S. Donald Sussman Fellowship Fireside Chat with Dr. James Simons — Chat 1 (Mathematics)

I'm Bob Millard, chairman of the MIT Corporation. Welcome to this special event. By the way, I know this is the business school, so you think corporation means profit-making. It's our board of trustees, and we're not profit-making here at MIT. So thanks are in order to many people, I'm sure. But let me offer particular appreciation to the donors of the Sussman Fellowship and to the MIT Sloan Finance Group for organizing what is, I understand, a series of events, and tonight being the first of the three in the Donald Sussman Fellowship Award Series. I'm delighted to be here because I know both of the men being honored this evening, Donald and Jim, and I consider them friends. Tonight, we honor Donald Sussman's 30-year career in alternative investments and fund management focused on both quantitative and fundamental strategies.

Donald is the founder of, and I've known him for 40 years, is founder of Trust Asset Management Company, Paloma Funds, and the China Capital Management LLC. And the Sussman Fellowship is awarded every two years to an individual or group who best exemplifies Donald's impressive career as an investor in quantitative investment strategies. And this academic year, the selection committee chose an MIT graduate, MIT Course 18 alumnus, Dr. Jim Simons, as the recipient of the award. For most of us, Jim needs no introduction, but let me do so anyway because it's always uplifting to review his accomplishments. By the way, I should say he's done an enormous amount for MIT in so many ways, not just in the business school.

He's had an extraordinary career and continues to, as I should say, three careers as a mathematician, an investor, and a philanthropist. I have to say, he's done all three at world-class status. He's the founder and chairman of Renaissance Technologies. Jim and his wife Marilyn lead the Simons Foundation, which is dedicated to advancing the frontiers of research in mathematics and the basic sciences. The Simons Foundation does this in many different ways, including Quanta Magazine, which if you don't subscribe, you should. I read it every day. It's online and it's free. And it's some of maybe the best scientific reporting on subjects which are invariably, you know, rivetingly interesting. And the Flatiron Institute in New York, which I have visited, which is applying

I guess Jim's incredible skills in fields of mathematics and computation to other fields like astrophysics and biology. Jim is the founder and chairman of Math for America, whose mission is to substantially improve math education in our nation's public schools. And Jim and Marilyn have also had substantial, as I said, impact at MIT, not only in the math department, but through the Simons Center for the Social Brain, which is part of our brain and cognitive science endeavors, and many other things around the Institute. We are forever and will forever be grateful to the Simons. So that was a very, very abridged list of Jim's accomplishments. As I said, I'm proud to count Jim not only as among my friends, but as among MIT Corporation's life members.

And I'm proud to introduce him this evening. Tonight's conversation will be moderated by MIT math professor who used to run the department, Tom Mrowka, and we'll focus on Jim's career in mathematics. Gentlemen.

Okay, so this conversation is going to focus on Jim's contributions to mathematics. So if you have questions at the end, no questions on finance, no question on philanthropy, just math. Hopefully. So why don't we begin at the beginning? How did you get interested in mathematics as a child?

I was always interested in mathematics, even when I was a little kid. There were two things that stand out in my mind. When I was very little, I wanted to know all the powers of two, so I just kept: two, four, eight, 16. Those are some of the powers. And I got up to 1,024, but I was only, I don't know, two or three years old, so that was pretty high up, and I figured that's far enough. But I had an experience when I was not much older, maybe three, three and a half or so. And I was in the car with my father. And he said, "We have to stop for gas." I said, "What do you mean?" He said, "Well, we have to put in some more gas. Otherwise, we'll run out." That was concerning to me, running out of gas. I didn't know quite what it meant, except we didn't have any more.

And I said, "But you don't have to run out of gas. All you have to do is use half of what you presently have. Then use half of that and keep going." Now, that was a pretty profound thought. Of course, I didn't realize, "Yeah, but you really wouldn't get anywhere either." But it was a kind of sophisticated thought for a four-year-old or a three-year-old. But I always liked math. And that was pretty much, except for reading. I loved reading. But math was the only subject I really cared about in school.

Did you enjoy math in high school?

Yeah. Yeah, I liked math in high school. I loved plane geometry. That was theorems and proofs. Most of you, I'm sure, took plane geometry. I hope you took plane geometry. The idea of theorems and proofs really appealed to me. And I liked that very much. I liked it. I liked it better than calculus, which we learned. I mean, I was fine at calculus, but plane geometry, the idea of theorems and proofs really appealed to me very much.

So you were an MIT undergraduate. Can you tell us a little bit about some of your experiences here? What was MIT like back then?

Well, I was an MIT undergraduate. I grew up in Brookline and Newton, not very far away from MIT. And I came here already knowing a fair amount of math. So I was taking sophomore courses in my freshman year. But in the second semester, I saw there was a graduate course in algebra, abstract algebra. It was a first-year graduate course. But the important thing, it said, "No prerequisites." Okay, "No prerequisites." So I took this course and found it extremely puzzling. I mean, I managed to get through it, and there were theorems and proofs, but I just couldn't grasp the essence of this subject. And that summer, I got a book on the subject, and within a week, it was perfectly clear. All the things that were puzzling me—why would one do this?

In particular, the fundamental theorem of homomorphisms, which I don't think will mean much to most of you, or maybe to some of you. But it's a basic theorem in all kinds of algebra. You make a map from one place to another. And then there's some things that go into zero, let's say. You can divide out by those things and get a structure on the left-hand side equal to the image on the right-hand side. But then after that, I took all the graduate algebra classes, and they were a breeze. I was only 17 when I was taking the first graduate course, so it's not surprising. I was a little puzzled, but nonetheless. So I was fine at MIT. I graduated in three years, so that was good. But the subject that really turned me on to my field in mathematics was something called differential geometry, which is the study of surfaces, high-dimensional surfaces.

