Next, Kesten shuffled on to the stage. He has an unusual face, in that he has a white beard of the kind that looks as though it is never trimmed — indeed, I think that is probably the case, given the way it grows out sideways as well as down — and looks slightly fake, to the point where one cannot help imagining what he would look like without it, to which the answer is that he would probably look a lot younger as the hair on the top of his head is black. (As I write this, in a large room with dozens of terminals, he has just walked in.)
He began by telling us that Smirnov had got perfect scores in the International Mathematical Olympiads in 1986 and 1987, just in case we were in any doubt about his mathematical talents. His next remark came as a very pleasant surprise: even since the committee had made its decision — in fact, in the last two months — Smirnov had proved a major result. Let me begin by saying what that was.
A self-avoiding walk is a path in a graph that never visits the same vertex twice. (I suppose that is what a graph theorist would just call a path, but here we are thinking of random walks that are conditioned on avoiding themselves.) Self-avoiding walks lead to some of the most fascinating open problems in mathematics, since (i) much of their behaviour is extremely well-understood and (ii) very little of this understanding is rigorously proved. For example, it is known to physicists that a typical self-avoiding walk in of length has end-to-end distance about but nobody has rigorously proved an upper bound of for any and even more remarkably, nobody has proved a lower bound of even (I’ll leave it as an exercise to see that the “obvious counting argument” doesn’t work, even though the idea of its not working is fairly preposterous.)
One thing it is easy to show is that the number of self-avoiding walks of length starting at the origin is of the form where is a function that grows subexponentially. Equivalently, if is the number of walks of length then tends to a limit (Sketch of proof: it is easy to see that for all and and it is a general fact that submultiplicative functions have the desired property.) The constant is called the connective constant of the lattice on which the walk is taking place.
In theory, you can work out the connective constant to any desired accuracy by working out what is for larger and larger However, in practice the calculation gets pretty big pretty quickly. Exact values of have not until now been discovered, but Smirnov and his former (I think) student Hugo Duminil-Copin have proved that the connective constant of the planar hexagonal lattice is Apparently the result itself was the predicted answer, and it had been shown that was an upper bound, but proving the opposite inequality was much harder. Where does a strange number like come from? This may be obvious to some people but it wasn’t to me. Fortunately, I bumped into Smirnov later and he told me it was When I told him it wasn’t, he went very pale indeed …
Actually, that’s not true. He corrected himself and said As I think about it, I see that even that is false, but that makes me not quite sure that that is what he said. More likely, it is closely related to in some way. That’ll be a nice problem to work on if I’m short of something to do during a boring talk. (Incidentally, the talks I’ve been to so far — I’m writing this before the first talk on Sunday — have not been boring at all. I’ll try to write about some of them in due course.)
I should add that even in cases where physicists don’t know the connective constant they can still tell us the extraordinary fact that the right behaviour of is Fascinatingly, the is independent of the lattice, and in fact independent of a whole lot more — basically, any model of self-avoiding walks that is even remotely reasonable leads to this term. This is the mysterious phenomenon of universality. One of the holy grails of mathematics is to prove universality, which basically says that if you stand back from a self-avoiding walk so that the mesh of the lattice is too small for you to see it, then you will not be able to tell what the model is. Universality appears in many other contexts too, and non-trivial universality results are always hugely interesting. (The central-limit theorem can be viewed as such a result. And if you think about it, it is remarkable that a random walk in should have a distribution that, in the large scale, is rotation invariant, which is just one small consequence of universality.)
I said that hearing about this result was a pleasant surprise. I’ve explained the surprise part, I hope. The pleasantness was simply that as a member of the committee one wants reassurance that one has made the right choices (to the extent that the notion of “right choices” makes any sense, which is an important qualification). Part of the point of the medal is to reward promise, so I greet with grateful amazement the fact that Smirnov has been fulfilling his promise in a big way even before the ICM. (I’ve often wondered how the members of the committee that chose Witten felt when the Seiberg-Witten invariants burst on to the scene. Up to then there had been some criticism of the choice, on the grounds that Witten did not tend to publish formal theorems. Afterwards, the critics were silenced.)
