My Life in Mathematics

Good evening, everyone. Today's lecture will be presented by Dr. James Harris Simons. Now, please allow me to introduce to you our special lecture speaker. My college friend, Hyungjoo Park, chairman of the Seoul ICM, suggested me to be your emcee in this lecture, thinking my story is somewhat similar to Dr. Simons. As a youth, I dreamt of science and majored in astronomy in college. 18 years ago, I changed my career into e-commerce business. My company, Interpark, has been regarded as quite a successful one. Recently, together with my friend mathematicians, I started a science foundation to support public scientific knowledge and education spreading. You see, apparently, similar to Dr. Simons,

But if you look into his story a little bit more, you can easily notice that he's a supernova whose luminosity outshines all the small stars like me. Firstly, Dr. Simons's achievement as a mathematician was gigantic. He received a B.S. degree from MIT and a Ph.D. from UC Berkeley in mathematics. He became a professor at Harvard at the age of 23. In 1974, he made a major breakthrough in three-dimensional topological quantum field theory, which is called the Chern-Simons theory. Secondly, Dr. Simons's achievements in business has been phenomenally successful. In 1982, when I was a freshman in college, he founded Renaissance Technologies, a hedge fund company. He executed financial trading thoroughly based on the mathematical analysis and modeling.

This idea was daring and revolutionary at the time, hence invited much doubt from traditionalists. But he adhered to his principle and proved he was right. Year by year, he and his employees, most of whom are PhD scientists in mathematics, physics, and statistics, has been yielding phenomenal record in financial trading, even in market tumble. With an estimated net worth of $12.5 billion, he is ranked 91st richest person in the world by Forbes 2014. Thirdly, Dr. Simons has been a great philanthropist. In 1994, Dr. Simons and his wife, Marilyn Simons, co-founded the Simons Foundation to support scientific research, education, and health. If I list a few of his donations to the society, $13 million to the Brookhaven National Laboratory and $25 million to Stony Brook University, $60 million to found the Simons Center for Geometry and Physics at Stony Brook,

150 million to the Stony Brook University and 60 million to establish the Simons Institute for the Theory of Computing at UC Berkeley. It's huge. Ladies and gentlemen, it is a great honor to have this great mathematician and great businessman and great philanthropist giving a special lecture here at Seoul ICM 2014.

Well, kamsahamnida. For those of you who don't know Korean, that means thank you. And thanks for the very kind introduction. I'm very honored to be here. This is my first time in Korea, and I have to say, I first heard of Korea when I was, I guess, 12, when the Korean War began. And, well, it ended a few years after that. I'm sure the country was a mess. But in those 60 years, the progress that I've seen here today is just mind-blowing. It's incredible. And I think every Korean should be very, very proud of the country that's been built here in those 60 years. So, yeah, well, go ahead, clap. So I'm going to talk a little bit about myself and my careers, the role that mathematics played in my life, plays in the world.

And then at the end, I will take some questions. If everyone asks one question, we'll be here for a month. So that won't happen. But there will be microphones passed around if people want to ask questions. So this is the International Congress of Mathematicians. And well, I always liked math. When I was a little boy, I would think about numbers and shapes and even logic. And I was a kid who just liked thinking, and I thought a lot about it. When I was eight, the family doctor, Dr. Kaplan, who knew I was a bright boy, said, well, you know, a bright boy should be a doctor. I said, well, I don't want to be a doctor. He said, no, no, it's a wonderful profession. If you like science, you'll make a lot of money.

It's a very secure position. And I said, well, I want to do something with mathematics and physics. I didn't know so much what I was talking about, but I knew perfectly well I did not want to be a doctor. And he said, listen, he said, you can't make any money in mathematics. He assured me of that. Well, you know, he was right in a way. This was 1946. You know, there was mathematics and engineering, of course, and physics. But there was no computers and there was no mathematics in commerce. There was no commercial applications of mathematics. And, you know, being a doctor was a good, stable life. Fortunately, I would have made the worst doctor in the world. I'm certain of that. So I paid no attention to Dr. Kaplan and went on.

