An Interview with Richard Taylor

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.)

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(\sqrt(-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(\sqrt(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.

This department is an extremely friendly department. People just seem to talk to each other more than they do in many places.