that the boundary between this inside and that outside formed by this letter O is actually the curve itself. So if you're standing inside the O, and you want to get to the outside of the O, you've got across that letter O, which is the curve. Okay, so that doesn’t sound very profound. KK: It’s obvious. It’s just completely obvious. EL: Any of us who are big doodlers—like, when I was a kid, at church, I was always doodling inside the letters in the church bulletin. That’s the thing. I know that there's an inside and outside to the letter O. SD: You do. Yes. And you could ask your kid brother, kid, sister, whoever. Anyone—you probably didn't need a big mathematical theorem to assure you of this somewhat obvious statement when it comes to the letter O. Okay, so, I do want to tell you why I think it's really interesting beyond this fact that it seems obvious. But before I do, I just want to make two quick notes. And one is that you really do need the simple part, and you really do need the closed part of the theorem because, for example, if you think about a non-closed curve, like the letter C, and you're standing on the piece of paper around that letter C, maybe even inside, like where the C is surrounding you, it actually doesn't separate the piece of paper into an inside and an outside. And then you also need the non-simple part because if you think about the letter P, which is not simple because it intersects itself, if you think about the segment of the P that's not the loop, so the vertical bottom part of that P, that is part of the curve, the letter P, and that piece of the curve doesn't separate—so even though that P seems to have a little bit of a bubble up there, in the in the loop of the P, the bottom part of the P is part of the curve, and it's not the boundary between the inside, what you might consider the inside of the P, and the outside of the P. So you really do need the simple part and the closed part. KK: Right, right. SD: Okay, so the reason I think it's interesting, in spite of the fact that it seems obvious, is because it actually isn't very obvious. And it's not obvious when you talk about what mathematicians love to call pathological curves. KK: Yeah. Okay. No, I know, I know, the theorem I just wanted to shrug my shoulders and say, “Oh, look, it's just a special case of Alexander duality.” Right? And so surely it works. But yeah, okay. SD: And there are other poorly-behaved curves, or misbehaved curves, like another curve you might think about is the Koch snowflake. So one way of thinking about the Koch snowflake is—again, I'm going to wave my hands a little bit here because we're in audio and I can't draw you a picture—but if you think about the outline of a snowflake, and there's a prescribed way to draw the Koch snowflake, but I'm going to simplify it a little bit. Imagine the outline of a snowflake, so not the inside or the outside of the snowflake, just the outline of it. And on a Koch snowflake, that snowflake is going to have jagged edges. It's going to zig and zag as it goes along the outline of the snowflake. The Koch snowflake actually has an infinitely jagged curve, line, to draw it. So it's not that it has 1000 zigs and zags or 1 million or even 1 billion. It has an infinite number of zigs and zags going back and forth. So you know, it's a little bit easier to imagine the— what could loosely be defined as the inside of the Koch snowflake, and the outside of the Koch snowflake when you imagine one being drawn on a piece of paper. You know, right in the heart of the very dead center of that Koch snowflake, you could probably feel pretty confident saying, “Hey, I'm inside the Koch snowflake.” And then far outside, you could be confident saying, “I'm outside of the snowflake.” But if you think about yourself right up against the edge of this Koch snowflake. And put yourself right there. Then as you think about this boundary of the Koch snowflake, the boundary is supposed to be what separates the inside from the outside, but if you're right up close to that