by Anne-Marie Oreskovich and Dmitry Sagalovskiy
Last semester, the math club had the privilege of interviewing Harvard Professor and recent Fields medalist Curtis McMullen. During the hour-long interview, Professor McMullen discussed his background, his research, his experiences at various universities across the country, and the Fields medal. The math club would like to thank Professor McMullen for taking the time to let us get to know him better. To find out more about Professor McMullen, see his webpage at http://math.harvard.edu/~ctm
Q: How long have you been at Harvard?
M: A year and a half if you don't count my graduate student days.
Q: So you were a graduate student here?
Q: And where were you an undergraduate?
M: I was at Williams College in Western Massachusetts, and then I spent a year in Cambridge, England.
Q: Where are you from?
M: That's sort of a hard question to answer. I basically grew up in Charlotte, Vermont, but I was actually born in Berkeley, California. We moved around a little bit too, but I think of myself as being from Vermont.
Q: So could you tell us a little bit about the medal?
M: I believe it was started in 1930's. It was established by a Canadian, Fields, and I know that Ahlfors and Douglas were given the first two. It's given every four years at the ICM, and in the more recent years they've been giving it to three or four people. So let's see, who else got it this year? Kontsevich, Gowers, and Borcherds. Actually all of them except for Gowers have spent time in Berkeley, which is where I was for the last seven years before I came here. So I knew both Borcherds and Kontsevich from Berkeley.
Q: Where were you when you find out?
M: I was here. You find out a couple of months in advance, and it's supposed to be kept secret until the actual day of the ceremony. So actually I didn't tell anyone, which was quite difficult, because there were rumors circulating, and I'd constantly have to be denying them.
Q: Can you tell us a little bit about what your research was on that gave you the medal?
M: Let me begin with the direction of my research. First, I wrote my thesis at Harvard, but I didn't work with a Harvard professor. I had been doing some computer work with David Mumford on Kleinian groups before I graduated, and I got interested in that subject. But I actually ended up writing my thesis with Dennis Sullivan, who at that time was a professor at City University in New York and IHES at France. So I was very lucky that Mumford introduced me to him in the last year of my graduate career, at which point I had no advisor and no thesis topic. And I went to France and worked with Sullivan at IHES for a semester, and I met Steve Smale there who gave me this nice thesis problem on solving polynomial equations by iteration.
You've probably heard of Newton's method for solving polynomials. If you apply Newton's method for a cubic polynomial, it may not work. You may get stuck under a local minimum. And if you change the initial guess a little bit, it might still not converge to a root. So Newton's method is not reliable for solving polynomial equations. The problem I worked on was whether or not there was any algorithm like Newton's method, involving iteration of just one rational function, that can reliably solve polynomial equations. I was able to prove the answer is no for degree 4 or more, and actually I found a new algorithm for solving cubics, which is reliable.
Then I went to MSRI and was at MIT for a semester, then Princeton for four years. Peter Doyle and I worked in Princeton on solving fifth degree equations, and we found this beautiful unexpected algorithm for solving quintic polynomials. But it's not contradicted by my thesis because it's a tower of iterations; that is, you iterate one rational function, take the thing to which it converges, and plug that into another one.
As you may know, solving the quintic is bound up with the Galois group A5, and the fact that A5 is a simple group. This was used by Galois to prove you can't solve the quintic equation by radicals.