But my first job was not as a mathematician. I was 14 years old, and it was Christmas season. And I was hired. I got a job from the local garden supply company for the Christmas season to work in their basement down below in the stock room. And there my job was to put away things as they came in and bring them out as they needed. But I was not very good at that. I could not remember where anything went. There seemed to be no rhyme nor reason. And I was not succeeding as a stock boy in the basement. So I was demoted to floor sweeper. And they gave me a broom, and they said, okay, sweep the floors. So for a couple of weeks, I had a very happy time. I liked sweeping floors because it let me think.

And I would sweep the floor and think. I suppose I thought some about math. I'm certain I thought about girls. But I thought, you know, whatever a 14-year-old might think about. Now, the time was up. Christmas had come. And so I was going to leave this job. And the couple, a man and a woman, they weren't married. But the man and the woman who ran the basement said goodbye to me and tried to be nice and say, well, what are you going to do when you grow up? I said, oh, I'm going to study mathematics and go to MIT. They thought that was the funniest thing they ever heard. Here was the guy who couldn't remember where the sheep manure belonged, was going to go to MIT and study mathematics. But I did.

I did. I did go to MIT, and I did study mathematics. And I liked it a lot. But there was a moment when I realized how great it was and how wonderful it was that I had chosen to study mathematics. It was like an epiphany. I don't know how well that word translates, but it's like it was a flash of light. And it's when I learned Stokes' theorem. Now, for those in the audience, and I'm sure there are some who don't know Stokes' theorem, everyone should know Stokes' theorem, but I know some people don't. It's a tremendous generalization of Isaac Newton's fundamental theorem of calculus. And some, I know, are high school students here, maybe you know the fundamental theorem of calculus. It was a big generalization into many dimensions.

It used algebra. It used calculus, of course, and geometry. It worked in all dimensions, and it didn't have to be flat spaces. It could be curved spaces. And it was a very simple statement. And it just struck me how incredibly beautiful this Stokes' theorem was. And it made me appreciate the beauty of mathematics. And it also made me go into, choose the field of differential geometry as the area of mathematics that I followed. So I was at MIT, and again, I needed some kind of a little job. And I got a job as a programmer in the Instrumentation Lab. Now, I didn't know anything about programming, but it paid, I don't know, $2 an hour or whatever it was, and I figured I could learn. Now, they had a computer there, of course, and it occupied a very large, very hot room.

And it was full of vacuum tubes. It had no chips. These days—those days—it was these hot vacuum tubes, which tended to burn out at the least moment, any moment. The memory was a rotating drum. And there was no random access memory at all. So I was supposed to learn the programming language that they had there and write programs. I wasn't really any better at programming than I was at putting the stock away in the garden supply company. It was not for me. Finally, I learned enough to write a program and passed it in to my boss. And he said, well, he said, 'This program will work, but it's about the ugliest program I ever saw.' And I said, 'Yes, I agree.' And I didn't enjoy one minute of writing this program.

And, 'I'm going to retire from my job as programmer.' And well, I never programmed since. I think I tried one other time. It was a complete failure. So whatever it is about mathematicians' brains, they're not all organized to be computer programmers. But in any event, I finished MIT. I finished it early. In fact, stayed one year as a graduate student. But then my mentor, Is Singer, and a fellow named Warren Ambrose—if any of you remember Ambrose—said that I should go out to Berkeley. Chern, the great geometer, was showing up there that year. And I ought to have a change from MIT and learn geometry from Chern. Well, I went. And I got a nice fellowship. I went out to Berkeley. The only thing missing was Chern, because Chern had celebrated his first year at Berkeley by taking a sabbatical.

So I'd gone out there to work with him, but he wasn't there. Fortunately, there were other people on the faculty, and I worked with a mathematician named Kostant, Bert Kostant, and started to write a thesis. I got an idea. And I came to Kostant and I said, 'I have this idea. It's interesting.' He said, 'It is interesting. It's a nice idea.' He says, 'It might relate to such-and-such a problem,' which I won't say. And I said, 'Oh, yeah, I never heard of that problem.' He said, 'Well, that's the problem.' But he says, 'Don't work on that because that's too hard.' I said, 'Okay.' Well, you know, when you're very young and someone says that's too hard for you, all it does is make you want to do it. And so I did.