boundary, and in the process of drawing an infinite number of constructions to get the ultimate Koch snowflake. You continue zigging and zagging, you add more zigs and zags every time. Then even in the steps that it takes you to get to your drawing of the Koch snowflake, at some point, it might seem like “Hey, I'm inside. Oh wait, now they zigged and zagged and I’m outside. Oh, wait, they zigged and zagged some more. Now I'm inside again.” So it seems like even in the finite steps that you need to take to draw that Koch snowflake, to imagine what the it is in its infinite world, it seems like that boundary is not really clear. So again, another place where it makes you stop and say, “Wait a minute, maybe the Jordan curve theorem is not as obvious as it first looked.” KK: Right. Why do you love this theorem so much? SD: Yeah, so I love it. It actually it kind of goes along with your question of what do you pair it well with? So maybe I'll just jump ahead to what's sugar. Yeah. So, um, because even in my book and in the chapter that in which I discuss the Jordan curve theorem, I actually paired it with a poem. And the poem is by a New Hampshire native, Robert Frost, who actually went to Dartmouth, which is where I got my doctorate. And one of my favorite poems by Frost is called “The Road Not Taken.” And in the beginning of the poem, he's standing in front of this fork in the road, essentially, and he's looking at both options, realizing, “Okay, I've got to go left or I've got to go right.” You know, he starts off: Two roads diverged in a yellow wood, And sorry I could not travel both And be one traveler, long I stood And looked down one as far as I could To where it bent in the undergrowth;So he's standing here and he's saying, “Well, which path should I take?” And he notices one that he calls you know, “it was grassy and wanted wear” and had no leaves—what was what was the line—“in leaves no step had trodden black.” And he ultimately comes to the conclusion that he's going to take the past path less traveled. You know, at the very end of the poem, he says, “Two roads diverged in a wood and I—/ I took the one less traveled by,/ And that has made all the difference.” And it strikes me that what Frost is telling us, and what the Jordan curve theorem is telling us, is take the paths that are more unusual, that aren't well trodden, that people don't always look at first, that aren't as obvious or as paved for us. Maybe it's a path that's going to make you question whether you're inside or outside. Or maybe it’s going to have what feels like this amorphous boundary that you can't quite put your finger on. I guess it reminds me that sometimes making a non-traditional choice in life, or looking at pathological objects in math, is actually something very engaging to do, and can can make a life a little bit more interesting. You know, when I first heard about this theorem, I had the same reaction that most everybody else does: Okay, so I can just draw a curve—you know, you say a curve and you think, “Oh, I can just draw a curve.” I'm just going to do a squiggle on a piece of paper. And as long as I make it simple and closed, then it might be the letter O or it might be some blob that doesn't intersect, but at least starts and ends where it ends where it started. You know, I remember thinking, wait, why does this theorem get its own name? Why isn’t it just lemma 113.7? EL: An observation. KK: Clearly. SD: Why did it get its own name? A I remember asking, and a lot of people, at first everybody was happy to recite the theorem and and say what it was and laugh at how obvious it was, but then later, I kept searching and searching, and then finally I ended up discovering that in fact, it wasn't as obvious, but in order to appreciate how it’s not that obvious, you needed to look at the paths not taken, the more unusual lines and curves. EL: Yeah, so this is a theorem that, of course, I I feel like I've known for a long time, not just in the “it's obvious” sense, but in the sense that it's been stated in cla…