So it involved topology, but it also involved calculus and functions and all that kind of stuff. And well, you know Stokes' theorem. Now, Stokes' theorem is a generalization of the fundamental theorem of calculus. You probably remember the fundamental theorem of calculus as the integral of the derivative gets you back where you started, the integral of f-prime from a to b equals f of b minus f of a. That generalizes, as you know, to all kinds, high dimensions, where you're gonna integrate something over a boundary of something, and that's equal to the integral of something else over the interior of that. And it's such a beautiful theorem that I was, I wouldn't say transformed by it, but it really,

It really, I loved it. So that was the direction that I went. And I stayed at MIT one more year. I graduated in three years. I stayed as a graduate student. I worked under a guy named Iz Singer. Now, some of you who are on the faculty here might know who Singer is, but he was a great mathematician. And I visited with him this morning. He's 94, and he's a little doddering mentally as well as physically. But he's still in pretty good shape. He can swing a tennis racket. And if he holds onto the side of a chair, he can do that. But we had a good talk this morning. So I worked with him. But there was a great mathematician in the field, my field, differential geometry, named Chern. And he was named Chern.

And he was just leaving University of Chicago to go to Berkeley. And Singer, and also another guy, Ambrose, who I worked with, suggested that I go out to Berkeley. I'd been here at MIT long enough. Go to Berkeley and work with Chern, because he's just going there. So, well, I went out there to work with Chern. But the only problem was Chern wasn't there. He had, I mean, well, he wasn't there. He was there in principle. But he was also on sabbatical, which meant that he wasn't there in fact. In principle, he was there. So I didn't work with Chern. But I found someone else to work with. And he was good. He was a professor here at MIT for many years after Berkeley. He came back to the East Coast, and his name was Kostant.

And he was a very, very good mathematician, Bert Kostant. And that's who I ended up working with, because Chern wasn't there. And that went well.

So what did you work on in your thesis?

In my thesis?

Well.

My thesis was in a subject called holonomy. Now, I could describe it. Could I use the blackboard? Go for it. OK. It's a real lecture. Ready? Yeah. So there's chalk. OK. Chalk there. So suppose you're in the plane. Here, I'll try to sit here. And you have a curve. Let's say a closed curve. And you take a vector. It doesn't matter what vector it is. But you move it around this curve, keeping it parallel to itself. Wherever it is, it's parallel to itself. Over here, it looks like this. And finally, it gets back to where it started. And lo and behold, it's the same vector. It didn't change. We just moved it parallel to itself, and there it was. So no big deal. But now suppose that I had a sphere.

And here's the equator. And here's the North Pole, and here's the South Pole. It's a sphere. Now I'll take a vector tangent to the sphere, let's say this one, and I'll draw a great circle right down to here. And it meets, of course, because it's coming from the North Pole, so it meets this perpendicularly. Move this vector parallel to itself. Well, it just goes down here. And finally, it sticks out over here. See? Parallel. And I say, OK. Now I'm going to start moving it this way, keeping it parallel to itself, parallel to itself, parallel to itself. OK. Now I'm going to go back up, draw another line up here, another great circle. And let's say tangent to this one. And now I'll move the vector up.

It's staying parallel to itself, parallel to itself, parallel to itself. But when it gets back here, it's not the same vector anymore. It's turned around. It's turned around through an angle. So in general, when one has surfaces or high-dimensional things that have a metric on them, as this has a metric, if you take a tangent vector, take it around a loop, and come back to itself, it does not, in general, get back to where it started. And if you use, consider all the loops that you can use from any point, you'll get a whole bunch of transformations, a whole group of transformations of the tangent space to itself by going around closed loops. And that set of transformations is called the holonomy group.

It's a group. Some of you might know what a group is. But a group is something you can multiply two things, and you're still in the group. And two transformations, you follow them by each other, and it's still a transformation. So it's called the holonomy group. Now, in fact, a lot of early research was done on this group by Singer right here at MIT shortly after I arrived. Now, a guy named Marcel Berger had made a list shortly before this time frame of all the possible holonomy groups, all the possible groups that could be holonomy groups. And well, he had a list. I don't know. It was eight or 10 of them. And they were all kind of well-known groups. But they all had one thing in common. They acted transitively.

on unit vectors. So if you had any unit vector here, and there was another one over here of the same length, there would be a closed curve somewhere that would take this one to this one when you parallel translated it around. So that's called transitive. Any element could be translated into any other element—transitive. Now, not all groups—these were Lie groups, but that doesn't matter—not all groups act transitively. So that was a commonality that all of these things had. So it begged the question, well, it raised the question, well, why is that? And can you prove this intrinsically without having to appeal to some list of things and say it's true for everything on the list? What was intrinsic about this?

So when I was, I got to Berkeley and working with—I'll sit down again—working with this guy Kostant, I had come up with some little idea. And he looked at it and he says, 'Oh, that's cute. That could relate to this question about the transitivity of holonomy groups.' I said, 'Oh, yeah, what's that?' He told me the question. But he said, 'Don't work on that. Because Borel, he was a famous mathematician, has tried it. Singer has tried it. And don't work on that.' But of course, that just got me going. So I did work on it. And by gosh, I solved it. In fact, at one point, I was stuck. And I consulted Singer. In fact, back here. I had come back for Christmas. And I said, 'I'm stuck. I can't get past this.'

He says, 'But you're not using your own hypothesis.' 'You're right. I'm not.' There was some piece of the hypothesis I was not using. So I got through that. And so it was a good thesis. And so that was my thesis. That was my thesis. Well done. It appeared in the best journal. And I was very pleased. And MIT hired me to be an instructor. So then I was back at MIT.

So although you didn't work with Chern, I think you have an interesting story about meeting him for the first time in Berkeley.