I didn't pay much attention to him and proceeded and made pretty good progress in that problem and actually solved it. And after the two years I was there, and that was my thesis, and it was a good one. I was very happy with it. Now during the second year that I was there, Chern did show up and we became very good friends. Many people think I was his student. No one thinks he was my student. But I was his student because we were associated there and did some work together later. But I wasn't his student, but I became his friend. And we stayed friends all the rest of his life till he died at 93 years old about 10 years ago. So from Berkeley, I came to MIT again to teach, and then to Harvard to teach.

And my research at that time was on a subject called minimal varieties. Now, minimal varieties are a high-dimensional version of what were called minimal surfaces. A minimal surface is a surface of smallest area with respect to its boundary. If you take a twisted wire frame and you dip it in a soap-sud solution and pull it out, you'll get a soap film that fills that frame. And that film has the least area of any other surface with that as boundary. That's called a minimal surface. And minimal varieties are higher-dimensional versions of that kind of construction. They're very interesting, and I approached it from a geometric point of view and started trying to learn how those things worked and everything about them.

And I was there in Cambridge for three years. The work was going slowly. I was at Harvard, and I didn't like Harvard. I don't know why I didn't like Harvard, but for some reason or other it didn't agree with me. And I decided I needed some kind of a change. And I'd heard about this place in Princeton called IDA, the Institute for Defense Analyses. They hired mathematicians. They paid them well. And they worked. I didn't even know what they worked on. But I learned that they worked on cryptology, codes and ciphers, all secret stuff. And they had huge computers there, as it turned out. But the best thing initially was you needed to spend 50% of your time on their work, but you could spend the other 50% of your time on your work, on my own mathematics.

So that seemed great, and they also paid quite well. Now, this is 1964, and computers had become much more advanced than the one I first saw. First of all, they were solid-state. They didn't have vacuum tubes. They still took up a pretty large room, as the one there in Princeton did. Now they had disk drives instead of a rotating drum. And best of all, they had this particular computer, which was the largest computer in the United States, had eight megabytes of random-access memory, eight megabytes. Now today, a cell phone has 16,000 megabytes, and that's a little cell phone, but eight megabytes. But everyone was extremely impressed with those eight megabytes. And, well, I learned a lot there.

I learned statistics. I learned model-building. And I really enjoyed them. It was great fun to come up with an algorithm that might be useful for something. Codes and ciphers in those days had become all mathematics, just strings of zeros and ones generated perhaps from some unknown cipher machine. You'd have to look at this and try to determine where did it come from, what kind of machine might this have been, and so on. So it was—it involved statistics. And I liked coming up with some algorithm that might, you know, show something about this string of zeros and ones and let it run on the computer that someone else programmed. And it was great. Everything was top secret. I couldn't tell anyone, including my wife, what I was doing.

She would say, "How did it go today?" And I'd say, "Good." And, "What did you do today?" And I'd say, "The usual." So it was all hush-hush. But on the other hand, I was working at mathematics, which I'll get to in a minute. But I'll tell you, today, cryptology is out in the public domain. I don't know if there are actually departments of it, but it's a big field in computer science, how to make secure communications. But in those days, it was all inside. There was no work outside. It was all pretty much classified and government work. And there wasn't much of a theory either. I mean, the idea, too, was, if you're going to design a machine, well, let's make it as complicated as possible. And the more complicated, the better.

That'll be harder to decipher. There was no public key, as kind of notion has been developed. And computational complexity as a field hadn't begun. Immanuel Blum had written a paper in this time frame, which was a seminal paper. And I looked at it then, because I was trying to figure out, well, how can you tell whether something is sufficiently complicated to be difficult for the other guy to figure out? And well, Manuel Blum gave some hints in that direction. While I was there, in the course of the work, a famous algorithm was developed that some of you might know of today. It's called the EM algorithm, the EM algorithm was developed in the 60s at IDA to attack their work. But now that algorithm is used in speech recognition, genetics, it has all kinds of applications.

And it was nice to see that a piece of work that was developed for one purpose ended up having so many, such a variety of purposes. At that time—this is an algorithm which keeps re-estimating parameters and increasing the probability that the observed output came from a set of parameters, such as Markov processes. Anyway, the point is they developed the algorithm there, but it was very difficult. In fact, it was impossible for a couple of years for them to prove that it actually kept increasing the probability of the output. It obviously did, because on a computer, it just kept making it better. Finally, a tortuous proof was published in, I don't know, two or three papers. But now it's a one-page proof.

