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THIS ISSUE:
* EXCLUSIVE INTERVIEW WITH RICHARD TAYLOR, including Math-123-level
overview of his and Wiles' proof of FLT.
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.
RICHARD TAYLOR, interviewed by Scott Sheffield
HOW LONG HAVE YOU BEEN AT HARVARD?
Just over two weeks.
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
SYSTEM?
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.