Yeah. So this is a funny story. So beginning of my second year at Berkeley, I was giving a seminar. And right at the beginning of it, this tall Chinese guy walks in. And I said to the fellow next to me, 'Who's that?' He goes, 'That's Chern.' I says, 'Chern?' I had never seen a picture of him. I assumed with a name like Chern, it was short for Chernowski or something. It was some guy who came over from Poland, and he shortened his name from something to Chern. If it had been Chen or Chan, I would have realized he was Chinese. And well, okay, so that's how I met Chern. And we became friends in my second year there at Berkeley. But most people, because we worked together later in life, most people think I was his student.

I wasn't. I was supposed to be his student, but I wasn't his student. So that's the story.

So you spent some time at Harvard in the early days? Yeah. Anything interesting about that period of time? I was here.

It was a strange time in my life. I had come here to teach at MIT, but I was—I was kind of restless. And I had two MIT friends who were down in Bogota, Colombia. They were Colombians. And I had been there once to visit them. And they were very smart guys. They weren't scientists particularly, but they were very smart guys and good businessmen, I felt. So I said, 'You guys ought to start a company down in Colombia.' And at the end of the first semester when I was back here teaching, I flew down there and I told them, 'I'm not going to leave until you find a business. And then I'll invest something in the business,' although I didn't have any money at all at the time. But I figured maybe lightning would strike and I'd get a little money.

So they did find a business, and it was making, of all things, vinyl floor tile. You put vinyl tile on the floor. And there was a lot of construction going on at that time. And so that's what they were going to do. And I came back to MIT. I was starting to do some mathematics. But in the back of my mind was, gee, they were going to put up this tile factory. And I thought, 'You know, I should go down to Colombia and help work in the tile factory.' The craziest thought! Because it occurred to me almost too late: I don't know anything about business, I don't speak Spanish, and I certainly don't know anything about vinyl tile. How could I possibly contribute to this business? But I had already stepped down from my Moore Instructorship, thinking that I was going to do this.

And as the summer went on, because I was going to go at the end of the summer, and I had some crazy job, which I hated, I realized I was not going to go to Colombia. And I told this to Singer and Bott. Bott was a great mathematician at Harvard. And Bott said, 'Oh, fine, I'll put you on my contract.' So in those days, you had an NSF grant, so you could just hire someone on the contract. So I was at Harvard then, and then they made me an assistant professor, and I was there for one more year. But I didn't like Harvard, actually. I don't know. I mean, I liked MIT. I wasn't so crazy about Harvard. I was working in an area, which I'll describe in a little bit, called minimal surfaces, minimal varieties, which I will actually explain what that is.

And the progress was slow. It was exciting, but it was slow. And I needed money, because I had borrowed some money to invest in the crazy Colombian business. So I needed some money. And there was this place in Princeton called the Institute for Defense Analyses. And this place hired mathematicians to attack Russian secret codes. It was a very highly classified place. It was on the Princeton campus at the time. And they paid very well. So I applied. And they said, 'Okay, this guy's okay.' And I had to go through a security clearance. And so they hired me. I was getting now double the salary that I was getting as an assistant professor at Harvard, a place I didn't like anyway. So that was a very good experience.

And I was there four years. And the rule was you could spend up to half your time at your own mathematics, and at least the other half your time on their stuff, which was trying to break the secret codes. And I found both fun. My math started going very well. And I liked the idea. It was the early days of computers. This was in the mid-'60s. And they had computers there. Not that I could program them. I was terrible at programming. They had programmers who would do it. But you came up with ideas, and you would test them on the computer. On the computer, you test ideas. Oh, maybe this would work to break this code. So you write an algorithm up that you think might work, and someone else programs it, and the computer chugged away.

Usually, it failed. Once in a while, it worked. But I found that a lot of fun. And at the same time, I was getting deeper and deeper into minimal varieties. So should I tell people what a—I think it's time to tell them about—you'd like to know yourself, probably.

Yeah, absolutely.

He knows. He knows. Well, a minimal variety, let's take it in two dimensions. Suppose you take a wire frame, a loop, but twist it around. It doesn't matter. And you dip it into soap suds. When you pull it out, there'll be a film of soap spanning that boundary. You've probably seen that. If it was just a little circle, kids blow bubbles through them. But it could be twisted. Anyway, that film minimizes the area among all surfaces that have this as boundary. Every other surface has larger area than this soap film. So that's called a minimal surface. And 20 or 30 years earlier, it had been proved by a guy named Jesse Douglas, who won the Fields Medal. In fact, I think he won the first Fields Medal.

He proved that no matter how crazy the boundary was, you could always get a two-dimensional minimal surface which was smooth. It didn't have some point or something like that. A nice, smooth, minimal surface would span any boundary, which had to be a curve. It couldn't maybe be like that. And that took a lot of work on his part. That's why he won the Fields Medal. And it's called the Plateau problem, not because it's the top of a plateau, because it was conceived as a problem by a guy named Plateau. Sometimes there's a name and a word you think you know. Have you ever heard of Price Club? Everyone heard of Price Club? You know where it got its name? Mr. Price. Mr. Price started Price Club. So you think it has to do with it.

It had nothing whatever to do with money. But anyway, Plateau's problem. But the higher dimensional versions were up for grabs. Okay, it's a two-dimensional surface in three-dimensional space. What about a three-dimensional surface in four-dimensional space or five-dimensional space, six-dimensional space? So anyway, I was studying the general principles of minimal, minimal surfaces in high dimensions and proved a number of interesting theorems about these things and was getting revved up and learning a lot. But at the end of the day, I wanted to attack this problem, which was a big open problem. And, well, it had been solved one dimension higher: three-dimensional things in four-dimensional space.

