My Peregrinations through Mathematics

Well, I'm certainly honored to be a member of this organization and to represent my class. The last time I represented my class, I was president of my eighth-grade graduating class. And I forgot my speech in the middle. I've never—and I remembered it ever since. So hopefully... but I had notes this time. So, you know, I've had a variegated life in science. Loved it when I was a little boy. And I went to MIT and was mentored by Is Singer as an undergraduate and went out to Berkeley to get a PhD under Bert Kostant. That was in 1961. And my thesis and—the field that I worked in was differential geometry. My thesis was published in a good journal. I was very happy about that. And I taught at MIT and at Harvard for the next three years and then spent four years at the Institute for Defense Analyses.

That was a super-secret, code-cracking think tank in Princeton. But it was not a bad place, and I could do mathematics there as well as try to crack codes. But the first five years after my PhD, I worked in one area, which is high-dimensional versions of what are called minimal surfaces, which we'll get to in a moment. I didn't publish anything during those five years, but it was worth it because I learned a lot about that stuff. Here's a two-dimensional minimal surface. There's a picture of it there. Minimal surface is a surface that minimizes area with respect to its boundary. If you take a wireframe, a loop, maybe twist it up and so on, dip it in a soap film, and you pull it out, you'll get this nice surface that spans it.

That minimizes area among any competing surfaces. That was quite a time-honored subject in mathematics, and it was heavily analytic, a lot of partial differential equations and stuff that I was not especially fond of, but I took a geometric approach. I wanted to understand the geometry in full generalization, if you could, so the ambient manifold here, this is in three-dimensional space, it's a two-dimensional surface, but I was trying to understand the general concept of k-dimensional minimal varieties in n-dimensional Riemannian manifolds, not necessarily Euclidean space. But always, they minimized their area. Of course, it was a high-dimensional analog with respect to their boundary, and proved some nice general theorems.

And finally, turned to a problem called Plateau's problem. And Plateau's problem is back to—back to this one. Here's another minimal surface, you'll get an idea. See, that's one. Its boundary are those two rings, the one on the top and the one on the bottom. The surface is what you see inside that's all colored. That's minimizing the area for anything that's bounded by those two disjoint rings. So, the question was, can you always do this? Can you always, given any boundary, let's say a simple closed curve or maybe a pair of loops like that, get a surface that really does the job and doesn't have any singularities? It's smooth. It's differentiable. It doesn't have points or creases or anything like that.

And this problem was first solved in 1930 by Jesse Douglas. And in fact, he won the first Fields Medal for that work. He spent his whole life teaching at CCNY, where they had no graduate school at all. But he was a terrific mathematician. This was an active field ever since then, but these dimensions didn't get up. Finally, a guy named Almgren raised that by one dimension. He showed that in four-dimensional space, given a three-dimensional boundary, you could always find... Sorry, yeah. In four-dimensional space, three-dimensional boundary—no, no, four-dimensional space. A two-dimensional boundary, yeah, sorry. Two-dimensional boundary, you can always get a three-dimensional spanning thing.

The inside has to be higher dimension than the boundary. So that was the first step in raising the dimension of the solutions to this problem, and it spanned it, and it was smooth—no creases, no mess. I went through this, my studies, and then finally turned to this problem. And a lot of work had been done so that geometry could really play a role, not differential equations. So I proved that this is true. You can always do this through dimension seven. So up through dimension seven, you'd have a five-dimensional boundary and a six-dimensional surface. This is Euclidean space. That was all fine. So I gave one proof that went all the way up through dimension seven. And in dimension eight, my proof didn't work anymore.

And I found a counterexample, or what was proposed as a counterexample, which was a cone. So you have to be in eight-dimensional space. You take, for those who might have some notion of what I'm saying, you take the three-sphere cross the three-sphere, sitting there in eight-dimensional space—a three-sphere in one four dimensions and another three-sphere in the other four dimensions. You take that cross product. You get a six-dimensional boundary. You connect all the points in that boundary to the origin of the eight-dimensional Euclidean space, and you get a cone. It has a point, but that turned out to be minimal among all local competitors. It was a proposed counterexample. That was as far as I could get.

The paper got published with all this stuff in it. It was a good paper, and a year later, three Italians—Bombieri, Giusti, and De Giorgi (I think De Giorgi led the charge)—they proved that that cone, which was locally minimum, was actually globally minimum, and that was the end of the problem. So Plateau's problem was—Plateau was a guy, I said Plateau's problem, he was a person—was killed. That was the end of that line of questioning. You couldn't do it anymore; there'd be singularities there. So after that work, I went to Stony Brook University, a new university, as chair. I was pretty young to be chair, but they had reached the bottom of the barrel, I think, in looking for a chairman, so they picked me.