It turns out that to be able to solve an equation using an iterated rational map, what you have to do is find a rational map whose symmetry group is the Galois group of the polynomial. Now there is only a small set of groups that can be symmetry groups on the Riemann sphere, and the interesting ones come from the Platonic solids. So A5 , the symmetry group of the dodecahedron, is the most complicated one you can get. We used this rational map with A5 symmetry to give a new algorithm for solving the quintic equation reliably. And by the same token, since S6 or A6 does not operate on the Riemann sphere, there is no similar algorithm to solve equations of degree 6, or more. So that was my first area of research: solving polynomials, and dynamics of rational maps. Link
Now, the next thing I worked on when I was at Princeton was Thurston's theory of hyperbolic 3-manifolds. Thurston has a research program, which has been very successful, to try to find a canonical geometry for three-dimensional objects. For example, if you imagine you have some manifold, that is secretly a 3-sphere, if you could somehow find a round metric on it, then you would suddenly recognize it as the 3-sphere. So if you can find a metric that gives the manifold a good shape, then you can recognize what the manifold is. It turns out that most three-dimensional manifolds admit these metrics, but the metrics are not positively curved like the 3-sphere, they are negatively curved. For example, if you take the outside of a knot in S3, a knot complement, then it almost always admits one of these so-called hyperbolic metrics of constant negative curvature. Because of that, there are now computer programs, where you can just draw a knot at random with a mouse, and click, and within one or two seconds it will tell you exactly what knot it is. And if you give it two knots, it will immediately recognize whether or not they are the same knot. This is amazing because the problem of classifying knots was classically extremely difficult to solve.
While at Princeton I found a new, analytic proof of Thurston's theorem that provides hyperbolic structures on many 3-manifolds, including most knot complements. This new proof has to do with Poincaré series, a classical topic in complex analysis, and it also lead to the solution of conjectures of Kra and Bers. Later at Berkeley I began to see parallels between the theory of 3-manifolds that fiber over the circle; this topic is worked out in 2 books that appeared in the Princeton "Annals of Math. Studies". The Fields medal was, I imagine, in recognition of these projects.
So I worked on the dynamics of rational maps, and I worked on hyperbolic 3-manifolds, and I worked on Riemann surfaces per se , and I've also worked on topology of surfaces and knots. And the thing I'd like to emphasize is that for me all of those fields are really the same field. You very easily start working on a problem in dynamics, and find yourself a few months later working on a problem in knot theory or topology, because they are all very interconnected -- knots, complex analysis, polynomials, Riemann surfaces, hyperbolic 3-manifolds, etc. There is not really a name for this field, but that's the field I work in.
Q: So you've been at arguably the four best schools in America for mathematics: Princeton, Berkeley, MIT, and Harvard. Can you compare and contrast them in terms of the atmosphere, friendliness, pace people work at, etc., for undergraduates thinking of going on to graduate school?
M: They are really different. Let me leave out MIT, because I only spent a semester there. Princeton is a terrific department, but the town is a little stuffy and boring for a young person. It has the highest density of people from "Who Is Who", and it's very cultured. There is nothing unexpected ever happening. So it doesn't seem very lively to me. But I wasn't there as a graduate student. Princeton is a wonderful place to go to if you know you are not going to be there forever. I look back very fondly on my years at Princeton.
Princeton and Harvard both treat their graduate students very well. There is a good ratio of the number of students per faculty. Students are well-funded, departments are small enough that students get a lot of individual attention. And I think the students learn a lot from each other in both of the places. That's a big component of graduate education.
Berkeley is also really wonderful. It's a place that has a huge department, one hundred faculty if you count emereti. I really loved it, but it takes a lot of energy to find a good place to live, to find a good advisor, and to get into the right niche, mathematically and so on. But as you do that, it pays you back very much. And the weather is beautiful. You can walk from campus into Strawberry Canyon then into Tilden Park, and be completely out of view of humanity within 40 minutes. (At Harvard, on the other hand, I found I could bicycle for an hour, and still be in suburbia...) In Berkeley the swimming pools are outdoors, it's very lively, and it's also very tolerant -- to all sorts of different lifestyles, different kinds of people. You feel a sense of freedom. You don't feel any qualm about trying out a new idea, and not worrying so much about whether or not it's going to work. One of the great things about Berkeley is that there are so many graduate students, and so many postdocs in the area, especially with MSRI, that you can have a working group on any mathematical topic you can think of. There's a lot of mathematical interest there.
I really enjoyed being a graduate student at Harvard too. Cambridge and Berkeley both have advantages over Princeton, in the sense that they're young communities, there is a lot going on, they're close to a major city. You can tell a little bit from my graduate experience that although I think Harvard is really great, the fact that its faculty is small might make it hard to find an advisor who is in the area you want to work in. And I think that the real key to success in graduate school is finding something that you are interested in enough to keep you going for four or five years.
