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 study of all metrics on a space. And so, yeah, again, I haven't thought of an application so much to other fields, but something that is a beautiful and unexpected theorem that opened up whole vistas of mathematics, I think, in the last whatever. I don't remember when Riemann stated this problem. When did he live, in the 1840s?

KK: Yeah, middle 1800s.

HM: So it’s been 175 years or something that people studying, have been studying? consequences in some sense of, of this or analogs of this?

EL: Yeah. Well, and so this is something: I never wonder it at the right time to check and see, but is there a place where you can go and say, like, this is my domain 1 — and maybe it's a square or maybe it's the flag of Nepal or something — and this is my domain 2, or just the unit circle, and here is the conformal map between them. Is that something that exists?

HM: Typically not. There are certainly examples where you can, but it's very, very rare that you can write down an explicit formula for the map. That's again, maybe why it's such a beautiful theorem, but you cannot, I don't let’s see, I hope I'm not — maybe you can do it for a circle to an ellipse. Um, maybe. I'm not 100% sure. There are people who know much, obviously know much, much more about finding something explicit. But in general, no, if you take some crazy Jordan curve, no way, do you know an explicit formula.

EL: You just know it's there.

HM: You know it's there.

KK: Well, that's important, though, right? If you're going to go looking for a needle in the haystack, you do, in fact, want to know there's a needle in it.

EL: Yeah.

KK: All right. So another fun part of our podcast is we ask our guests to pair their theorem with something. So we're dying to know what pairs well with the Riemann mapping theorem?

HM: Well, I thought about that a lot.

KK: This is the harder part.

HM: I tried desperately to find food, but I couldn't think of really the right thing. So I do love music. And this is maybe crazy far-fetched, but I paired it with Stravinsky's Rite of Spring only because to me, this Riemann mapping theorem revolutionized geometry and complex analysis. And I think of the Rite of Spring of Stravinsky, which was premiered in the early 20th century, revolutionized modern music, contemporary music. That's the best I can do.

EL: I like that.

KK: Well, I do too. And for all we know, there were riots after Riemann published his theorem.

HM: Could have been.

KK: You know, “there’s no way this is true!” Mathematicians stormed out.

HM: Maybe he gave a lecture and people threw tomatoes at him.

EL: Yeah, well, I must say when I was thinking about asking you to be on the podcast, I did think about the many wonderful meals that we have shared together, and I know that Howie is a great appreciator of the finer things in life, including music, too. I think we've gone to a concert together. And yeah, so I thought that this would be an excellent thing. You know, I was I was talking with Jon earlier about, what is Howie going to pair with it? And my first thought was actually pancakes, which I think are a little pedestrian, but that you can make them into so many different shapes. And there's even, there are people who will do these things where, you know, if you pour the batter on, in a certain way, you know, you can get these beautiful things. I mean, part of it is part of the batter cooks longer than the rest of it. And so you've got shading based you know, how they do I've seen, I think, you know, Yoda and like, I don't know, all sorts of different things. There’s this Instagram account. But that was one, all these different shapes you can do. I'll say that Jon actually suggested jigsaw puzzles. Oh, no, sorry, he first suggested jigsaw puzzles, but there’s only one right way to do that. But then he said tangrams, you know those things with all the shapes, you know, there's a square and triangles and stuff. And then you can rearrange them to make all these different shapes, although those are non-continuous maps. So it wouldn't be quite as good. But, I do like the Rite of Spring. And it means that Stravinsky is doing really great on My Favorite Theorem because Eriko Hironaka actually picked Stravinsky also.

KK: Firebird.

HM: She picked the Firebird. I’ll have to look at her podcast. Maybe I'll give her a zoom meeting and we can compare music and the math.

EL: Yes. But I like that. And I am now going to ret-con in some riots following Riemann declaring that you can make these conformal maps.

KK: Well, this has been great fun. I do love the Riemann mapping theorem and Howie, thanks for joining us this

HM: Well, thank you for having me. It was a pleasure.

On this episode of My Favorite Theorem, we were happy to talk with Howard Masur, a math professor at the University of Chicago, about the Riemann mapping theorem. Here are some links you might find interesting as you listen.
Masur's website
Evelyn's article about the Koch snowflake
Jeremy Gray's article about the history of the Riemann mapping theorem (pdf)
A recording of Stravinsky conducting the Rite of Spring
Did the Rite of Spring really cause a riot at its premiere?