Full show notes at the publisher

Episode 57 - Annalisa Crannell Aug 13, 2020

Show notes

Episode 57 - Annalisa Crannell Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Welcome to My Favorite Theorem, joining forces today with Talk Math With Your Friends. I'm Evelyn Lamb. I co-host this podcast. 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, where it is boiling hot today, and I’m very happy to be in this—how would they put this on on TV?—crossover event, right? EL: Yeah. KK: So like, I think last night on NBC, on Wednesday nights, there are all these shows that take place in Chicago: Chicago Med and Chicago PD and Chicago Fire, Chicago Uber, who knows what. Anyway, sometimes they'll just merge them all into one three-hour super show, right? So here we go. This is the math version of this, right? EL: Yes. And I realized today that our very first episode of My Favorite Theorem, we published that in late July 2017. So this is our early third birthday! And we're so glad that people came to join us! And we are very happy today to have our guest Annalisa Crannell with us. Hi, Annalisa. Can you introduce yourself and tell us a little bit about yourself? Annalisa Crannell: So hi, my name is Annalisa Crannell. I profess mathematics at Franklin and Marshall College, which is in south-central, southeastern Pennsylvania. It's a small liberal arts college. I got my PhD working in differential equations, partial differential equations, nonlinear differential equations, switched into discrete dynamical systems, topological dynamical systems, but for the past 10 or 15 years have been really thinking hard about projective geometry applied to perspective art. KK: That’s quite the Odyssey. AC: Yeah, I was really influenced by by Paul Halmos saying that one of the marks of a really good mathematician is that they can change fields. And so yeah, I feel like I'm trying to enjoy so many different aspects of what this profession allows us to do. EL: And a fun story, at least it was fun for me, is that one time you were here in Utah giving a talk at BYU, which is down the street. And we went to an art gallery and you pulled out your chopsticks and showed me how you use your chopsticks to help you know where to stand to best appreciate art, and it was just so amazing to me that that was this thing that you could do. So that was that was a lot of fun. And I think it just, to me, sums up the Annalisa experience. AC: Thank you. Yeah, summing, I guess, is a good thing for mathematicians. I think everybody should carry chopsticks with them. I mean, it's great. It's frugal. It helps you avoid to trash, but it also helps you do really cool mathematics. So what's what's not to love about them? EL: Yeah. So what is your favorite theorem? AC: So if you had asked me about five years ago, I would have said the intermediate value theorem. But today, I am going to say no, Desargues’ theorem. So Desargues’ theorem first came into human knowledge in the 1640s. And it's a theorem that sounds like it's sort of about planar geometry, but I really think of it as being about perspective. So is this when I'm supposed to tell you what the theorem says? KK: Yes, please. EL: Yeah. Okay, should we all get out our—so this is one, I feel like I always need like a piece of paper. (I’m trying to hold it up, but I’ve got a Zoom background.) But I got my piece of paper out so I can hopefully follow along at home. AC: Yeah. If you had a piece of paper or a chalkboard right behind you, you could imagine that you would have a triangle, like, standing up on a glass pane. And then on one side of this glass pane would be maybe a magician or somebody holding a light. Maybe your granddaughter drew the magician. (Okay, for people in the podcast, I'm showing a picture that my granddaughter drew on the chalkboard.) If this light shines on the triangle, then it casts a shadow, and the shadow is also a triangle. And so we say those two triangles are perspective from a point, the point is the light source. And we say that because the individual corners, the corresponding corners, are colinear with the light source. So A and the shadow of A are collinear with a light. B and the shadow of B are colinear with a light. But it turns out that those shadows, the triangle and its shadow, are also perspective from a line. And what that means is that if you think not about the points on the triangles, but the three lines on the triangles, and you really think of them as lines, not line segments, so going on forever, then the corresponding lines will also intersect along a line. And you can think of that second line, which we call the axis, as the intersection between the plane of glass that's sitting up in the air and the ground. So the interesting thing to me about Desargues’ theorem is that it pretends like it's a theorem about planar geometry, because this theorem holds when the two triangles are both in the same plane, in R2 or something, but the best ways of proving it, the most standard ways of proving it, are using essentially perspective, going out into three dimensions and proving it for two completely different planes and then pushing them back down into the regular plane. And so to