Kevin Knudson: Welcome to My Favorite Theorem. I’m Kevin Knudson, professor of mathematics at the University of Florida. I’m flying solo in this episode. I’m at the Geometry in Gerrymandering workshop at Tufts University, sponsored by the Metric Geometry, what is it called, Metric Geometry and Gerrymandering Group, MGGG. It’s been a fantastic week. I’m without my cohost Evelyn Lamb in this episode because I’m on location, and I’m currently sitting in the lobby of my bed and breakfast with my very old friend, not old as in age, just going way back, friend, Jeanne Nielsen Clelland.
Jeanne Clelland: Hi Kevin. Thanks for having me.
KK: So you’re at the University of Colorado, yes?
JC: University of Colorado at Boulder, yes.
KK: Tell everyone about yourself.
JC: Well, as you said, we’re old friends, going all the way back to grad school.
KK: Indeed. Let’s not say how long.
JC: Let’s not say how long. That’s a good idea. We went to graduate school together. My area is differential geometry and applications of geometry to differential equations. I’m a professor at the University of Colorado at Boulder, and I’m also really enjoying this gerrymandering conference, and I’m really happy to be here.
KK: Let’s see if we can solve that problem. Although, as we learned today, it appears to be NP-hard.
KK: That shouldn’t be surprising in some sense. Anyway, hey, let’s put math to work for democracy. Whether we can solve the problem or not, maybe we can make it better. So I know your favorite theorem, but why don’t you tell our listeners. What’s your favorite theorem?
JC: My favorite theorem is the Gauss-Bonnet theorem.
KK: That’s awesome because if anybody’s gone to our Facebook page, My Favorite Theorem, or our Twitter feed, @myfavethm, the banner picture, the theorem stated there is the Gauss-Bonnet theorem. That’s accidental. I just thought the statement looked pretty.
JC: Yeah, and when I first looked at your page, I saw that. And I thought, well, I guess my favorite theorem is already taken since it’s your banner page, so I was really excited to hear that I could talk about it.
KK: No, no, no. In fact, I was doing one last week, and the person mentioned they might do Gauss-Bonnet, and I said no, no, no. I have an expert on Gauss-Bonnet who’s going to do it for us. So why don’t you tell us what Gauss-Bonnet is?
JC: OK. So Gauss-Bonnet is about a relationship between, so it’s in differential geometry. It comes from the geometry of surfaces, and you can start with surfaces in 3-dimensional space that are easy to visualize. And there are several notions of curvature for surfaces. One of these notions is called the Gauss curvature, and roughly it measures whether a surface is bowl-shaped or saddle-shaped. So if the Gauss curvature is positive, then you think the surface looks more like a bowl, like a sphere is the prototypical example of positive Gauss curvature. If the Gauss curvature is negative, then your surface is shaped more like a saddle, and if the Gauss curvature is zero, then you think your surface, well the prototypical example is a plane, a surface that’s flat, but in fact this is a notion that is metrically invariant, which means if you take a surface and bend it without stretching it, you won’t change the Gauss curvature.
JC: So for instance I could take a flat piece of paper and wrap it up into a cylinder.
JC: And since that doesn’t change how I measure distance, at least small distances on that piece of paper, a cylinder also has Gauss curvature zero.
KK: So this is a global condition?
JC: No, it’s local.
JC: It’s a function on the surface, so at every point you can talk about the Gauss curvature at a point. So of course the examples I’ve given you, the sphere, the plane, those are surfaces where the Gauss curvature is constant, but on most surfaces this is a function, it varies from point to point.
KK: Right, so a donut, a torus, on the inside it would be negative, right?
KK: But on the outside,
JC: That’s exactly right, and that’s a great example. We’re going to come back to the example of the torus.
JC: So at the other extreme for surface, particularly for compact surfaces, you have topology, which is your area. And there’s a fundamental invariant of surfaces called the Euler characteristic. And the way you can compute this is really fun. You draw a graph, and the mathematical notion of a graph is basically you have points, which are called vertices, you have edges joining your vertices, and then you have regions enclosed by these edges, which are called faces.
JC: And if you take a surface, you can draw a graph on it any way you like. You count the number of vertices V, the number of edges E, and the number of faces F. You compute the number V-E+F, and no matter how you drew your graph, that number will be the same for any graph on a given surface.
