I ended up writing more than I expected to about Avila. I’ll try not to fall into the same trap with Bhargava, not because there isn’t lots to write about him, but simply because if I keep writing at this length then by the time I get on to some of the talks I’ve been to subsequently I’ll have forgotten about them.
Dick Gross also gave an excellent talk. He began with some of the basic theory of binary quadratic forms over the integers, that is, expressions of the form . One assumes that they are primitive (meaning that , and don’t have some common factor). The discriminant of a binary quadratic form is the quantity . The group SL then acts on these by a change of basis. For example, if we take the matrix , we’ll replace by and end up with the form , which can be rearranged to
(modulo any mistakes I may have made). Because the matrix is invertible over the integers, the new form can be transformed back to the old one by another change of basis, and hence takes the same set of values. Two such forms are called equivalent.
For some purposes it is more transparent to write a binary quadratic form as
If we do that, then it is easy to see that replacing a form by an equivalent form does not change its discriminant since it is just -4 times the determinant of the matrix of coefficients, which gets multiplied by a couple of matrices of determinant 1 (the base-change matrix and its transpose).
Given any equivalence relation it is good if one can find nice representatives of each equivalence class. In the case of binary quadratic forms, there is a unique representative such that or . From this it follows that up to equivalence there are finitely many forms with any given discriminant. The question of how many there are with discriminant is a very interesting one.
Even more interesting is that the equivalence classes form an Abelian group under a certain composition law that was defined by Gauss. Apparently it occupied about 30 pages of the Disquisitiones, which are possibly the most difficult part of the book.
Going back to the number of forms of discriminant , Gauss did some calculations and stated (without proof) the formula
There was, however, a heuristic justification for the formula. (I can’t remember whether Dick Gross said that Gauss had explicitly stated this justification or whether it was simply a reconstruction of what he must have been thinking.) It turns out that the sum on the left-hand side works out as the number of integer points in a certain region of (or at least I assume it is since the binary form has three coefficients), and this region has volume . Unfortunately, however, the region is not convex, or even bounded, so this does not by itself prove anything. What one has to do is show that certain cusps don’t accidentally contain lots of integer points, and that is quite delicate.
One rather amazing thing that Bhargava did, though it isn’t his main result, was show that if a binary quadratic form represents all the positive integers up to 290 then it represents all positive integers, and that this bound is best possible. (I may have misremembered the numbers. Also, one doesn’t have to know that it represents every single number up to 290 in order to prove the result: there is some proper subset of that does the job.)
But the first of his Fields-medal-earning results was quite extraordinary. As a PhD student, he decided to do what few people do, and actually read the Disquisitiones. He then did what even fewer people do: he decided that he could improve on Gauss. More precisely, he felt that Gauss’s definition of the composition law was hard to understand and that it should be possible to replace it by something better and more transparent.
I should say that there are more modern ways of understanding the composition law, but they are also more abstract. Bhargava was interested in a definition that would be computational but better than Gauss’s. I suppose it isn’t completely surprising that Gauss might have produced something suboptimal, but what is surprising is that it was suboptimal and nobody had improved it in 200 years.
The key insight came to Bhargava, if we are to believe the story he tells us, when he was playing with a Rubik’s cube. He realized that if he put the letters to at the vertices of the cube, then there were three ways of slicing the cube to produce two matrices. One could then do something with their determinants, the details of which I have forgotten, and end up producing three binary quadratic forms that are related, and this relationship leads to a natural way of defining Gauss’s composition law. Unfortunately, I couldn’t keep the precise definitions in my head.
Here’s a fancier way that Dick Gross put it. Bhargava reinvented the composition law by studying the action of SL on . The orbits are in bijection with triples of ideal classes for the ring that satisfy . That’s basically the abstract way of thinking about what Bhargava did computationally.
In this way, Bhargava found a symmetric reformulation of Gauss composition. And having found the right way of thinking about it, he was able to do what Gauss couldn’t, namely generalize it. He found 14 more integral representations on objects like above, which gave composition laws for higher degree forms.
He was also able to enumerate number fields of small degree, showing that the number of fields of degree and discriminant less than grows like . This Gross described as a fantastic generalization of Gauss’s work.
I spent the academic years 2000-2002 at Princeton and as a result had the privilege of attending Bhargava’s thesis defence, at which he presented these results. It must have been one of the best PhD theses ever written. Are there any reasonable candidates for better ones? Perhaps Simon Donaldson’s would offer decent competition.
It’s not clear whether those results would have warranted a Fields medal on their own, but the matter was put beyond the slightest doubt when Bhargava and Shankar proved a spectacular result about elliptic curves. Famously, an elliptic curve comes with a group law: given two points, you take the line through them, see where it cuts the elliptic curve again, and define that to be the inverse of the product. This gives an Abelian group. (Associativity is not obvious: it can be proved by direct computation, but I don’t know what the most conceptual argument is.) The group law takes rational points to rational points, and a famous theorem of Mordell states that the rational points form a finitely generated subgroup. The structure theorem for Abelian groups tells us that for some it must be a product of with a finite group. The integer is called the rank of the curve.
It is conjectured that the rank can be arbitrarily large, but not everyone agrees with that conjecture. The record so far is held by the curve
discovered by Noam Elkies (who else?) and shown to have rank 19. According to Wikipedia, from which I stole that formula, there are curves of unknown rank that are known to have rank at least 28, so in another sense the record is 28, in that that is the highest known integer for which there is proved to be an elliptic curve of rank at least that integer.
Bhargava and Shankar proved that the average rank is less than 1. Previously this was not even known to be finite. They also showed that at least 80% of elliptic curves have rank 0 or 1.
The Birch–Swinnerton-Dyer conjecture concerns ranks of elliptic curves, and one consequence of their results (or perhaps it is a further result — I’m not quite sure) is that the conjecture is true for at least 66% of elliptic curves. Gross said that there was some hope of improving 66% to 100%, but cautioned that that would not prove the conjecture, since 0% of all elliptic curves doesn’t mean no elliptic curves. But it is still a stunning advance. As far as I know, nobody had even thought of trying to prove average statements like these.
I think I also picked up that there were connections between the delicate methods that Bhargava used to enumerate number fields (which again involved counting lattice points in unbounded sets) and his more recent work with Shankar.
Finally, Gross reminded us that Faltings showed that for hyperelliptic curves (a curve of the form for a polynomial — when is a cubic you get an elliptic curve) the number of rational points is finite. Another result of Bhargava is that for almost all hyperelliptic curves there are in fact no rational points.
While it is clear from what people have said about the work of the four medallists that they have all proved amazing results and changed their fields, I think that in Bhargava’s case it is easiest for the non-expert to understand just why his work is so amazing. I can’t wait to see what he does next.
Update. Andrew Granville emailed me some corrections to what I had written above, which I reproduce with his permission.
A couple of major things — certainly composition was much better understood by Dirichlet (Gauss’s student) and his version is quite palatable (in fact rather easier to understand, I would say, than that of Bhargava). It also led, fairly easily, to re-interpretation in terms of ideals, and inspired Dedekind’s development of (modern) algebraic number theory. Where Bhargava’s version is interesting is that
1) It is the most extraordinarily surprising re-interpretation.
2) It is a beautiful example of an algebraic phenomenon (involving group actions on representations) that he has been able to develop in many extraordinary and surprising directions.
2/ 66% was proved by Bhargava, Skinner and Wei Zhang and goes some way beyond Bhargava/Shankar, involving some very deep ideas of Skinner (whereas most of Bhargava’s work is accessible to a widish audience).