me, this is a really interesting example of sort of how art informs math rather than the other way around. Or maybe they both inform each other. EL: So going back a little bit, to me when I've I've looked at Desargues’ theorem before, somehow there's this big conceptual leap to me between perspective from a point and perspective from a line. Perspective from a point seems really easy to think about, and perspective from a line, I just have trouble getting it into my brain. AC: Yeah, I do think perspective from a point is so much more intuitive. And so, so the minorly intuitive, the somewhat intuitive way of thinking of this axis, is you can sort of pretend like it's a hinge. So if these two triangles will sort of fold on to each other from the hinge—the triangle on the glass and the triangle on the ground can fold along this hinge—then they’re perspective from a line. So if you think about something that's in the real world, a flat thing in the real world, and its mirror image, then those two, it's hard to say whether they're perspective from a point, but the lines in the real world thing and the lines in the mirror will intersect along the line where the mirror hits the ground. And so that's that's another way of thinking of this axis, sort of three dimensionally. KK: So I want to think about this in projective space, which probably isn't correct. Or maybe it is. I don't know. I mean, so these lines are points in projective spaces. This is this, how one might go at this in some other way? I asked the wrong question. I'm sorry. AC: So that's not exactly the way that I think of it because I think of the line as a line in projective space. KK: Okay. AC: And the point is a point in projective space. So the point comes from, you could say, from a one-dimensional line in R-whatever. KK: Okay. AC: And so here's one of the interesting things about this theorem and about me loving this theorem. In 2011, one of my coauthors and I wrote a book on the mathematics of perspective art, and we used Euclidean geometry all the way through. We were giving a MathFest mini-course on this and a young mathematician came up to us and said, “I just love how you use projective geometry in art because I learned projective geometry and felt like it had to have something to do with art. And you guys are the ones that explained to me how it does.” And Mark and I turned to each other. We're like, “What kind of geometry?” So neither of us had ever taken a projective geometry class. Neither of us had ever learned any projective geometry. We did not know that it existed. And so this young mathematician ended up changing our lives. We ended up working with her and really learning a bunch of projective geometry in order to come out with our most recent book, which came out last December. And so when you ask questions that get into really deep projective geometry, I'm like, “Ooh, I have to write that one down because that's something else I have to go learn.” So for those of you young mathematicians out there, I just want to say learning new stuff and not knowing stuff is is really so much fun! Don't be afraid of starting something new, even if you don't know it all. EL: And how did you first encounter Desargues’ theorem? AC: Oh, man, so I first encountered Desargues’ theorem when, Fumiko Futamura, this young mathematician, had convinced me I needed to learn it. So I bribed an undergraduate to go through Coxeter’s Projective Geometry with me because it seemed like that was the standard book. And Coxeter is, like, the famous guy in this realm, and he is completely non-intuitive. So I found Desargues’ theorem in there, and I'm like, “I have no idea what this means.” The notation is awful. The diagrams are awful. Everything about this is awful. And so I read through his book trying to say, “What does this have to do with art?” And that was a really fun way to read it. So we just decided Desargues’ theorem is about shadows. EL: Well, I was wondering. So I remember you have also given a talk about squares that kind of blew my mind, where I guess the the thesis of the talk is that all configurations of four points are a square, if you look at it from the right way. Is Desargues’ theorem related to that theorem? I feel like when you said the word shadow that is what reminded me of that other talk. AC: Yeah, thank you. So that's really cool. So most of us know what the fundamental theorem of calculus says. Most of us know what the fundamental theorem of algebra says. The fundamental theorem of projective geometry in one sense really ought to be Desargues’ theorem. So you can think about these triangles, these points, these lines as objects. For mathematicians, we care about verbs. So a verb is the function. So you can think of a perspective mapping as mapping one set of points and lines to another set of points and lines with this particular rule that says that corresponding points have to line up with the sun, which you call the center, and corresponding lines have to line up with the axis, this hinge. But there's other functions that take points to points and lines to lines. So we know in linear algebra, you can do this all the time, and in linear algebra sets of