KK: Which is remarkable enough.
JC: That is remarkable enough, right, that’s hugely remarkable. That’s a very famous theorem that makes this number a topological invariant, so for instance the Euler characteristic is 2, the Euler characteristic of a donut is zero. If you were to take, say, a donut with multiple holes, my son really loves these things called two-tone knots, which are donuts. A two-tone has Euler characteristic of -2, and generally the more holes you add, the more negative the Euler characteristic.
KK: Right, so the formula is 2 minus two times the number of holes, or 2-2g.
JC: Yes, and that’s for a compact surface.
KK: Compact surfaces.
JC: And it gets more complicated for non-compact. So the Gauss-Bonnet theorem in its simplest form, and let me just state it for compact surfaces, so I’m not worried about boundary, it says if you take the Gauss curvature, which is this function, and you integrate that function over the surface, the number that you get is 2π times the Euler characteristic.
KK: This blew my mind the first time I saw it.
JC: This is an incredible relationship, a very surprising relationship between geometry and topology. So for instance, if you take your surface and you wiggle it, you bend it, you can change that Gauss curvature a lot.
JC: You can introduce all sorts of wiggles in it from point to point. What this theorem says is that however you do that, all those wiggles have to cancel out because the integral of that function does not change if you wiggle the surface. It’s this absolutely incredible fact.
KK: So for example take a sphere. So we would get 4π.
KK: A sphere has constant sectional curvature 1. I guess, can you change that? You can, right?
KK: But if you maybe stretch it into an ellipsoid, the curvature is still maybe going to be positive, it’s going to be really steep at the pointy ends but flatter in the middle. So the way I always visualized this was that yeah, you might bend and stretch, which topologists don’t care about, and this integral—and the way we think about integrals is that they’re just big sums, right?—so you increase some of the numbers and decrease some of the numbers, so they’re just canceling out.
JC: Not only that, these numbers are scale invariant. So if you take a big sphere versus a small sphere, the big sphere has more area, but the absolute value of the curvature function is smaller, and those things cancel out. So the integral remains 4π.
KK: Right, so the surface of the Earth, for example, we can’t really see the curvature.
KK: But it is curved.
JC: It is curved, and the area is so big that the integral of that very small function over that very large area would still be 4π.
KK: Right. So on the donut, right, we’re getting this cancelation. On the inside it’s negative, and it’s going to be 0 in some points, and on the outside it’s positive.
JC: Right. That’s really the amazing thing about the donut. It’s this unique surface where you get zero. So you have this outer part of the donut where the Gauss curvature is positive, the inner part where it’s negative, and no matter what you do to your donut, how irregularly shaped you make it, just the fact that it’s donut shaped means that those regions of positive and negative curvature exactly cancel each other out.
KK: Wow. Yeah, it’s a remarkable theorem. Great connection between geometry and topology. Do you want to talk about the noncompact case?
JC: This also gets interesting for surfaces with boundary. It actually starts, when I teach this in a differential geometry class, where this starts is a very classical idea called the angle excess theorem. And this goes back to Euclidean geometry. So everybody knows in flat Euclidean geometry, if you draw a triangle, what’s the sum of the angles inside the triangle?
KK: Yeah, 180 degrees.
JC: 180 degrees, π, depending on whether you want to work in degrees or radians. This is a consequence of the parallel postulate, and in the history of developing non-Euclidean geometry, what happened is people had developed alternate ideas of geometry with alternate versions of the parallel postulate. So in spherical geometry, imagine you draw a triangle on the sphere. Say you’ve got a globe. Take a triangle with points: one vertex is at the north pole, and two vertices are at the Equator. Say you’ve moved a quarter of the way around the circle, and the straight lines in this geometry are great circles.
JC: So draw a triangle between those three points with great circles. That’s a triangle with three right angles.
KK: 270 degrees.
JC: Right, 270 degrees. What the angle excess theorem says is that the difference, and we use radians, so that has 3π/2 angle, instead of π. So it says that the difference of those two numbers is the integral of the Gauss curvature over that triangle.
KK: Oh wow, OK. OK, I believe that.