Three-dimensional surfaces in four-dimensional space were—you could always find a minimal one. Well, I managed to prove that you could do this all the way through six-dimensional surfaces in seven-dimensional space. But the proof didn't work when you were in eight-dimensional space. My proof didn't work. Not only that, I found a counterexample. Now, a counterexample to a theorem is an example that shows that your theorem is wrong, because here's an example of something that violates your theorem. I found what I proposed to be a counterexample. I didn't know how to prove it, but it was a surface that looked like it was absolutely minimal, but it had a point. It was like a cone. So it had a singularity to it.

It wasn't smooth. And I could show that it was locally minimal. If you perturbed it any way, it would get bigger. But maybe there was something way out there with this same boundary that would be smaller. I didn't know. That was the paper. It was an extremely successful paper. It's had 1,500 citations. We published that paper. And then immediately, a very famous couple of mathematicians saw the result, but with the counterexample unproven to be a counterexample. And this was Bombieri and De Giorgi in Italy. And years later, Bombieri told me, he said, De Giorgi woke him up and said, 'We can do this. We can show that this is really a counterexample.' And he says, they worked day and night for three days.

And he said, De Giorgi kept cracking the whip. And they did. They proved it. I could never have done that. But they proved it. So that put the whole problem to bed. And I was very pleased, because I wouldn't have been able to do that. So that was my main piece of work in those few years. And I did that mostly while I was at the Institute for Defense Analyses.

Did you use any of the computers to actually do real math? Say it again? Did you use computers to do your research there? only their research was done on the computers. No, no, I was allowed. What do you mean? I mean, you had access to all these computers.

Yes, but I didn't use them in research. I used them in code cracking. Yeah. Yeah, I didn't use. I mean, we had a third of the computing power in the state of New Jersey, I was informed, when this huge computer came into the place. And it had a million words of memory, of random access memory. Now the words were 60 bits, so it was like, I don't know, seven million, what do you call it, bytes. Seven million. And we were thinking, what are we gonna do with all this memory? How could we possibly use seven million bytes of memory? But within two weeks, the machine was saturated. People figured out awfully fast how to use this memory. You have 10,000 times that in your iPhone or something like that much memory.

But in those days, that was a big deal.

Do you have any interesting stories about the mathematics that you did for IDA? For them? Yeah.

It's all classified. It's true. It is all classified. It is all classified. A fellow came by here a few minutes ago. Who was the guy I gave the challenge? Oh, there he is. He's a mathematician. So I gave him a challenge. I got a fast algorithm to do something that was important. And it was easy to describe the problem. And so that was my main accomplishment when I was there. So I told him this algorithm, and he said he'd figure it out tonight and let me know the next day. Maybe you will, but I'm not so certain. So that was the kind of work that I did there.

And so when you were there, you also started to work with your first PhD student, Jeff Cheeger.

Oh, yeah, Cheeger, Jeff Cheeger. Yeah, I had a student. He was a wonderful student, a mathematics student. I showed him some papers. He read the papers and so on and so forth. And then one day, he came in and said, 'Hey, I proved this.' I said, 'You proved that?' 'Yeah.' And, 'Okay, well, you're done.' I mean, he really was. He embellished on it and so on, but usually, you know, a student, you have to lead him along and help him out and do this and that. So I've had several other students, some of them are pretty good, but no one as good as Cheeger. He just won the... the Steele Prize for Lifetime Achievement. So he became a very good mathematician. And yeah.

So after your four years at IDA, what made you leave, and then what happened?

Yeah, I left IDA, the Institute for the... Some of you might know the story. So the head of IDA, who was based in Washington, because he had a number of units under him, the Institute for Defense Analyses was a multi-unit organization, and the smallest of which was this little one in Princeton. So he was the big boss. And he wrote an article. This was during the Vietnam War. And he wrote a front cover article for The New York Times about how we were winning the war, and we had to stay the course, and it would all be great. So I didn't think much of that article. So I wrote a letter to the Times saying, 'Not everyone who works for General Taylor subscribes to his views.' And I wrote a letter saying, in my opinion, we should get the hell out of there as quickly as possible, in so many words.

No one said a word. No one said a word. But about three or four months later, a guy not much older than me—maybe he was younger, I was 29—came around and said he is a stringer for Newsweek magazine. A stringer means, you know, sort of a hanger-on. I don't know exactly what it means, except that he was doing an article for them on people who worked for the Defense Department who were opposed to the war. Now, he said, 'And I'm having trouble finding people,' which was not surprising, and he said, 'Could I interview you?' No one ever asked me if he wanted to interview me before. Now I'm being interviewed over here, I guess. So I said, 'Sure, you can interview me.' So he interviewed me. And he said, 'Well, what are you doing?'

'I'll tell you what I'm doing. We're supposed to work. We can work up to half our time on our own mathematics. And the rest of the time, you're supposed to work on their stuff. So right now, I'm working only on my stuff. When the war is over, I'll work an equal amount of time only on their stuff. See, so it'll all balance out.' That's what I told him. It sounded not totally unreasonable. So then I went back and told my local boss that I had given this interview. And he said, 'What did you say?' I said, 'Well, I said this and I said that.' He said, 'I have to call Taylor.' He went into the office, his office, he called Taylor, and he came out of the office and said, 'You're fired.' And I said, 'I'm fired?'

'You can't fire me.' He said, 'Why not?' I said, 'Because my title is permanent member.' See, that was my title. And he said, 'Well, you know the difference between a permanent member and a temporary member?' I said, 'No.' He says, 'A temporary member has a contract.' But of course, a permanent member didn't. So I was out of there. I was out of there. But it was fine, because I knew I would get a good job somewhere, because I had done this minimal varieties stuff. And I had various offers. But then Stony Brook University came along and asked if I would be the chair of their math department, which was a weak department and needed strengthening. And that sounded like so much fun to be able to build something.