parallel lines go to other sets of parallel lines. But there's other kinds of functions that do this. They're called colineations. So the fundamental theorem of projective geometry says that if you have four points and their images, and you know that points go to points and lines go to lines, then the entire rest of the function is pre-determined, we know that. So Desargues’ theorem says that one kind of colineation is perspective mappings, right? Just, like, a shadow or mapping from the floor, this tiled floor onto your canvas through a window. We know from linear algebra, there are these other affine transformations. And so one really cool theorem that I totally love is if you have something that's not a linear algebra one, that's not an affine transformation, then it's automatically a perspective transformation together with an isometry. So you took a photograph and you moved it. That's this notion that every single thing that you do with four points going to four other points that determines a whole function. So yeah, so anytime you have four points connected by four lines, even if they look like a bow tie, or they look like Captain Kirk’s Star Trek logo, it turns out that's actually a weird perspective image of a square moved around somewhere. EL: And you just have to figure out where you should stand to see it as a square. AC: Yes, exactly. KK: Are you guaranteed to be able to—so if it's on the wall, say, could you have to, like, lift it up into a third dimension to be able to see it correctly? AC: So one of the weird things that happens is if you have a bow tie, we sort of think of a bow tie is that the inside of the bow tie, you would imagine that has to go to the inside of the square. And that is not actually the way it happens. So let's let's think about something that's much more familiar to us. Can you map a circle through perspective into other weird shapes like an ellipse? Well, Sure you can. So imagine that you've got a lampshade, and you've got a circular lampshade, and the shadow that it projects onto the wall is actually a hyperbola. We know that. And the light from the inside of the shadow goes to the part of the hyperbola that goes off towards infinity. Well, if you have the bow tie, think about the area outside of the “x” as almost a hyperbola. So this is when it would be so wonderful if I could actually draw pictures, but it's a podcast. On the on the bow tie, there's two sides that are parallel to each other, and then there's this weird “x” in the middle. The parallel sides, extend them out towards infinity from the bow tie. Right? That turns out to be where the square goes, so if you had a square lampshade, it would cast a shadow that would look like this outside of the bow tie. So the same way that a circular lampshade casts a shadow that looks like the outside of the hyperbola, the U shape of the hyperbola. KK: My desk lamp is a rectangle, so I’m trying to see if it’s casting the right shadow here. EL: Yeah, some experiments you can do. I feel like it's this “expand your mind on what a square is” kind of idea. KK: Got to get rid of those old ideas, man. EL: Yeah. I know that we we traveled a little bit from Desargues’ theorem and I want to give you a chance to circle back and for me KK: Or square back. Sorry. EL: Or square back, or projectively bow tie back to Desargues’ theorem, and I guess what do you love so much about it? AC: What do I love so much about Desargues’ theorem? One of the things that I love is that it really tightly connects mathematics informing art and art informing mathematics. So Desargues himself, we don't know if he actually wrote this up and published it. We don't have a copy of his original manuscript. We do have something that came out from one of his, sort of acolytes, one of his followers, a guy named Bosse. And if you look at Bosse, okay, to draw Desargues’ diagram, you need 10 points: the three on the first triangle, the three on the second triangle, the sun, that gives you seven, and then the three along the axis. You also need 10 lines: the three on the triangle, the three on the other triangle, the three light rays, that gives you nine, and then the axis. When Bosse first published his diagram, his diagram was incomprehensible. It had 20 lines and 14 points, and it was just really a mess. And it was hard to even figure out where the heck the triangles were. KK: Yeah, I don’t see them. AC: And he ended up proving this not using sort of standard geometry, using using numerical stuff called cross-ratios. But the the proofs that make the most sense, that are convincing our proofs that allow us to think about things in three dimensions and use art. So that's one of the cool things, is that actually drawing, if you go and you shade in Bosse’s diagram in a cool artistic way, all of a sudden it sort of pops into 3-d and you can see it, but his original diagram not so much. The same is true of a lot of different proofs. If you try to imagine them as three-dimensional, if you draw them as three-dimensional, the proof becomes more obvious. But also Desargues’ theorem is actually useful for artists because if you want to draw the shadow of something, if you want to draw the s…