Q: Why did you choose to come to Harvard from Berkeley?
M: I first came as a visitor. And I found it really fun to teach here. At Berkeley classes for undergraduates are often very large, and it was just very rewarding to have these really good students in a small class. And I really liked the fact that the department is small enough that it's easy to get to know other faculty members. And of course, since I was a graduate student here, I always looked up to Harvard as being this wonderful place. Actually I found it hard to imagine being a professor here, so I wanted to explore what it would be like. I enjoy the fact that my areas of interest are different from, but overlapping with, those of other people in the department. I am very interested in a lot of the stuff other people do here. So for me, in a way, it lets me continue my education.
Q: But doesn't this decrease your opportunities for collaboration with other faculty members?
M: In the first place I travel quite a bit, so I see the people that are in my field in France, or in Stonybrook, or elsewhere. However, most research is done on your own; I do my best research by myself. Its very useful to be able to run an argument by an expert in the field, but I don't really miss having someone who is exactly in my field to collaborate with. I have to admit, it was a tough decision to come here. I miss living in Berkeley, and I may spend a sabbatical there.
Q: Do you see yourself as a renaissance mathematician in the sense that your work encompasses a wide variety of areas of math?
M (laughing): No, I see myself more as a dilettante, someone who dabbles in many different areas and is interested in many different things; I certainly wouldn't say a renaissance mathematician. Now, I really enjoy lots of different kinds of math, and I enjoy working on something I'm not an expert in and learning about that subject. This field I've been describing is really wonderful that way, because its so broad that it makes contact with many different types of mathematics. When I came to Harvard, I found that for a lot of the theory (such as Hodge theory on complex manifolds, etc.), I didn't really understand it and I wasn't very motivated to study it. So I started with a subject I could learn really well: one real variable.
I took a real analysis course when I was an undergraduate; I went to Stanford for a year and took a great real analysis course from Benjamin Weiss who was a visiting professor from Jerusalem. And that really got me excited about analysis. Then I went back to Williams and I worked closely with Bill Oliver. He was very influential in my mathematical education; it was from him that I first learned this idea of using dictionaries in mathematics to use as a sort of analogy between different fields or different theoretical developments to try to guide my work. So those were my early influences.
When I came to Harvard and I was sort of casting about. I knew how to computer program -- I'd been working in the summers at IBM-Watson in Yorktown Heights -- and Mandelbrot and Mumford were almost collaborating; Mandelbrot was furnishing access to computers at Yorktown Heights to Mumford, who was drawing these beautiful pictures of limit sets of Kleinian groups. As somebody who was conversant with the computer world at Yorktown, I started working for him as his computer programmer, helping him draw these pictures and so forth. You have to imagine, in those days, we had to make a long-distance modem call and then work at a 30 character per second terminal writing programs in FORTRAN. Then we would draw a picture and we would have to wait a week for them to mail it to us from Yorktown to see if it came out right.
Then I got interested in Hausdorff dimension, and since I knew some real analysis, I tried working on that. My first paper ever was on a problem I learned when I first met Professor Hironaka, who was a Harvard professor at the time, although he'd been on leave in Japan. When he first came back from Japan, he told me this question which he hadn't been able to solve, which was to compute the fractal dimension of a particular set. This set is obtained by drawing the letter "M" and repeating the same figure, as shown here.
In the end you get a set with is not self-similar, but it is self-affine. Fractals whose dimensions are easy to compute have the property that if you take a small piece and re-scale it by the same factor in both dimensions, it looks like a larger piece. This one has the property that a very little gap can be scaled to the big gap, but you have to scale by a power of two in one direction and by a power of three in the other; because of that it's dimension is tricky to compute. In my first research paper, I computed it's dimension: D = log2 (1 + 2log3 2). That was a wonderful problem; I worked on it very hard. You can see that I liked to stay close to the ground of math I really understood.