And so I remember I was interviewed by the provost, who was a very distinguished biologist. And after he finished, he said, 'Well, Dr. Simons, you're the only one I've interviewed for this job who actually wants it.' And I said, 'I want it. I want it. It sounds good.' So I took this job. And it was terrific. We really had a lot of fun. And it was a time when the State University was flush with money, which is certainly not the case today. But they had a lot of money. Rockefeller was governor, and he really wanted to see the State University flourish. And I was able to make very good offers to people. And in the first year, I hired 10 people, one of whom was Cheeger, and one of whom was a very, very famous guy, and so on.

And I built it up. Over three years, I think we hired 30 people. And of course, regrettably, had to get rid of quite a number of people who didn't have tenure. And it turned from a weak department to a very strong one. It wasn't Princeton, but it was a very strong department, especially in differential geometry, which, of course, I knew the best people in that field. So I enjoyed that very much, being a chairman. But at the same time, I was, I was really productive because I was doing a lot of math. And that's when I told you I collaborated with Chern. And that's when I started that collaboration.

So how did you get together? You were in Stony Brook. He's in Berkeley. How did you guys work it out?

Well, it certainly wasn't over the Internet, which did not exist, of course, in 1970. Well, I was fooling around trying to understand something in the field of characteristic classes, which are geometric measurements. It's very hard to describe a characteristic class, so I won't. But, well, I will a little bit. Did you ever hear of the Euler characteristic? Anyone ever? Raise your hand if you've heard of the Euler characteristic. Well, that's enough hands, so I'll tell the rest of you what it is. So if you take a closed surface, let's say a sphere, and put triangles or triangles plus squares plus—you make it into, you tile it with all kinds of polygonal tiles. It's all, as long as it all fits together.

Now you say, OK, how many faces? The number of faces. Subtract from that the number of edges, and add to that the number of vertices. OK, you've got faces minus edges plus vertices. And I don't remember the sign, but let's say it comes out to be two. No matter how you do it, whatever shapes you use, and so on and so forth, it always comes out to be two. Now, if you did the same thing to the surface of a donut, which is called a torus, it would come out to be 0. And if you had a donut with two holes, it would come out to be minus 2, and three holes, minus 4, and so on. So this was a topological invariant. So it only depended on the shape of the thing. It did not depend at all on how you tiled it, right?

And this was discovered by a mathematician named Euler in probably the late 1700s, you think? That ilk. And that's called the Euler characteristic. Well, it turns out there are other characteristics of higher-dimensional things not calculated that way, but by calculating another way, by integrating certain special differential forms over them, and so on. You get numbers. And it's quite a field. It's a very important field. And I wanted to learn this field. And I decided I'm going to learn it by making a combinatorial formula for the next one of these things up. And it's called the signature. And it's a signature of a four-dimensional manifold. And there's an integer that's associated with every closed four-dimensional manifold called its signature.

Now, there's a way to calculate it, but it's an indirect way. And I thought I could calculate this like the Euler characteristic. So I triangulated the thing and tried various methods of getting a combinatorial, what would be called a combinatorial formula. And I was getting somewhere, but then I got stuck with one term. See, I was integrating something and trying to flatten things out and approximating the integral where things were flat and so on. But something came out, one term, that I couldn't get rid of. And the term looked interesting, actually. And it was a term that you could calculate on any three-dimensional manifold and get a number. Of course, if you go do a calculation, you end up with a number.

But the number was well-defined up to an integer. So it's like you look at everything to the right of the decimal point. Forget the integer. So you can calculate that kind of a number. And it was invariant under conformal changes, this number. Conformal changes means, well, it's a technical term. Anyway, you stretch it out. You stretch it out in various ways. I won't tell you what conformal means. But it was invariant under conformal changes. And it had various other very pretty consequences. And so I showed this to Chern. And he liked it a lot. But he said, well, you're only doing this in three dimensions. We could do this in all dimensions. So I said, OK, let's do that. You tell me how, and we'll do that.

But I figured out his approach. So that was my work with Chern. And that work has been very useful, right? I mean, you tell me you use it in your stuff.

I use it every day. I was just teaching my students today about the Chern-Simons function.

In honor of this event?

No, it's just I did tell them to come. I hope some of them are here.

Are any of Tom's students here?

I'll get an A. OK.

I'm sorry I asked that. This stuff was, I don't know, we thought it was very good math. And it got published in one thing or another. I won a prize partly because of that and also because of the other stuff, the minimal surfaces. But shortly after that, the physicists started using this. Witten, many of you maybe have heard of Witten, a famous physicist. He used it in string theory, and then people used it in what's called condensed matter physics. So this math, it's called Chern-Simons invariants or whatever, Chern-Simons, it's a term, started appearing all over physics. And this is really, you know, the great thing, and I didn't know any physics at all. I knew F equals ma, and that was about the extent of it.

Oh, I also knew the inverse square law for gravity. No, I knew two things. But I didn't know any physics. I thought it might apply to physics because physicists at the time were investigating structures of the sort that were being expanded upon in this work. And I mentioned to one physicist, C. N. Yang, who was a great physicist actually, and a friend of mine, maybe this could be used in physics. And he said, yeah, maybe. And that was the last I heard of him in that connection. But it was only just a shot in the dark, and I had no idea how physicists would really use it. But it's a wonderful example of basic science, where you never know where it's gonna go. It could just lie there like a lump and nothing ever happened.

Or it could go in places you never imagined. And my favorite story, and I think it's true, I think it's true, is about the physicist named I. I. Rabi. He discovered something called nuclear magnetic resonance. Now, nuclear magnetic resonance was a phenomenon. And I won't explain what it is, but I actually know what it is. But anyway, it's a phenomenon. And he discovered this, and he won the Nobel Prize. Now, a few years later, a couple of other people realized, actually, you could use this to get the composition of materials, mostly liquid materials, I think. And they won a Nobel Prize. And then two other guys said, hey, you can use this to make images. You can use this to make images. And they didn't want to call it nuclear magnetic resonance, because it sounded too threatening.