Full show notes at the publisher

Episode 56 - Belin Tsinnajinnie Jul 09, 2020

Show notes

Episode 56 - Belin Tsinnajinnie Kevin Knudson & Evelyn Lamb Download Evelyn Lamb: Hello, My Favorite Theorem listeners. This is Evelyn. Before we get to the episode, I wanted to let you know about a very special live virtual My Favorite Theorem taping. If you are listening to this episode before July 16, 2020, you’re in luck because you can join us. We will be recording an episode of the podcast on July 16 at 4 pm Eastern time as part of the Talk Math With Your Friends virtual seminar. Join us and our guest Annalisa Crannell to gush over triangles and Desargues’s theorem. You can find information about how to join us on the My Favorite Theorem twitter timeline, on the show notes for this episode at kpknudson.com, or go straight to the source: sites.google.com/southalabama.edu/tmwyf. That is, of course, for “talk math with your friends.” We hope to see you there! [intro music] Hello and welcome to my favorite theorem, the podcasts that will not give you coronavirus…like every podcast because they are podcasts. Just don't listen to it within six feet of anybody, and you'll be safe. I'm one of your hosts, 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. So if our listeners haven't figured out by now, we are recording this during peak COVID-19…I don’t want to use hysteria, but concern. EL: Yeah, well, we'll see if it’s peak concern or not. I feel like I could be more concerned. KK: I’m not personally that concerned, but being chair of a large department where the provost has suddenly said, “Yeah, you should think about getting all of your courses online.” Like all 8000 students taking our courses could be online anytime now… It's been a busy day for me. So I'm happy to be able to talk math a little bit. EL: Yeah, you know, normally my job where I work by myself in my basement all day would be perfect for this, but I do have some international travel plans. So we'll see what happens with that. KK: Good luck. EL: But luckily, it does not impact video conferencing. KK: That’s right. EL: So yeah, we are very happy today to be chatting with Belin Tsinnajinnie. Hi, will you introduce yourself? Belin Tsinnajinnie: Yes, hi. Yá’át’ééh. Shí éí Belin Tsinnajinnie yinishyé. Filipino nishłį́. Táchii’nii báshishchíín. Filipino dashicheii. Tsi'naajínii dashinalí. Hi, everyone. Hi, Evelyn. Hi, Kevin. My name is Belin Tsinnajinnie. I'm a full time faculty professor of mathematics at Santa Fe Community College in Santa Fe, New Mexico. I’m really excited to join you for today's podcast. EL: Yeah, I'm always excited to talk with someone else in the mountain time zone because it's like, one less time zone conversion I have to do. We're the smallest, I mean, I guess the least populated of the four major US time zones, and so it's a little rare. BT: Rare for the best timezone. EL: Yeah, most elevated timezone, probably. Yeah, Santa Fe is just beautiful. I'm sure it's wonderful this time of year. I've only been there in the fall. BT: Yeah, we're transitioning from our cold weather to weather where we can start using our sweaters and shorts if we want to. We're very excited for the warmer weather we had. We're always monitoring the snowfall that we get, and we had an okay to decent snowfall, and it was cold enough that we're looking forward to warm months now. EL: Yeah, Salt Lake is kind of the same. We had kind of a warm February, but we had a few big snow dumps earlier. So tell us a little bit about yourself. Like, where are you from? How did you get here? BT: Yeah. I am Navajo and Filipino. I introduced myself with the traditional greeting. My mother is Filipino, my father is Navajo, and I grew up here in New Mexico, in Na’Neelzhiin, New Mexico, which is over the Jemez mountains here in Santa Fe. I went to high school, elementary school, college here in New Mexico. I went to high school here in Santa Fe. I got my undergraduate degree from the University of New Mexico, and I ventured all the way out over to the next state over, to University of Arizona, to get my graduate degree. While I was over there, I got married and started a family with my wife. We’re both from New Mexico, and one of our biggest goals and dreams was to come back to New Mexico and live here and raise our families where our families are from and where we're from. And when the opportunity presented itself to take a position at the Institute of American Indian Arts here in Santa Fe, it's a tribal college serving indigenous communities from all over the all over the nation and North America, I wanted to take that. I feel very blessed to have been able to work for eight years at a tribal college. And then an opportunity came to serve a broader Santa Fe, New Mexico community, where I also serve communities that are near and dear to my heart, where I've been here for over 30 years. And I'm really excited to have this opportunity to serve my community in a community college setting. So, going into academia, and going into mathematics, it's not necessarily a typical track that a lot of people have opportunities to take on, but I feel very blessed to be doing math that I love serving communities that I love, and