JC: As we were saying for a sphere, the total Gauss curvature integral is 4π. This triangle I’ve just described takes up an eighth of the sphere, it’s an octant. So its area is π/2, so that’s the difference of its Gauss curvature. So that’s why the difference of sum of those angles and π is π/2. So that’s where this theorem starts, and ultimately the way you prove the angle excess theorem, basically it boils down to Green’s theorem, which I was very excited to hear Amie Wilkinson talk about in one of your previous episodes. It’s really just Green’s theorem to prove the angle excess theorem. So from there, the way you prove the global Gauss-Bonnet theorem is you triangulate your surface. You cut it up into geodesic triangles, you apply the angle excess theorem to each of those triangles, you add them all up, and you count very carefully based on the graph you have drawn of triangles how many vertices, how many edges, and how many faces. And when you count carefully, the Euler characteristic pops out on the other side.
KK: Right, OK.
JC: It’s this very neat combination of classical things, the angle excess theorem and combinatorics. It’s fun teaching an undergraduate course when you tell them counting is hard.
KK: It is hard.
JC: And they don’t believe you until you show them the ways it’s hard.
KK: There’s no way. I can’t count.
JC: So it’s a really fun theorem to do with students. It’s the culmination of the differential geometry class that I teach for undergraduates. I spend the whole semester saying, “Just wait until we get to Gauss-Bonnet! You’re going to think this is really cool!” And when we get there, it really does live up to the hype. They’re really excited by it.
KK: Yeah. So this leads to the question. We like to pair our theorems with something. What have you chosen to pair the Gauss-Bonnet theorem with?
JC: Well the obvious thing would be donuts.
JC: And in fact I do sometimes bring in donuts to class to celebrate the end of the class, but you know, this is such a culminating theorem, I really wanted to pair it with something celebratory, like a fireworks display or some sort of very celebratory piece of music.
KK: I can get on with that. It’s true, donuts seem awfully pedestrian.
JC: They do. Donuts are great because of the content of the theorem. They’re a little too pedestrian.
KK: So a fireworks display with what, 1812 Overture?
JC: Something like that.
KK: Really, this is the end. Bang!
JC: I think it deserves the 1812 Overture.
KK: That’s a really good one, OK. And maybe we’ll try to get that into the podcast.
JC: That would be great.
KK: A nice public domain thing if I can find it.
[1812 Overture plays]
JC: Sounds great.
KK: So we like to give our guests a chance to plug something. So you published a book recently?
JC: I did. I recently published a book. It’s called From Frenet to Cartan: The Method of Moving Frames. It’s published in the American Math Society’s graduate series, and it’s basically designed to be a second course in differential geometry, so for advanced undergraduates or beginning graduate students who have had a course in curves and surfaces. Hopefully it’s accessible at that level, and it was really fun. It largely grew out of working with students doing independent study, so I really wrote this book in a way that’s intended to be very student-friendly. It’s informal in style and written the way I would talk to a student in my office. I’m very happy with how it came out, so if this is a topic that’s interesting to any of your listeners, check it out.
KK: That’s great. I took curves and surfaces from your advisor, Robert Bryant, who’s the nicest guy you’ve ever met.
JC: Oh, he’s wonderful.
KK: Everybody loves Robert. That was the last differential geometry course I took, so maybe I should read your book.
JC: Let me give him credit, too. Where this originally came from, when I was a new Ph.D., well relatively new, three years post-Ph.D., Robert invited me to give a series of graduate lectures with him at MSRI, and this book grew out of notes I wrote for that workshop many, many years ago. And Robert, when I very naively said to him, “You know, I have all these lecture notes I should turn into a book,” Robert, having written a book, should have laughed at me, but instead he said, “Yeah, you should!” And it became a back burner project for a long time.
KK: More than a decade, probably.
JC: Yeah, but eventually, I’ve had so much fun working with students on this project.
KK: I’ve written two books, and it’s really, it’s so much work.
JC: You don’t do it for the money.
KK: You really don’t do it for the money, that’s for sure. And of course it’s great you had such a model in Robert, as a teacher and an expositor.
JC: I count myself extremely fortunate to have had him as my advisor.
KK: Well, Jeanne, this has been fun. Thanks for joining us.
JC: Thanks for having me.