When a mistake was found in Andrew Wiles' original proof, he called in Cambridge algebraist Richard Taylor. They worked together at Princeton for eight or nine months and emerged with a solution to the world's most famous open problem. Taylor is now at Harvard and will be teaching Math 121 in the spring.
WHERE ARE YOU FROM?
I spent most of my career in Cambridge, but last year I moved to Oxford for one year before coming here. Before that I did a one year post-doc in Paris, a PhD in Princeton and was an undergraduate in Cambridge.
WHAT ARE YOUR MAJOR RESEARCH INTERESTS AND ACHIEVEMENTS?
The great problem that motivates me is to understand the absolute Galois group of the rational numbers, that is, the group of all automorphisms of the field of algebraic numbers (complex numbers which are the roots of nonzero polynomials with rational coefficients). If you like you can talk about all Galois groups of finite extensions of the rational numbers, but this is a convenient way to put them all together. It doesn't make a lot of difference, but it is technically neater to put them all together. The question that has motivated almost everything I have done is, "What's the structure of that group?" One of the great achievements of mathematicians of the first half of this century is called class field theory, and one way of seeing it is as a description of all abelian quotients of the absolute Galois group of Q, or if you like, the classification of the abelian extensions of the field of the rational numbers. That's only a very small part of this group. The group is extremely complicated, and just describing the abelian part doesn't solve the problem. For instance John Thompson proved that the monster group is a quotient group of this group in infinitely many ways.
There is some sort of program to understand the rest of this group, often referred to as the Langlands Program. There's a huge mass of conjectures, of which we are only beginning to scratch the surface, which tell us what the structure is. The answer is to my mind extremely surprising; it invokes extremely different objects. You start out with this algebraic structure and end up using what are called modular forms, which relate to complex analysis.
There seems to be an answer to this question: what's the structure? And the answer is something completely unexpected in terms of these analytic objects, and I think that's what attracts me to the subject. When there is a great connection between two different areas of mathematics, it always seems to me indicative that something interesting is going on.
The other thing we can see--another indication that it's a powerful theory--is that one can answer questions one might have asked anyway, before one built up the theory. Maybe, the first example was a result proved by Barry Mazur; he provided a description of the possible torsion subgroups of elliptic curves defined over the rational numbers. It was a problem that had been knocking around for some time, and it's relatively easy to state. Using these sorts of ideas, Barry was able to settle it.
Other examples are the proof the main conjecture of Iwasawa theory by Barry Mazur and Andrew Wiles, and the work of Dick Gross and Don Zagier on rational points on elliptic curves. And I guess finally, there's Fermat's last theorem, which Andrew Wiles solved using these ideas again. So in fact, the story of Fermat's last theorem is that this German mathematician Frey realized that if you knew enough of this correspondence between modular forms and Galois groups, there is an extraordinarily quick proof of Fermat's last theorem. And at the time he realized this, not enough was known about this correspondence. What Andrew Wiles did and Andrew and I completed was prove enough about this correspondence for Frey's argument to go through. The thing that amuses me is that it seems that history could easily have been reversed. All these things could have been proved about the relationship between modular forms and Galois groups, and then Frey could have come along and given nearly a two-line proof of Fermat's last theorem.
Those four [torsion points, Iwasawa theory, Gross and Zagier, Fermat] are probably the obvious big applications of these sorts of ideas. It seems to me the applications have been extraordinarily successful--at least four things that would have been recognized as important problems irrespective of this theory, problems that people had thought about before modular forms.
Somehow applications of this theory have been going for some twenty-five years. Barry's result was in the early seventies. (I think.)
HOW OLD ARE MODULAR FORMS?
Certainly about thirty years. Sort of in this period the ideas have been becoming more and more fixed. The first indications maybe go back to maybe the late fifties. But the ideas didn't really start becoming definite until maybe 1970. These dates are very rough.
WHAT ARE MODULAR FORMS?
Modular forms are holomorphic functions defined on the upper half complex plane--only the part with positive imaginary part. The group SL_2(Z) acts on the upper complex plane by Mobius transformation; by composition, the group also acts on the set of holomorphic functions of the upper complex plane. Modular forms are functions which transform in a simple way under the action of that group.
WHAT WAS YOUR PERSONAL ROLE IN THE DEVELOPMENT OF MODULAR FORMS?
I've done various things, but they're all rather difficult to explain on this sort of level. Maybe the simplest thing to talk about is the following: There should be some sort of correspondence between certain of these modular forms and two-dimensional representations of the absolute Galois group of Q. In one direction, things have been known for twenty-five years or something. If one starts with a suitable modular form, the way to construct a representation of the Galois group has been known for twenty-five years. Now the big problem has been to start with a representation of the Galois group and try to produce a modular form. In fact, there's one result that's rather old, due to Langlands and Tunnell. Until rather recently, that has been the only isolated result. Recently, Andrew Wiles did much better. I guess I was involved with this in the end. It is probably well known that there was a mistake the first time he tried to do something like this. He spent a few months trying to fix the mistake himself. Then he rang me up one day and to my great surprise asked me to come help him work on the problem. We worked together for eight or nine months and eventually found a way to get the arguments to work.
And I guess before that, my main interests have been in certain generalizations of these questions. For instance, if instead of the rational numbers one took the Gaussian numbers, one can ask the same sorts of questions. It's slightly less obvious there what one should mean by a modular form. They turn out to be functions not on the upper complex plane but on hyperbolic three-space. So I spent a lot of time trying to copy as much of what was known for the rational numbers to other fields like Gaussian numbers and Q adjoined the root of a negative number, say Q(root(-d)).