It was fine. We built up a good department. I started studying something called characteristic classes, which I wanted to understand. I'll give you an example of the origins of characteristic classes. It goes back to 1750 or so, and Euler, a great mathematician, scientist in general—Euler. So here's what he observed. So try to picture this. I don't think I need another slide, no. Picture a sphere that you tile—you cover it with tiles. They could be triangular, hexagonal; they can all be different shapes, but as long as they're tiles, they sort of fit together, and you cover this whole sphere with tiles. And then you do a count. Well, there's points called vertices where edges come together.

Let's count the number of those. Let's count the number of edges, and let's count the number of tiles—the faces of this thing. And you take vertices, the number of those, subtract the number of edges, and add the number of faces, okay? Vertices minus edges plus faces, and you get a number. Of course, what number do you get? You get two. You get two. I mean, suppose I use triangles instead of hexagons or squares, or I mixed it all up and did 67 million of these tiles—you still get two. But if you did this with the surface of a donut, a so-called torus, you get zero. Once again, no matter how you did it, you'd get zero. And if you did it with a torus with two holes, like a pretzel—you know, a donut with two holes—you'd get minus two, and three holes, you'd get minus four, and so on.

So, these were, this was called a topological invariant, and it really distinguished between, well, spheres and toruses. But it was quite amazing that no matter how you did it, you always got the same answer. So, now... There was that result. Then the famous mathematician Gauss came along. Now, remember, going back, this didn't have to do with really the shape of the sphere. It could have been a squishy sphere. It could have been a football. It could have been a hot dog. As long as you could deform it into a sphere, it didn't really matter. Once you put the hole and made it a torus, that was a different story. Now, Gauss came along studying geometry. He was, you know, a fabulous mathematician.

He did all kinds of stuff, but he studied geometry. And he came up with a definition of a surface which was now geometricized. It had a metric on it. It was a real, it wasn't squishy. It was a rigid surface of some sort. The notion of curvature. Every point there was curvature. And this could be a positive number or a negative number, and it would vary all over the surface. I won't describe how you exactly define that, but a saddle point has negative curvature, and something like a point on a regular sphere would have positive curvature. So, okay. And then he did something interesting. He integrated that over the whole surface. And what did he get? He got 2 pi times what we've now called the Euler characteristic.

So, if you did it over a regular sphere, you integrated its curvature. It doesn't have to, you know, it could be a funny-shaped sphere, but it would have funny curvature. You integrate that, and what you were going to get is 2 pi times 2, or 4 pi. And if you did it over a torus, you were going to get zero and so on. So he related the curvature of a surface, which is a geometric notion at every point. It's a geometric notion, not a topological notion. He related the curvature of the surface to its topology, to this Euler characteristic. And that was really one of the most beautiful theorems in all of mathematics. Now, there were other topological invariants that were subsequently discovered and so on, but to generalize this Gauss-Bonnet theorem to higher dimensions,

which in principle one should have been able to do, was very awkward and had been not doable. You kind of knew what to integrate, but you couldn't prove that this was going to be the right answer. People who had tried had very complicated approaches. And then a young Chinese mathematician named Chern, 1944, he came along and he proved this formula, this high-dimensional formula—there has to be even-dimensional manifolds—in the most elegant imaginable way. The title of this paper I think had the word short in it, I'm not sure. It was simple, intrinsic, whatever it was. It was, I don't know, you could write it down in two pages. You could write down this proof in two pages, and he proved that theorem.

And that opened the floodgates, in a way, of... the theory of characteristic classes, which are topological measures, but how they could be interpreted geometrically. And he and André Weil established right after that, they were friends, what's called the Chern-Weil homomorphism, which related all kinds of so-called characteristic classes, topological measurements, to geometric, geometric formulas and measurements of that sort. That's where I started. I wanted to learn this subject. The first thing I wanted to do was find a combinatorial formula for the signature. Now, combinatorial is like that Euler's first thing, vertices minus edges, some formula just in terms of a triangulation or something that could explain a topological invariant of a four-dimensional manifold called its signature.