being able to raise my families around the communities that I love to. So I feel like you have a special kind of buy-in by engaging in a career that serves my communities and communities that are going to raise my families as well, too. KK: That’s great. EL: Nice. So I see over your shoulder a little bit of a Sierpinski triangle. Is that related to the kind of math you like to think about? Or is it just pretty? BT: Yeah. One, it’s pretty. When I was at the Institute of American Indian arts, most of the students there, they're there for art. They come from Native communities, and they're not there to do mathematics, necessarily. So part of my excitement was to think about ways to broaden the ideas of mathematics and to build off of their creative strengths. And that piece is a piece that one of my students did. They did their own take on a Sierpinski triangle. I have a few of those items from my office where they integrated visual arts and integrated creative aspects of mathematics from cultural aspects as well, too. KK: So I always think of Native American artists being kind of geometric in nature. It feels that way to me, I mean, at least the limited bit that I've seen. Is that sort of generally true? BT: The thing about Native art is that Native cultures are diverse in and of themselves too. So there are over 500 federally-recognized tribes, and in Mexico are over 20 tribes alone, 20 nations alone, and each of them have their own notions of geometry and their own notions of their kinds of mathematics that they engage in with respect to the place that their cultures, their identities, and their languages are rooted in. So, yeah, a lot of it is visual, and geometric, because that's what we see. But there's also many I imagine that we don't see, that's embedded in the languages and the practices. Part of my curiosity is seeing how we can recognize what we do and what our traditions are, how we can recognize that as mathematical. And it might be mathematical in the sense that we, as professional mathematicians, might not be accustomed to seeing or experiencing. And, you know, I'm still trying to understand my own cultures, languages and traditions too. So I know mathematics more than a lot of how I experience my own culture. So on one hand, I'm seeing things from a traditional mathematician brought through academia, but I’m also trying to understand things through the lens of someone who's trying to better understand my cultures and histories. EL: So what is your favorite theorem? BT: The theorem I chose today was Arrow’s impossibility theorem. KK: Nice. EL: Great. And this will be a timely one, at least for the US, because it will be airing—I mean, I guess the past two years basically have been part of the 2020 presidential season—but really in the thick of it. So yeah, tell us a little bit about what this is. BT: So I'll say more about why I'm kind of drawn to this theorem. So it's a theorem that basically says that there is no perfect ranked voting system, or no perfect way of choosing a winner and, by extension, for me, it kind of brings up conversations about how democracy itself isn’t perfect and that it's really hard to say that a democratic system can accurately represent the will of the people. And I was drawn to this theorem because as I started thinking about the cultural aspects of mathematics and mathematics education, I'm also interested in the power dynamics and the political dynamics and the sociopolitical aspects of mathematics and math education. And a lot of what's out there and written about math education talks about using quantitative reasoning and quantitative analysis and statistical analysis to really engage in critical dialogues and examining inequities and injustices in the world. And all of that is rich and engaging and needed and necessary ways that we can use mathematics to view the world. But the mathematician part of me still misses the definition-proof-lemma aspect of engaging in mathematics. So this theorem kind of represents a way of engaging in politics through some of the theorem-definition- lemma aspects of it. So the way that I understand Arrow’s theorem, and I mentioned this to you before, that I don't know the ins and outs of this theorem, I just really like the ramifications of it and the discussions that it generates. But it basically starts with the idea that we can describe functions where we're considering a way of choosing a winner of an election from a list of candidates. And we're taking each voter’s ranked preference of those candidates. So one thing that we're assuming is that each voter can rank a list of n candidates, A1 through An, and if everyone can rank their preferences, then a voting system would be a way to take all of those, those ranks, or those ballots, and choosing an overall ranking that is supposed to indicate an overall preference for the group of voters. And what Arrow’s impossibility theorem talks about is that we want values, and want to describe good ways of what a good voting system is. So we want to describe list of criteria that shows that we have a good voting system. So the list of criteria that involves Arrow’s impossibility theorem talks about 1) and unrestricted domain; 