So mathematics makes a lot of progress that way. My personal mathematics was also going well. I'd learned more about minimal varieties. And I was having a good time with that—ended up solving a major problem in the field. And I subsequently won a prize. So those four years—I was there four years—I had a wonderful time. I learned about models. I did mathematics that was successful. But then I ran into a problem. So this was the Vietnam War time. And while our unit there on the Princeton campus had nothing whatever to do with the Vietnam War, the head of IDA, who was down in Washington—there were other divisions of IDA—the head of it was a famous general named Maxwell Taylor. And in his old age, he was running this outfit.

And he wrote an article in The New York Times Magazine section. It was a cover article that said, in effect, 'Oh, we're doing wonderful in the Vietnam War. We'll win at any minute,' and so on and so forth. And it was very positive about our progress. Well, I thought that was a very stupid article. I thought the war was stupid. And I thought his arguments were stupid. And I wrote a letter to The Times, which they published, saying, in effect, 'Not everyone who works for General Taylor shares his views on the Vietnam War.' Nobody said anything. I mean, people read the letter, but no one in the organization said anything. But then, I don't know, two months later, three months later, a young fellow—because I was a young fellow, too—came to me and said, 'I'm a writer for Newsweek magazine, and we're doing an article on people who work for the Defense Department who are opposed to the war.'

"And I'm not finding many." "I'm not surprised." But he said, "I found you. Would you like to give me an interview?" Now, not knowing anything about interviews, I said, "Sure, I'll give you an interview. Why not?" And I gave him an interview, and he asked me a bunch of questions. But the bottom line was, I said, "My algorithm was—the rule is 50% of your time must be spent on their work, and 50% could be spent on your work—I said, 'My algorithm now is I'm doing 100% of my time on my work. When the war is over, I'll do an equal stint of 100% of my time on their work, and then it'll all be even.'" Then I did the first and only intelligent thing I had done that day. I went to my local boss, and I told him I gave this interview.

Now, if I told him in the morning I was going to be interviewed by someone, he would have told me, 'Don't do that.' And that would have been the end of that. But he said, 'You gave the interview, what did you say?' And I told him what I said. He says, 'I have to, let me call General Taylor.' And he picked up the phone and he called General Taylor and he told him what happened. And there was some silence on the other end of the phone. And then he put the phone down and he said, 'You're fired.' I said, 'I'm fired?' He said, 'Yes, you're fired.' I said, 'Well, that doesn't seem reasonable. My title is permanent member.' And that was my title, permanent member. He said, 'Well, I'll tell you the difference.' I said, 'I used to be a temporary member, and then I graduated, and now I'm a permanent member.'

'How can you fire me?' He says, 'I'll tell you the difference between a temporary member and a permanent member.' He said, 'A temporary member has a contract. Permanent member just had a title.' So, okay. I should have stayed a temporary member. Then they wouldn't have been able to fire me. But I graduated. In any event, they fired me. And, well, there I was. I had two kids, and I had no job. But I got lucky. I got lucky. You know, I was, you know, going to find an academic job of some sort or other. But I was offered the chairmanship at Stony Brook University of the math department. I had never heard of Stony Brook University before that, but I learned quickly. And I went up and I interviewed for the job.

It turned out they had been, had a series of acting chairmen for the previous five years, each year a different one, because they were unable to find someone to chair the department. And, you know, it was a new department and I wasn't, you know, in retrospect I'm not surprised, but when the, the provost interviewed me, and I expressed enthusiasm. He said, 'You know, Dr. Simons, you're the first person I've interviewed for this job who actually wants it.' And I said, 'Well, I do. I do want the job.' He said, 'Okay.' So I said, 'I'll be happy to take it. It sounds like fun.' I did take it, and it was fun. At that time, Stony Brook had a very weak math department. Not impossibly so, but pretty weak. But a very strong physics department.

C. N. Yang, the great Nobel Prize-winning physicist, was there. He had a very good group around him. And he was eager to see a good math department develop. So, you know, I went there. We hired a lot of people. A few of them are in this room. And... they're older than they were then. And we built a good department. I enjoyed it. I enjoyed it a great deal, although I was not a perfect chair. I'll give you one example of my lack of perfection. I refused to give Yau tenure. I said, 'Well, he worked for us. He was in the department. He was very young. He had just come. And after one year, he says, "I want tenure." And I said, "Well, you know, other geometers here."' Yau, for those of you who don't know, is a very famous mathematician.