So they called it magnetic resonance imaging, MRI. Now, who hasn't had an MRI scan at one point or other? One who hadn't had it was I. I. Rabi. And then one day, in his dotage, he went in and maybe his shoulder was bothering him. And he had an MRI scan. And how astounded he must have been to realize that this discovery of his 50 years earlier was now going to help his shoulder. Most of you are scientists, or many of you are going to be scientists out there. You just never know where good science might go. And if you're around for my third lecture, Marilyn and I do a lot of philanthropy, and we support basic science. And that's kind of our mantra, that good basic science is really valuable. Any other?

Why don't you tell us a little bit more about your interactions with Yang?

My interactions with Yang? With Yang and the sort of dictionary that... Oh yeah, well that's right. I told you one story. So Yang was a Nobel Prize-winning physicist who was at Stony Brook when I got there. He occupied the Einstein chair and he had won the Nobel Prize at quite a young age. He was a terrific physicist. And my first year, we were in the same building, and he invited me upstairs to show me what he had been doing. I came up, and he was at the board. We're doing this, we're doing that. I didn't understand a word. I didn't understand a word of what he was saying, but I was polite. I said, "Thank you very much. Very interesting." And I went back downstairs. The next year, we went through the same charade.

At least on my part, it was a charade. I had a faint understanding. And we went downstairs. The third time, however, he was covering the board. And all of a sudden, I realized what he was doing in mathematics. He was trying to do something in mathematics. And I said, "Stop. Stop right there. What you're trying to do has already been done." He said, "This has been done?" I said, "Yes, it was done 30 or 40 years ago." And he said, "Well, why would mathematicians have done that?" I said, "Well, it just came out of the math. It didn't come out of physics." And I said, "And where you're going, you're going in the wrong direction. You have to do it this way and not that way." So he got very excited. So he invited me to give a seminar to his faculty.

Now, he had a faculty that was eight people or 10 people of the very, very high-class physicists. It was certainly the best class I ever had. And so we had a discussion over lunch. And what I was really doing was telling them, "Well, in physics, it appears you say this. But in math, we say this. But it's really the same thing." And we went back and forth and got it all straight. In fact, Yang wrote a glossary afterwards of a few hundred words in Mathese and in Physics, the two things. And besides having a lot of fun teaching the smartest class I ever had, I acquired a dictionary. I'm a very bad speller. And at the end, you remember these dictionaries? They're this thick. It's the International Oxford, I don't know what.

You had to use two arms to hold this damn thing. So they gave me a dictionary, which I was very grateful for and lugged it home. But I didn't become any better a speller as a result. So that was my interaction with Yang. He's still alive. He lives in China. He's 93 or 94. And Marilyn and I visited him about a year ago. We visited him a few times. And he's going. I mean, China is a wonderful place in a certain sense, the standards. Some years earlier, when I was visiting with him, and he was healthy and hale, we were walking down the street. And on several occasions, kids would come up to us, push me aside, and say, "Can I have a picture with you, Dr. Yang?" Now, who's going to do that in the United States to some scientist, you know?

I mean, give me a break. It's not going to happen. But it happened in China. They really have huge respect for people of that ilk. And Chern himself, when he died, he lay in state and hundreds of thousands of people came by. For days, there were crowds coming by at Chern's funeral. And he died about the same age as Yang is now. But Yang is still there. So that's my experience with Yang.

So in 1976, you won the Veblen Prize. Say it again? In 1976, you won the Veblen Prize. I did. And then, more or less, you quit mathematics. How did that happen?

Yeah, well, it wasn't yes. I did quit. I was already in the process of quitting mathematics. By the way, Tom won the Veblen Prize, so it's, everyone wins, no, not everyone. Not everyone wins the Veblen Prize. It was a good prize. When I won the prize, I was very pleased, and then my relatives asked me, well, how much money is it? I don't know if there's any money. I wasn't so pleased because I was gonna get rich. I was pleased because I won the prize. It turned out there wasn't much money in it, maybe $500 or something like that. But it was something. Anyway, yeah, well, I was beginning to get frustrated with some of the math that I was doing. And the business in Colombia had become a success, and my father and I had a little bit of money from that business, and I thought it would be interesting to trade with it.

And I had an interest in foreign currencies. I don't know why. I did some trading, had some good luck, and gradually decided I went halftime, first for a year, and then full time in business. And that'll be the subject of my next talk, if any of you want to come back next Wednesday, and you'll hear about that. But that's what I did.

But you couldn't quit mathematics completely. You've gone back and worked with Sullivan.

In my old age, yes, I've gone back and done some mathematics. And after about 20 years, I started to do some math again. Anyway, I started doing some math. And there was work that this guy Cheeger and I had done, worked on. But one of the reasons I left math is because we were both stuck in something. So I come back to this subject. And I had an idea of how to proceed with what I wanted to prove. And I showed it to Jeff Cheeger. And he said, I don't want to talk about it. He says, you should talk to Dennis. Dennis knows about this guy, a guy named Dennis Sullivan, who is a very famous mathematician who had been here at MIT for a number of years but was then, well, not at MIT. He was partly at Stony Brook and partly in France.

Anyway, I saw him. And I said, is such and such true? And he said, yes. I said, oh, that's good. That's good. And I said, well, is such and other such true? He says, well, that's a much tougher question. Why are you asking me that? He said, why are you asking me that? I said, well, because I wanted to prove such and such. And he got very interested. And then the two of us worked together and wrote a paper titled An Axiomatic Characterization of Ordinary Differential Cohomology. Got it? You got that? Axiomatic? Quiz afterwards. But it was a nice piece of work in the following sense. There is this thing that actually Cheeger and I had constructed years earlier that had come to be called differential cohomology.