Then I started getting more interested in complex dynamics, so I went to one complex variable from one real variable; I always stayed close to stuff I could really understand. So now, twelve years after my Ph. D., I'm finally writing a paper that has to do with Kähler geometry; and I certainly didn't feel comfortable with Kähler metrics when I was in graduate school. I had to not only work up to the topics, but also see an internal motivation for getting to them, rather than having them plopped down in a "well this is what we're going to learn next"-manner.
Q: What was the "dictionary analogy" that you spoke of?
M: My biggest mathematical influence was my thesis advisor, Dennis Sullivan. Not only was he my thesis advisor, but when he was still at IHES in France, we would spend a couple of months together every summer there, and I would go to his seminar from New York or Princeton. He's a professor in Stony Brook, NY now, and I try to visit there about once a year.
Sullivan invented a beautiful dictionary between rational maps and Kleinian groups. A rational map is a map of the Riemann sphere to itself given by the quotient of two polynomials; for example x2 + c, where the polynomial in the denominator is 1. The interesting thing to study is iteration of these maps. When you have a compact hyperbolic 3-manifold, its universal cover turns out to be the solid (open) 3-ball. The quotient of the 3-ball by the action of the fundamental group of the original manifold is the manifold again. The 3-ball can be compactified by adding its boundary in R3, namely the sphere S2. The group action on the 3-ball extends to the boundary S2 as Möbius transformations (i.e. maps of the form (az+b)/(cz+d)). This is called a Kleinian group. Notice that we began by considering a 3-dimensional manifold and we ended up with a dynamical system on the sphere. This is how the two subjects are connected. There are many theorems making this connection explicit. I wrote a survey article ("The classification of conformal dynamical systems") for Yau's conference which laid out not only this dictionary, but a research program for proving results based on it. Understanding and developing this dictionary has been a big motivation in my work. For example, one big gap in the dictionary is reversing the process I described -- if we are given a dynamical system on the sphere, no one knows how to find a three-dimensional object associated to it. There is lots left to do in this exciting field!
Q: Where do you keep your Field's Medal? Do you keep it at home?
M (laughing): I cannot reveal that information!
Q: What was the situation when you won the Field's Medal? How did it feel?
M: My first reaction was one of complete astonishment; I was really aghast. I actually thought I wasn't qualified, in terms of age. I also knew so many great mathematicians here, and at Berkeley, and other places, that I couldn't believe that I'd been selected. Also, in 1991, I won the Salem Prize, which is a prize in Analysis; I was pleased to be recognized that way because I really love the field -- it was my first, as a mathematician. In fact, I had written my minor thesis as a graduate student on Salem numbers, and this prize is in honor of Raphael Salem, so it has personal meaning for me. I had never expected to get any recognition of that kind, so I certainly felt I had already had my share of recognition. (I was equally surprised I got an offer from Harvard; then again, I didn't know what to say.)
This brings to mind is a saying of Lipman Bers, who was one of my mentors; he said: "Mathematics is something that we do for the begrudging admiration of a few close friends." I think that's a good description of mathematics; you don't expect more than that, because the satisfaction of mathematics is really a personal thing. So I feel very lucky to have been selected for recognition by the Fields medal committee.
One of the wonderful things about math is that the community is fairly small. When I went to Berlin to receive this prize, many people I knew well from over the years were present -- a wonderful international community of friends of mine. It was really a nice thing.
Q: How were you able to contain your excitement?
M: Well, what happened was, I was so aghast that I quickly forgot about it, because I couldn't really believe it. And then every once in a while, I would remember. And I would think, that can't really be true (laughing), and of course, I would have no way to check, since it had to be a secret.
Q: Is there anything else you'd like to share with us about the medal?
Actually, I have a story about when I was coming back from
The security guard in the airport running the metal detector
stopped me when my backpack went through the machine.
She said, "Excuse me, what do you have in your backpack here?" I said,
"It's a gold medal." She said, a little dubiously, "Mmm hmm."
So I took it out of my pack. A little chagrined, she said
"Oh, very nice; is it yours?" I said "Mmm hmm!"