So a lot of other people have thought about totally real fields. A totally real field is a finite extension of the rational numbers such that whenever you embed it in the complex numbers, it actually lies in the real numbers. Q(root(2)) is an example. Q(cube-root(2)) is not an example, it can be embedded entirely in R, but it doesn't have to be. It turns out that totally real fields seem easiest for this theory. I thought about these for a bit. Then I turned to things like the Gaussian integers, Q(i), which are the simplest examples of non-totally real fields. This is probably what I was best known for in our little circle for before the work on Fermat.
Classical modular forms are holomorphic. There is no notion of functions on hyperbolic three-space being holomorphic. It's not a complex space--it's got three real dimensions. It's this lack of being able to talk about things that are holomorphic that make this case and anything that isn't totally real harder.
WHAT ARE YOUR MAJOR RESEARCH GOALS FOR THE FUTURE?
Certainly at the moment I'm thinking about the same sort of questions. This solution of the Fermat conjecture got so much publicity, but in a sense it's only a small way towards the goal of working out this correspondence between representations of Galois groups and modular forms and their generalizations. There is far more left to be done than has been done. There have been some big steps forward, but compared to what's left to do, there is still an awful lot left to do. We're only scratching the surface. To a large extent, we feel confident that we know what's true, but we're very far from proving most of it. It's very tantalizing, this big, beautiful picture that we can't get our hands on.
ARE YOU COLLABORATING WITH OTHER FACULTY AT HARVARD?
At the moment I'm working by myself, but I've only just arrived here. It's certainly a great place to do this sort of this thing. Barry Mazur, Dick Gross, Noam Elkies--you couldn't ask for a better group of colleagues in our subject.
WHAT COURSES DO YOU HOPE TO TEACH IN THE SPRING?
In the spring I'm teaching Math 121. I've yet really to discover what's in the course or anything. I am looking forward to teaching math majors in future years, and I'm sure I will. I'm sure I'll teach a variety of things, algebra, algebraic geometry, number theory. I'm sure I'll be teaching graduate courses.
DO YOU PLAN ON ADVISING ANY UNDERGRADUATE STUDENTS FOR THESES?
Chris Degni came to see me. He's doing something on some conjectures of Serre in this area. Senior theses are something that doesn't exist in England. It's a concept that's new to me, so I'll have to learn what's expected. Four graduate students have moved with me from England here, so I have four graduate students. Three of them that are relatively early in their graduate career and will get Harvard degrees, the fourth is in his final year and will get an Oxford degree.
WHY DID YOU CHOOSE TO COME TO HARVARD?
I guess I got the formal offer in the spring from the dean, but we'd obviously talked about it with the faculty here for some time before that. One strong personal reason is that my wife's American and would like to be in America. Also it's a great department. Like I say, it's difficult to imagine a better collection of colleagues in my subject than there is here. By all accounts, the students here are very bright. I don't really have personal experience, but I'm sure it's true.
WHAT DO YOU LIKE/DISLIKE ABOUT HARVARD LIFE?
I actually visited for six months a couple of years ago, and one thing I like is the sun. Somehow in Britain for half the year, it's extraordinarily dark. That's partly because it's further north and partly because there is more cloud cover. I've heard people complain that in the winter it's cold here, but at least you see the sun. And I like the energy; people are very energetic and enthusiastic here. Something I noticed is that in Britain it's cool to pretend you never do any work. Students there obviously do work because they learn the same stuff as anybody else, but they like to pretend they do nothing. Whereas here, people in Princeton would come to me and tell me they had spent the last twenty-four hours in the library. Here, they seem to pretend they work harder than they do. I suspect that people work the same in both places; it's just the gloss they put on it.
This department is an extremely friendly department. People just seem to talk to each other more than they do in many places.
HOW DOES THE AMERICAN SYSTEM OF EDUCATION COMPARE TO THE BRITISH
Undergraduates in England usually study one subject. Most mathematics students in Cambridge are only studying mathematics; they spend 100% of their time studying that. This makes teaching there a different experience from teaching people who are studying mathematics as part of a broader education. I have the impression that most teaching here is done in middle sized classes. In Cambridge there is a combination of very large classes--100 people or so-- or very small classes where one or two students meet with one professor. The continuous assessment is also different. In England, the only assessment is at the end of the year. Through the year you get no grades at all, and everything depends on how your perform over two days during the large exam at the end. I don't yet have enough experience with the American system to know which I prefer, but there are these differences of style.
WHEN DID YOU BECOME INTERESTED IN MATH?
Very early, I suspect. My father is a theoretical physicist. There was always a culture of mathematical science in the family. I don't remember exactly, but certainly as a teenager I was interested in mathematics. I just enjoyed reading recreational books on mathematics and trying to do math problems and finding out about more advanced mathematics. There wasn't any one thing that struck me as particularly interesting.
WHEN DID YOU FIRST DISCOVER YOU HAD TALENT IN MATHEMATICS?
Well, I guess already in high school it was clear that I was better than most of the other kids in mathematics. But as you go on, you're always mixing with people who are more talented in mathematics. It is never clear if you have a real talent or just appear talented in the group you are currently mixing with. I really enjoy mathematics. I think great interest in mathematics and determination to persevere accounts for more than people often give credit for. If you are very keen on working on mathematical problems, you usually get good at it, and I think this can make up for a fair amount of mathematical talent. I have certainly know people who are far brighter mathematicians than I am, but if they have thought about a problem for two days and can't solve it, they get bored with it and want to move on. But that is not a recipe for good research; you have to just keep going on and on.