And so it was a new object, a new mathematical object that we had. And it satisfied a bunch of rules. You could map it from here to here to something familiar and something else familiar mapped into it. So it was surrounded by six things, each of which mapped into it or it mapped into. And they were all familiar objects in math. And this was a new object. And so my guess was that any functor, such a new object that satisfied this diagram with all these familiar things mapping into it and out of it, had to be this. There couldn't be anything else. So it should be characterized just by the way it sat in between the maps in and out of all this stuff. That's what I wanted to prove. And with Dennis, we proved that, and I was happy.

I've tried to do some math since, both with Dennis and myself. About seven years ago, I got a really good result. I was extremely pleased. Well, Dennis and I got this result. He had one proof, I had another proof. We were very happy, and I was starting to write it up. But it occurred to me, well, maybe this was known. So I Googled it. And sure enough, it was known. It had been proven 15 years earlier by a guy at Stony Brook who I knew. So I was still pleased that I had discovered and proved this nice result. And of course, not so happy that I couldn't cheer and say, 'This new mathematics,' because it wasn't new mathematics. The funny thing is that about two years ago, I learned, and Claude LeBrun, who had done this 15 years earlier, learned that in 1950, a French mathematician had gotten the same result, disappeared from sight, and his work just sort of faded away, and no one paid attention to it.

The so-called first guy to get the result was not the first guy. It was this obscure Frenchman whose fate is unknown to us all. Do any of you know that Frenchman? I don't think so. So I've done some math, but I'm not... It gets harder. Indeed. It gets harder. Well, I think maybe we could... Anyone have a question? Anyone have a question? There's a hand. There's two hands. I think they'll bring you a microphone.

Awesome. Hello. I'm Miles. I'm a junior at Harvard studying applied mathematics. And my question is related. During your long career in mathematics, how has your view or the way in which you've approached this subject evolved? And then how has the subject itself evolved with regard to the various emphases that have been placed on it over the years?

Well, I think the question is, over the years, how has mathematics evolved, you say, and how does one's approach to it?

And how has your approach to it evolved personally?

Yeah, well, my approach to it was pretty much the same as it always was. I think mathematics has flourished in the past, let's say, 30 or 40 years since I—I really stopped doing it when I was in my 30s and now in my early 80s, so it was quite a while ago. It's flourished since then. All kinds of new mathematics is really wonderful because it grows up. Someone comes up with an idea. The next thing you know, there's a whole concept. There's a new definition. Hey, look at this stuff. Like this differential cohomology that I mentioned. That did not exist when I was a kid. And it came along, differential cohomology. And now it's part of the fabric of mathematics, along with so many other things that have come along.

So it's a wonderful field. Getting back into it, the fact that that work that I did with Dennis was of value was kind of surprising in a way, because I was using stuff from way back then. And so mathematics keeps flourishing. Obviously, some of it is getting computational more than before. With computers and all, one can test hypotheses in number theory, let's say, or maybe in other areas. So computers are playing a bigger role in mathematics. I don't think computers will replace creative mathematics. But maybe they'll be able to prove a theorem just by trying a million, billion approaches and, 'Oh, this works.' But are you a mathematician? Are you studying mathematics?

I do applied mathematics with a focus in economics, so less on the theory or pure math. Economics.

You're studying economics.

Applied mathematics with applications in economics, so less on the pure math theory side.

Yeah. Well, my wife, who's sitting up there, was an economist who applied mathematics to her work. And that's good. That's good. Thank you.

OK. Hi. I'm Ken Schoenberg. I'm a Sloan and MIT alumnus. So my question is, if you—if you could talk a little bit about the most challenging, either cryptographic or mathematical problems that you've been involved with solving, and maybe if you could talk a little bit about how the difficulty of that compares to the difficulty of the challenge of what you created to make Medallion what it is today.

Well, you asked me about the hardest thing that I'd succeeded in. But it's more interesting to give you the hardest thing that I haven't succeeded in. So I've been working on a problem for, I don't know, 10 or 15 years. And in fact, a partial result in that problem is the one I just described, which had been proved before. When I came to Stony Brook in 1968 to be chair of a department, there was an older—well, everyone was older than I was—professor there who had just published a paper proving there could be no complex structure, whatever that is, on the six-dimensional sphere, whatever that is. Now, the six-dimensional sphere is easy to describe. It's unit vectors in seven-dimensional space.

That's the six-dimensional sphere, just like the circle is unit vectors in the plane. So the six-dimensional sphere, a complex structure has to do with complex numbers. And I'm not going to explain all that. But a complex structure is overlapping neighborhoods where the transition functions are what's called holomorphic. Anyway, it's a very simple manifold, a very simple structure. It's just a sphere, a six-dimensional sphere. It doesn't have a complex structure. And this fellow had claimed to prove it didn't. It couldn't have a complex structure. But the paper was controversial. One journal had rejected it, and another journal took it, and it was out there. So I said to myself, well, I'm the chair.

I'll read this paper and decide for myself whether it was right. But it was so bloody complicated, I finally gave up. And I said, well, posterity will judge it. Later on, I asked Chern about the paper, and he said, no, it's wrong. How do you know? Did you find an error? He said, no. He said, but it had no new idea in it. Well, that was an argument. And it did turn out to be wrong. Subsequently, someone did actually find an error. So it was wrong. But it's a great question. It's hard to describe why is this such a great question. But for all kinds of reasons, this is a great question. And I have been working on this without success for the better part of 10 years. And people keep working on it.

In fact, Chern, just before he died, was working on this problem. And he made some progress. He advanced the issue. He didn't solve the problem, but he made progress in it. And that was in his 90s. So I have another 12 years to go on this problem. Any other questions?

Paul. Hi. Thank you. So I guess my question, first of all, am I right in thinking that the French mathematician you mentioned is Alexander Grothendieck, right? I can't hear you. Sorry. I was just guessing the name of that French mathematician that you mentioned disappeared from the public life. Oh, the name of the French mathematician? Was he Grothendieck you were referring to? I don't recall the name. Do you know what it was? I think Alexander Grothendieck, a member of the Bourbaki group. And he kind of disappeared from the public life because he was against the military investment in mathematical sciences and fundamental sciences.