It was easily defined, the signature was easily defined in terms of some other things, but not in terms of these triangulations or whatever. So I worked on that just to sort of get my feet wet in this field. And it was going good. I had this integral. I was trying to get rid of the various pieces and turn them into combinatorial things. But then there was one piece that was resistant. It was not malleable to whatever hammers I hit it with. And then I realized that this piece had some interest. I called it this pesky term, but it had some independent interest, and with it, I was able to define a function of a three-dimensional manifold, which was a very interesting function. It depended on the metric on the manifold, but it was what was called a conformal invariant, which doesn't matter.

Anyway, I got this number. Actually, it wasn't a number. It was a number only up to an integer. You didn't know the integer part, so you could think of it as a point on a circle, if you like, but it was a number modulo the integers. It was 0.723 or 1.723 or 6.723. It was the 0.723 that mattered. This was a geometric invariant. It led to some nice results. You calculate it in two different ways. I showed this to Chern and he took one look and said, 'We can generalize this to all dimensions.' And he had the technique to do that. I didn't even know what he was talking about at first, but I soon learned. And together, we wrote this paper, creating these invariants in all dimensions. And here, I think...

There, there's Chern, he's the older guy, and me, there's the younger guy. You can see I'm not Chinese, and that he is. Underneath, all this is pointing—this is slides from his first exposition of this at an international congress a few years after we did the work that somehow got rescued. There it was. However, here's what was interesting, or what else was interesting. Ten years later or so, Ed Witten and some other physicists found that this stuff was useful in physics. Not only in string theory and quantum field theory, other people applied it to condensed matter physics, and now these Chern-Simons invariants are really kind of ubiquitous in many aspects of physics. The quantum Hall effect, whatever that is.

Even... I think you have to get something cold, but I'm not sure. And even in quantum computing. I was just out on the steps with someone who was working on quantum computing, and he told me that these terms are used in that. Now, so who knew? I mean, as you can tell, I don't know any physics. I didn't know any physics then. I don't know any physics now. But it was pure serendipity. But as many of you know in the audience, you do something in basic mathematics or basic science, and you really don't know where it's going to go. You don't know what application it might find. It might find none. It might just be a beautiful result or experiment. But often, it finds its way to applications that may never have occurred to you.

Well, after that, I worked with another partner, Jeff Cheeger. Here, I'll show you something which I won't even explain. You see that? That's a diagram, right? It's a very pretty one, I think, but I won't tell you what's inside. Sort of started a little mini field called differential cohomology. But it led to some questions that I found extremely frustrating. So frustrating, in fact, that I didn't even want to look at pure mathematics anymore. And in 1977, I left academia, I left pure mathematics, and I didn't do any mathematics for, it says here, 26 years, which I think is approximately right. And instead, I started a hedge fund. Now, if you can think of a jump from, you know, Chern-Simons invariants to hedge funds, that was a big jump.

But anyway, I thought that would be something interesting to do, and we built models of financial markets. We modeled financial markets, and that actually seemed to work pretty well. We brought in mathematicians and physicists, astronomers, computer scientists, all kinds of people who actually wanted to make some money, I think. That's always a motive. But the work was fun. It turned out to be a huge exercise in what today is known as machine learning. We just kept finding anomalies in the data that would allow you to better and better predict what was coming next. And luckily, this was enormously successful. And today, right here, thriving enterprise, 300 people, 100 PhDs, still churning away, predicting markets that way.

Well, while I was learning to make money, my wife, Marilyn, was learning to give it away. And I think she didn't hope to give it away quite as fast as it was coming in, but she was pushing pretty hard there. In 1993, she started a foundation. At first, it was... Oh, yes. That's easier to understand than the diagram that was there before. There's Marilyn and that other guy. First, it was kind of a general purpose foundation, gave to various good causes, but gradually we converged on supporting basic science. That's what we do today. We support life science and physical science and mathematics of all sorts, neuroscience, oceanography. I think there are some people in this room who may have been beneficiaries of that support.

And after I left the business, the Renaissance money-making business, I joined Marilyn at the foundation. And somehow or other, after those 27 years, I went back to doing some pure math with a guy named Dennis Sullivan, who was a famous mathematician. That's pretty much my talk. I added this morning, in reflecting on what it is that I've accomplished, I've really always had outstanding collaborators, really outstanding. In math, it was Chern, of course, Cheeger, Sullivan, all members of the National Academy at one point or another. In business, it was Jim Ax and Henry Laufer and Peter Brown and Bob Mercer in that machine learning exercise. And in philanthropy and in life, of course, Marilyn.

So there it is. There's my talk.