I’ve got to get to David Aldous’s plenary talk, so I’m going to speed up. The other highlights of Smirnov’s mathematical output are that percolation on the triangular lattice has a continuum limit that is conformally invariant (which then has amazing consequences thanks to the work of Lawler, Werner and Schramm). See Tao’s blog post for more on this. And more recently Smirnov has proved conformal invariance for the Ising model in the plane — another truly remarkable result. And if I understand correctly (this certainly seems to be claimed in Julie Rehmeyer’s account) he has proved universality in this case, jointly with Dmitry Chelkak.
In case you are not familiar with it, the Ising model is a model for the behaviour of arrays of magnetic particles. There is a tendency for these particles to want to align with each other, but there is also a “temperature” parameter that goes in the opposite direction. (One can think of it as follows: the more the particles are vibrating, the less strong is their tendency to align.) To model this, one takes a grid of numbers, each equal to 1 or -1, and associates with it an energy, which is the number of neighbouring pairs of numbers that are different. Call this number One then chooses a parameter and defines a probability distribution on the set of all possible assignments of 1s and -1s, in such a way that the probability of any given assignment is proportional to To understand this, it is helpful to look at the extreme cases. If then all probabilities are the same, so we have the uniform distribution, and there is no tendency for neighbouring signs to be the same. This corresponds to the case of infinite temperature (and indeed, the temperature is thought of as ). If is very large, then the probability is very heavily weighted in favour of low-energy configurations, which means that one expects to see big areas of 1s and big areas of -1s. For intermediate low-energy configurations are also heavily favoured, but this is compensated for by the fact that there are far fewer of them, so a typical configuration will not obviously have low energy — whether it does depends on the value of And there is a critical value of where the system changes from having big patches of 1s and -1s to being much more random in appearance. The behaviour of the Ising model near the critical probability is particularly interesting, and exhibits the kind of conformal invariance and universality that gets people excited. Up to now, we have “known” this without actually knowing it. Thanks to Smirnov and his collaborators, we know it.
More on Ngo.
I received an email last night from Kartik Prasanna, who was a graduate student in Princeton when I was there as a visiting professor from 2000-2002 and is now at Ann Arbor, with an answer to Assaf Naor’s question about what the Langlands programme, if successful, would tell us about more down-to-earth matters such as solutions to Diophantine equations. With his permission, I reproduce his very interesting thoughts. (I don’t understand all the details, but the general points are made very clearly and cogently.)
[I] felt compelled to write in to respond to the question that Assaf Naor raised about whether the Langlands program is useful in any way in understanding Diophantine equations. Broadly, the answer is, yes, it should provide some basic tools to do so, but there is significant work required beyond ….
To describe what might be involved, it’s best to look at one concrete example where fantastic progress has been made – that of the Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves over the rationals of analytic rank less than or equal to one. (I’d say is as concrete as you could get !) In this case, the basic conjecture, namely that the analytic rank equals the algebraic rank is known to be true. The proof itself is a combination of many remarkable results, the main landmarks being (not in chronological order, so I’ve added the years these results were roughly proved ):
1. (1994) The theorem of Wiles ( and Taylor-Wiles, Breuil-Conrad-Diamond-Taylor) that elliptic curves are modular. This tells us that the p-adic Galois representation associated to the Tate module of an elliptic curve E is isomorphic to the Galois representation coming from a modular form for a particular congruence subgroup of In particular, this Galois representation occurs in the cohomology of an associated modular curve where is the so-called conductor of the elliptic curve ( a measure of its bad reduction.)
2. (1982) The theorem of Faltings on the Tate conjecture for divisors on a product of two curves, one of the results for which Faltings got a Fields medal. (He proved something more general, and also the Mordell conjecture in the same article.) This theorem implies from 1 above that there is an actual geometric map
of algebraic curves over the rationals, that is defined over the rationals as well.
3. (1984) The theorem of Gross-Zagier. They look at certain specific points on constructed using the theory of complex multiplication – the so-called Heegner points, and study the images of these points under the map Roughly what they show is that when the analytic rank is 1, then the image of a suitably chosen Heegner point is nontorsion, by proving a precise formula relating the derivative of the L-function to the “height” of the point. This implies that the algebraic rank is greater than or equal to 1, which equals the analytic rank.