He was one of the winners of the Fields Prize years ago. Very distinguished guy. I said, 'Well, but we have other people. You'll have to wait your turn.' He says, 'I'm not waiting.' And I said, 'Well, so long.' So that was not a brilliant move, but there it was. While I was at Stony Brook, in addition to working and building the department, I continued to do mathematics. And I went back, well, not back, I collaborated with Chern. I had had some ideas in low dimensions of some things. He looked at that and said, 'Oh, you're just scratching the surface. We can do this in all dimensions. We can really build this up.' And we did. And I was driving back and forth to Berkeley. He was at Berkeley, of course, and I was on the East Coast.

And we wrote a very nice paper. I was very happy about it. And it was mathematics that had begun to do with certain invariants, certain geometric invariants of things called fiber bundles and connections, something the high schoolers will hopefully, some of you, learn one of these days. And it was invariants. You've got measurements, you've got numbers out. And it turned out that these things called Chern-Simons invariants have had a great application in physics, both in condensed matter physics and theoretical physics, quantum field theory, and so on. Now, I didn't know any physics at the time, and I still don't, to speak of. So I won't speak of it. But it did have this application. And it's a wonderful thing about mathematics.

You create some new mathematics, it looks beautiful. But whether it's ever going to apply to anything except mathematics, you have no idea. In any case, you don't care. It's beautiful mathematics. It's advancing the field. You're happy to have done it. But increasingly often, and I think always, mathematics has found its way, much mathematics has found its way to applications that the inventor of the mathematics did not have the foggiest notion that that would be the case. And that's really, in a way, the way it is with basic science in general. Discoveries are made in basic science because people really want to know the answer. That's why basic science is so important. They're not so concerned, will this cure cancer or will this send rockets to the moon?

They just want to know the answer. But then, of course, sometimes it does cure cancer and it does send rockets to the moon in later incarnations. And I also collaborated with another guy named Jeff Cheeger, and we invented something called differential characters, and that's now become the root of a little subfield in mathematics called differential cohomology. Now, I said that word, cohomology, I've now said it twice. Someone yesterday told me, don't use the word cohomology in your talk, but I now have used it three times. I said, "It's a great word, you know, why shouldn't I use it?" Uh, and that was, uh, uh, something different. It was creating, making a definition that seemed to be—still very proud of that definition—and, uh, building some mathematics around it.

It was related to the work with Chern. So it didn't come out of nowhere, but so anyway, uh, that was, that was very good up to a point, but then Cheeger and I wanted to apply this to a problem, some problems about volumes and whether they're rational or they're irrational or they're transcendental. And it's a whole set of questions which are still wide open, and they're beautiful questions. We didn't have the tools to solve them in any event. I tried for a couple years, and I was getting frustrated. I'd been at Stony Brook then for maybe seven or eight years. I'm not sure how long. I was getting very frustrated. I was in between wives, as it turned out. I was at least on the wrong side of the first one.

And I had just taken up with the person who became the second one. But I didn't know that at the time. I had come into a little money from an investment that my father had made. And I decided to try something else altogether. I decided to try investing. And I got a few people around me. And well, I was lucky. The investing worked well. We weren't investing in stocks at that time. We were investing in foreign currencies, trading foreign currencies, trading interest rate instruments like bonds and treasury bills and so on. And the markets were pretty kind to us. And we were doing very well. But it was very stomach-churning. I mean, you'd go in one morning, you'd think you were a genius, everything was your way.

You'd come in the next day, it was in the wrong direction, you'd think you were an idiot. And looking at the data, looking at data made me think there might be some way to model this data mathematically and produce a more or less stomach churning way of approaching this subject. So we did. I'd had the experience at IDA in Princeton of building models, so it was not unfamiliar to me. I hired a few mathematicians and a programmer. We bought a nice computer. Now, it's much later, computers wasn't quite PC time yet, but they were just on the verge, but computers had gotten much cheaper and much more powerful, and we could do a great deal with them. And so we hired some people, mathematicians, gradually some physicists, computer scientists, and slowly we built these models, and slowly they took the place of the fundamental trading that we had been doing.

