# Richard Taylor - research interests and achievements

In 1996, just after being appointed to Harvard University, Richard Taylor was interviewed by Scott Sheffield. See

http://math.mit.edu/~sheffield/interview.html

We give Taylor's reply to the question: 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 $\mathbb{Q}$, 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 realised 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 realised 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 recognised as important problems irrespective of this theory, problems that people had thought about before modular forms.

Last Updated November 2019