4. Waldspurger’s theorem (1980) on nonvanishing on central values of L-functions of quadratic twists for modular forms on GL_2. This is a key ingredient in proving nonvanishing of the Heegner point above.
5. (late 80′s) The work of Kolyvagin on Euler systems that proves the reverse inequalities in the case of analytic rank 0 and 1.
What role did automorphic forms and the Langlands program play in this ?
(a) Certainly, a very important role in 1. For instance, the starting point for Wiles’ work is the Langlands-Tunnell theorem on modularity of certain Artin representations that uses Langland’s theory of base change for a particular case of the general Langlands’ conjectures.
(b) Waldspurger’s result is based on a deep study of automophic forms on and their transfers to metaplectic groups. Interestingly, this is not part of the original Langlands’ conjectures, since metaplectic groups are not algebraic. But there are close connections nevertheless.
(c) Most of the above can be generalized to the case of elliptic curves over a totally real field F. In this case, step 1 (due to Skinner-Wiles, Fujiwara etc.) would produce a Hilbert modular form, which under a suitable hypothesis admits a “Jacquet-Langlands transfer” to an automorphic form on the multiplicative group of a quaternion algebra B that is split at precisely one infinite place of F. The associated “Shimura variety” is actually a curve, which I will call X. Then Faltings’ theorem says there is a map
as before. Shouwu Zhang (around 1999) showed the analogs of 3 and 5 in this context. Again the Langlands program played a crucial role, in allowing one to move our modular form to a different group, namely the multiplicative group of the quaternion algebra B, where the geometry of the associated Shimura variety is simpler and particularly well adapted to constructing rational points.
You might note that the cases of the Langlands program used here were all those proved many many years ago. Also, while it provides some key inputs and basic tools, one needs to go significantly beyond what is predicted by the Langlands program in order to understand Diophantine questions. For instance, one of the key themes in the work of Wiles is the study of the INTEGRAL structure of Hecke algebras – these appeared in your post. One might say that the Langlands program is about understanding the eigencharacters of the Hecke algebra – in other words the structure of the Hecke algebra tensor Q – but it doesn’t say anything about their fine integral structure, the study of which requires understanding CONGRUENCES between modular forms. This study of congruences is in a sense one of the main themes of Wiles’ work, that led to the proof of 1.
Going forward, one might ask: what Diophantine questions might one study and what role will the Langlands program play ? A big open problem is of course the BSD conjecture for curves of analytic rank greater than 1. Another is possible generalizations to higher dimensional varieties. The analog of the BSD conjecture is the Bloch-Beilinson conjecture that predicts that ranks of Chow groups of homologically trivial cycles should be related to orders of vanishing of L-functions. These are difficult questions, but there are reasons to believe that progress on them will require some analogs of steps 1-4 above. Namely, let’s say one is interested in studying algebraic cycles on some variety V. Very speculatively, one would need:
1′. Modularity results – showing that under suitable conditions, the Galois representations associated with the cohomology of V come from automorphic forms that live on Shimura varieties. (This would only work for varieties closely related to Shimura varieties.)
2′: The Tate conjecture ! (for the product of the variety V and the Shimura variety X of part 1.)
3′: Studying special cycles on X and their relation to derivatives of
3”: Finally, study the cycles on V that one gets by correspondence via the Tate cycle of part 2′.
4′: Nonvanishing results for twists of automorphic L-functions.
5′: Some analog of the Euler system argument, which I cannot even begin to imagine …..
What is likely is that the Langlands’ program will provide some key basic tools in trying to understand these questions. However, the point I want to make is that as before, one needs to go SIGNIFICANTLY beyond the Langlands program to understand the rich arithmetic of Shimura varieties and their associated Diophantine geometry. So even if one assumed all the Langlands conjectures, it would just be a starting point. Nevertheless, if one doesn’t have the basic tools that are provided by the Langlands program, one cannot even begin in most cases.
By the way, there is all kinds of interesting work going on, especially on steps 1′ (eg. Taylor and his students) and 3′ (eg. Kudla and his collaborators, Shouwu Zhang and his students.) There is even a notion of an “arithmetic fundamental lemma” – see the paper of Wei Zhang by the same title on his webpage:
There is also lots of interesting related work from the p-adic side that would take too long to describe !