And this process took almost ten years, maybe eight. So by 1988, everything we did was models. And ever since, that business, the way we ran it, has been 100% computer models. Today, that company is called Renaissance Technologies. I've recently retired from it, but it's been enormously successful. Has 300 employees, 90 have PhDs in math and physics, astronomy, computer science, statistics, all the hard sciences. And I think they've recently hired a biologist. What the biologist is going to do, I'm not sure. But he seems to think he can do something there. And we trade, or the company trades, publicly traded instruments only, all around the world. It's 100% model-driven. The model is never overridden.

We slavishly follow the model, even though it might be making predictions that someone in the company thinks are stupid. It doesn't matter. Whatever the model says, they do. And it's been an extremely successful business. The researchers there, those 90 people, are continually developing new algorithms, trying to find new anomalies. It's really a very big exercise in machine learning, if you want to look at it that way. You keep studying the past, understand what happens, how it might impinge non-randomly on the future. The only difference is that when you do physics, you assume you're in a stable world. You believe that the physical constant G, you know, the gravitational constant, is not going to change tomorrow.

It will stay the same. And what we do, well, did, well, they still do, is kind of do physics, but in a gradually changing world. Maybe the gravitational constant does change, maybe from 32 feet per second per second to 31 feet per second per second, but it's a slow precession, slow enough so that your understanding of the past is a good guide to the future. It's not perfect, and some of the signals, the predictive signals and so on, gradually become eroded as other people have perhaps traded them out, or the market is just changing. But that's the nature of that business, and it was great. So it turns out that Dr. Kaplan was wrong. He was just wrong. You can make money from mathematics, and not just in finance.

I mean, mathematics today is at the heart of so many things, just search engines and social networks and chip design and what have you. It all depends on mathematics, as various people have pointed out. Tomorrow, Emmanuel Candès will speak on some math, and then another prize winner, well, a prize winner here today, Stanley Osher, also contributed to this general field of reducing MRI imaging time. And in fact, Emmanuel told me he reduced it by a factor of six. Now, an MRI, you know, is a, well, you go in there, you're in this machine, it's surrounding you, you have to lie still or sit still or whatever you're doing, and then it takes the pictures and does that. But as a result of this mathematics,

The time needed is cut by a factor of six, which means that a little child who can't possibly sit still for 12 minutes might be able to sit still for two minutes and have such an exam done. But this was only math, so the machines didn't change. There was no change in the hardware. Only the software, only the algorithms interpreting the data were what made this development occur. Now, we could go on, but I want—I'm going to read something from the National Academy of Sciences in the U.S., but I want first for everyone to hold up his cell phone. Does everyone have a cell phone? I'm sure everyone does. Just hold it up in the air for a minute. Okay. I'm waving mine around. Yeah, come on. We must have acres of cell—oh, okay, there they come.

There they—everyone has a cell phone. Okay. So, okay, now you can put them away. So this language I'm going to read you is in the words of the National Academy of Sciences. It says, to make a cell phone call—oh, okay. I must have said something funny, but I don't know what it was. But in any event, to make a cell phone call, you enter numbers in the decimal system, which are then converted into a series of bits, zeros and ones. Next comes the conversion to an electromagnetic signal. Then a receiver is located, and finally the signal is translated and finally converted into the sound of your voice. Now, every step of this process depends on mathematically dependent technology: error-correcting codes, linear and nonlinear filtering, hypothesis testing, spatial multiplexing, and statistical waveform estimation, all of which are based on pure mathematics such as matrix analysis, linear algebra, graphical models, etc.

Now, all of this mathematics permits you, while walking home, to tell your wife or husband that you might be late for dinner. Now, that's a lot of stuff to give such a message, but we all know that cell phones have transformed our lives in many ways, and at the root of it is all these applications of mathematics. So, speaking of wives, in 1994, my wife Marilyn and I started a foundation. I think the fellow who introduced me alluded to that. She ran the foundation, and it had a broad mission. A foundation, a charitable foundation, for some people might not know, is an organization that gives away money. Of course, you first have to give it money. And then it gives it away to worthy causes, all kinds of causes.