2) social ordering; 3) weak Pareto or unanimity; 4) a non-dictatorship; and 5) independence of irrelevant alternatives. And I'll go through what each one means. So basically, an unrestricted domain means that we want a voting system or a way of choosing a winner to be able to take any set of ballots with any number of candidates and be able to give some overall ordering, that these functions are well-defined. So the unanimity condition talks about if everyone prefers one candidate over another, where every single voter has one candidate ranked over another candidate, then the overall function that turns the ballots into an overall social ordering should indicate that that candidate is preferred over the other candidate. And we also don't want a dictatorship, right? And the idea of that mathematically defined is that we don't want one voter deciding exclusively what the overall social ordering is of the candidates. And so we don't want a dictatorship. And we want an independence of irrelevant alternatives, and what that what a lot of people think about as an example of is a “spoiler” candidate or a third party candidate, where even if everyone prefers one candidate over another, that a change in order of a third or other candidate, without disrupting that other order, shouldn't change the overall outcome of an election. They relate that to how sometimes third party candidates can be a spoiler for an election even though overall, it looks like a plurality of voters might prefer one candidate over another. But certain voting systems can have that characteristic where third or other other set of candidates can disrupt the outcome of that election. KK: I’ve never heard of that. EL: Wouldn’t it be terrible if that ever happened? [Note: These statements were delivered somewhat sarcastically, presumably referring to the 2000 Presidential election in the US] BT: Right, right, right. So what Arrow’s impossibility theorem says is that those all may be desired characteristics of a voting system or a social choice function, but that it's impossible to have all of those criteria in a voting system. So the general outline of the proof is that if we have a system that has the unanimity criterion, and an independence of irrelevant alternatives, that if we have those two criteria in a social choice function, then the voting system must be a dictatorship. So if we add those assumptions, then we can go through and show that there is a voter whose sole ordering determines the overall ordering of the voting group, of the voters. KK: That’s how I always learned this theorem, is that you set down these minimal criteria, and the only thing that works as a dictatorship, right? BT: Right. KK: These criteria are completely reasonable, right? EL: You can’t have it all. BT: Right, right. They're not outlandish. They're what we might think of as things that we might value in a democracy. And, of course, these, these things don't perfectly replicate what's going on in the real world, but the outcome is still fascinating to me that mathematically, we can show that we can’t have all these sets of what we think are reasonable criteria in a voting system. KK: Recently, maybe in the last two years, I’ve been getting interested in gerrymandering questions. And there's there's a similar sort of theorem that got proved in the last year or two, which essentially says that, you know, people don't like these sort of weird-shaped districts, they think that's bad somehow, because it's on unpleasing to the eye. But apparently — and there’s also this idea of the efficiency gap, where you sort of want to minimize wastage. So if you laid out some simple criteria, like you want compact districts, and you want to make the efficiency gap, minimized that, then the theorem is you have to have weird shape districts, right? So it’s sort of an impossibility theorem in that way too. So these these kinds of ideas propagate through all of these these kinds of systems, EL: The real world is impossible. BT: Right. And even by extension, you know, in many voting theory classes, there's a districting problem, which relates to a good metric for measuring compactness. But then the apportionment issue as well, that it's very hard, if not impossible, to find a fair way of apportioning a whole number of representatives that's proportionate to the state's population, relative to the overall population of the country. KK: Yeah. BT: And so yeah, this is one of my favorite theorems because it kind of opens the door to those conversations and gives me another way of thinking about when representatives, or people who talk about the outcomes of elections, say things like “the people have spoken,” “this is the will of the people,” “we have a mandate now,” that I think these outcomes really complicate those claims and should really give us a critical eye and a critical way of really discussing what the will of the people is, and how those discourses really perpetuate the idea that voting, and voting alone, can accurately indicate the will of the people and that that's to be accepted, and that we mo…

Full show notes at the publisher

Our TOPPODCAST Picks

Follow Us

Stay Connected

Related Podcasts

Links

Stay Connected