OK. But I'm just guessing here. I don't know. I mean, it wasn't Grothendieck, of course. But I don't know who it was. But maybe you can. That's not my question, though. OK, well, it may have been he, but he disappeared, this guy, whoever he was, if that's his name. I saw his name, but I don't remember. I don't remember.

So my question was, I guess, given what you described in terms of your early and mid-career concentration on pure mathematics, differential geometry, topology, I guess maybe graph theory, all the abstract algebra and all that, and then later on, moving on to the world of, I guess, for lack of a better word, business, but particularly finance and quantitative finance, how did you get from that point to the other in the sense that what helped you achieve in both areas so well? I guess my question is like, yeah, these are, I mean, mathematics in general and science in general help one with doing the analysis and being able to explore ideas and logical thinking and that thought process, of course.

But then, a kind of pure math concentration, did it at any point in time lend itself to the way you needed to be successful in the world of business and quantitative finance, I guess?

So, how did I get into finance, and is that the question?

Right. That's the gist of it, yeah.

OK. How did I get into finance? See, I could make an abbreviated... Well, for one thing, I got into finance because I liked money, and I thought it would be fun to have some. But my policy is to really work with the smartest people you can. And when I went into that business, I found good partners. And at first, we weren't using math in it at all. And then we began to bring statistics and things like that and build models. But that's all. You have to come next Wednesday for this. But that was it.

Hi. Can you tell us about the Center for Computational Mathematics at the Flatiron Institute and what sort of mathematical work you hope to accomplish there?

Yes. So at our foundation, which we'll talk all about in the third lecture, we support basic science by giving grants and so on. But we also have started in-house science, and it's computational science. We started with what was computational biology. We brought in a few people to work in-house, and it went great. They built up a group of 40 or 45 people doing various things in biology, coming up with some beautiful algorithms for making certain measurements in one thing or another. So we thought that was very good, Marilyn and I, and we decided to generalize the whole notion—do this with other fields. So the second field was astrophysics, and the third field was quantum physics. And the fourth field is about to be computational mathematics.

And there's going to be a fifth field: computational neuroscience. So we have a building to do this. We have about 150 people in the building. And what are we going to do, you ask, computational mathematics, which sounds like something of a, you know, redundant—I mean, computational mathematics, isn't all mathematics computation? But this will involve machine learning, computer science, statistics, algorithms, and and numerical analysis. So it's really applications of mathematics writ large. That's what it will be. And are you interested in a job there? That's a serious question, actually. Anyway, that's what it will be. A few people have been hired. We're just bringing on staff.

Hello, thank you. So after you went to industry, the development of geometry is actually very fascinating and exciting. Have you ever thought about, like, what kind of geometry would you like to continue if you were staying in academia? Or equivalently, like, can you name some geometric theory developed after your work in industry that most excites you or made an impression?

Well, geometry, I guess, has flourished in various ways since I was doing it. And I don't know enough about geometry—some of the developments in geometry recently. I suppose Tom knows more than I do. So I don't really know how to answer your question, except I'm pretty much ignorant of the subject that you're asking about. So I can't really say anything.

A little bit off topic. How did the Colombian tile company work out?

How did the tile company work out? I'll tell you how it worked out. Not well. Not well. Our goal was a million square meters a year. And we thought there'd be dividends in the second year. And we never got above 300,000 or 400,000. We were limping along. We—plastic pipe came in, you know, PVC pipe, plumbing, you know, pipes. And there was a lot of construction in Bogotá. And some people were making pipe, but they only extruded the pipe. The fittings, all the complicated fittings, the elbows and the T-joints and so on, they couldn't make that because that had to be injection molding equipment, and that was expensive equipment. But since we were already using vinyl, PVC, we knew how to work with PVC.

I say we—the boys down there. So they decided to invest in the injection molding machine and go into the pipe business. And it was boom time. First of all, anyone else who was extruding pipe had to buy their fittings from us. So they went out of business after a while. And we had more or less a monopoly on pipe. And it was 24 hours a day, seven days a week. For one reason or another, this kind of pipe was just very, very desirable. And there was a lot of construction. So it's, you know, and I think businesses are often that way. You start off doing one thing and it doesn't work so well, but you're there. You're in the—you know, you see what's going on in the world and you realize, hey, I could change somewhat.

I already have a plant. I already have this. I already have that. I could change and do something else. It's not so often that the original plan just works perfectly. But the key thing was that the two guys who were running the business were very smart. And they could roll with the punches and see what they could do. So the answer is poor and then wonderful.

Hi. So during your Cold War code-breaking work, were there any results in number theory or cryptography or any other mathematical fields that were discovered that still aren't known in the public domain?

That are still not known in the public domain?

Right, as in haven't been researched or arrived at by the general academic community.

Oh. Yeah. See, in the days that I was doing it, no one was doing cryptography or cryptanalysis outside the classified world. It wasn't a field. But of course, now today, there's a lot of research that goes on in that. And a lot of it, some of it, is very sophisticated number theory of various sorts. So I'm not up to date at all on what's been happening in that world, and certainly not what's happening in the classified world, because they won't tell me. Yeah, imagine that. They won't tell me. And in the unclassified world, it's not a field that I know very much about. I know a little about it, but not very much. And I guess that's the last question. So thank you.

Well, hi, everyone. I'm Leonid Kogan. I'm chair of the finance group here. I just wanted to thank everyone for coming today. It was a fascinating discussion. We have two more to come. So thank you very much, Tom and Jim, for this conversation. We have two more fireside chats, March 6th and March 13th, covering kind of the other two topics, Jim's experience in finance and philanthropy. And we hope to see you there. So thank you very much. Thank you again.