And at first, in fact, the foundation had a very broad mission. But over time, it grew larger. It grew substantially larger. And the mission narrowed to the support of mathematics and science, research, and education. Basic research. That's the primary mission. 90% of the money from the foundation goes to the support of basic scientific research. I'll mention a few. In fact, they already have, but I'll say them again because it's written right here. A few things that the foundation has done over the past several years. It established a center for geometry and physics at Stony Brook University, a center for theoretical computer science at Berkeley, research projects in autism, the origins of life, in condensed matter physics, in the workings of the human brain, et cetera.

Those are all large-scale collaborative programs and many other things. By the way, at the suggestion of Ingrid Daubechies, your outgoing president of the IMU, established inside the walls of the foundation a data analysis center where we've hired some mathematicians, applied mathematicians, a couple of neuroscientists and geneticists to do research inside the walls of the foundation rather than give money to do it outside. It's become a very interesting thing, this foundation. I finally retired from Renaissance and became full-time at the foundation, doubtless annoying my wife, but having a pretty good time. We've had to learn to work together, which has been occasionally difficult, but most of the time a lot of fun.

And we never lack for things to talk about as a result. Now, I'd been away from academics for a long time, and I'd been away from the research, the academic research world. When I came back in, a lot of things had changed. First of all, science just seems to be moving much faster. And that's presumably due to the Internet and better communications and easier ways to get your work published and circulated and so on. But things seem to be just accelerating. As an example of that, more mathematics research has been published in the last 25 years, I think by page count or whatever, than in all the history of time up until then. And I'm assured by John Ewing, who knows things like this, that it's considerably more than 25% than all of that before.

So mathematics, like the rest of science, is growing, and maybe even faster than other branches of science. A whole field that didn't exist when I left, theoretical computer science, is now a full-blown field. They seem to be continually asking the same question. But nonetheless, a lot of good stuff has come out of theoretical computer science. Biology is increasingly quantitative. When I studied biology, it was extremely boring. And it was just memorization of I don't know what. And as you know from my work in the garden supply store, my memory is not terrific for seemingly random facts. But now biology is more and more becoming quantitative. There's an area of biology that's referred to as quantitative biology, but I'll bet that maybe in 20 or 30 years, you wouldn't say quantitative biology anymore, any more than you'd say quantitative physics, because I think it's going to be all quantitative.

And that's a good move. And another dramatic thing is that there are far more women in science, certainly, than there were when I was a practitioner in academic science, and especially in life science, but more and more in physics and mathematics. And of course, the Fields Prize today, the first woman to receive a Fields Prize, is just a harbinger of things to come. Women are catching up everywhere, and that's great. Well, completing the circle in my old age, I've come back to doing some pure mathematics. And in fact, working in that field, now I'll say the word a fourth time, differential cohomology. And it's been very gratifying. We've actually been able to jump back in and get some results.

Now, I've been collaborating with Dennis Sullivan, one of the great mathematicians of my generation. So it might not be so surprising that we've accomplished some good things. But there it is. And in spite of a long absence, we've done some decent work. And I'm very pleased. So I'm going to conclude with a few guiding principles. And you can take them for what you will. But I think that these are principles which, even without articulating them to myself, I have followed. And the first is just to do something that everybody else isn't doing. If you run with the pack, you'll probably lose. You might win, but more likely you'll lose. Try to come up with a different way of doing things or a different set of problems to attack.

And I think, anyway, that's what I did. Partner with the very best people you can find. And, well, some people just want to work alone their whole lives, but partnering is a great thing. And it amplifies what you can do. But you have to choose your partners carefully. I've been lucky enough to have wonderful partners, both in the financial business and in mathematics, and now in the foundation. So work with the best people you can find, because that's going to make your work better. Now, the third thing is, be guided by beauty. Now, I talked about the beauty of, in my mind, of Stokes' theorem. But we all know as mathematicians that there's a great deal of beauty in mathematics and that aesthetic is what can push you along.

But that aesthetic of doing something that's beautiful really works outside of mathematics. Creating an organization which is really running smoothly and where all the pieces fit together and everybody's happy, that's kind of a beautiful thing. It's worth striving for, and it can make you feel almost that same kind of feeling, that kind of aesthetic feeling that you would in doing mathematics. Don't give up easily. Stick with things, even when it really hurts. To do good things can take time. And finally, hope for some good luck, because there's no substitute sometimes for just plain good luck. All right. Well, thank you.

I've told you.