Before I continue with brief descriptions of the laudationes, let me mention that Julie Rehmeyer has written descriptions of their work for a general audience and Terence Tao has now posted about the work of the Fields medallists and the other prizewinners. And as I have already said, the ICM website has links to the full texts of the laudationes themselves. So anybody now wanting to understand the mathematics has an excellent starting point, and I am free to concentrate on the more frivolous details of the talks, perhaps slipping in the odd mathematical comment as I do so.
Jim Arthur went next. His was the terrifying task (though much less terrifying for him than for most) of explaining the work of Ngô Bảo Châu to a general mathematical audience. I’d say that he did about as well as it is possible to do, which meant that he was able to convey some of the flavour, but obviously without managing to transmit to the non-expert the sort of wisdom tht it takes the experts in this particular area years to accumulate.
Ngô’s big result is a proof of the so-called Fundamental Lemma, conjectured by Langlands. Not bad to get a Fields medal for a lemma, you might think. The story here is that Langlands, while working out his famous programme, recognised the need for this lemma, and thought that it would be reasonably straightforward to prove. However, it turned out that he had wildly underestimated its difficulty. As Arthur put it, what Langlands was seeing was the visible part of the iceberg, and to prove the lemma it was necessary to uncover and understand the iceberg in its entirety. (I can’t remember his exact words, but he did not use the phrase “tip of the iceberg” and definitely did talk about icebergs.)
Arthur then gave a general description of the Langlands programme (having apologized in advance that much of what he was going to say would be “murky”). After the laudationes I found myself having coffee with Assaf Naor and Irit Dinur. (If you haven’t heard of them, they are both fabulous mathematicians, and both speaking here. In fact, Irit has just given her talk, the theoretical computer science plenary lecture, about the famous PCP theorem, of which she has a famous new proof.) The conversation turned to the topic of how many Fields medals could in theory be given for advances in the Langlands programme. My view was I suppose the official line, which is that it is such a deep and difficult area that any major advance is huge news, though I couldn’t resist a joke comparison to pole vault records, where people who are in a position to beat them deliberately don’t beat them by much because you get big money for beating world records. (I’m not seriously suggesting that somebody who had a proof of all the Langlands conjectures would sit on it, or release it only gradually.) Assaf (and I hope he won’t mind my making his views public) was more sceptical, maintaining that all mathematicians have their icebergs to explore and that the Langlands programme was not as unusual in this respect as perhaps it is sometimes conveyed as being. He said that he likes to ask the experts whether if they could assume all the results they wanted that are currently conjectural, they would know more about any concrete Diophantine equations. Apparently they don’t particularly like this question. Whether it is an appropriate criterion to judge the area is of course a matter for debate. In fact, that is what prompted me to say that perhaps the iceberg was the true and fascinating object of study in that area. I didn’t think of saying it at the time, but after a while there is not much interest in solving more and more Diophantine equations (not that Assaf was claiming that there was), and attention must turn to more global phenomena somehow. Perhaps that is what algebraic number theory is. I’m not sure why I’m musing on this at such length, but one more thought is that the question, “What is the most general statement of which Gauss’s law of quadratic reciprocity is a particular example?” is an obviously entirely valid and interesting one, and if I understand correctly, one of the Langlands conjectures is more or less a proposed answer to it.
I don’t understand what an automorphic form is, but there are levels of non-understanding (I would be enjoying several deeper ones later in the talk) and Jim Arthur lifted me to a slightly higher one — by which I mean that I had a slightly better idea what automorphic forms were after the next section of his talk. Before, I just thought of them as particularly nice kinds of functions that number theorists liked, and often mentioned in the same breath as modular forms (which I understand slightly better but still by no means fully). Anyhow, automorphic forms are eigenforms of natural operators on arithmetic symmetric spaces. He then said that these natural operators were Hecke operators, which themselves were Laplace-Beltrami operators on … er … I can’t remember. Hang on, we’ve got some spaces around — those symmetric spaces — so that’s OK.
What does one get out of a portion of talk like that? That is, what does one get out of hearing one concept one does not understand explained in terms of others? Let me try to say in this case. I don’t know exactly what an eigenform is but I presume it’s an eigenvector (and in fact he described them as simultaneous eigenvectors, so perhaps they were simultaneous eigenvectors for all Hecke operators — hmm, not sure about that). He talks about natural operators, which is obviously not meant to be precise and can therefore be understood in a non-precise way. Then he said “arithmetic symmetric spaces”. I don’t know what those are, though I imagine they are one of those definitions that is rather simple when you finally get told it. (I had that experience with algebraic groups, objects that I was afraid of until I learned that they were just groups where the set and the group operation are defined by means of polynomials.) I happen to know that a Laplace-Beltrami operator is what you get when you ask what the right analogue of the Laplace operator should be for a function defined on a manifold. (These last two definitions I know only as a result of editing the Princeton Companion to Mathematics, which forced me to pick up quite a lot of this kind of general knowledge.) And I’ve heard Hecke operators mentioned numerous times without ever actually finding out what they are. As a result of all that I now know that automorphic forms are eigenfunctions of operators that come up in a nice natural way in a number-theoretic context and that relate to all sorts of buzzwords I’ve heard many times. That doesn’t tell me exactly what an automorphic form is but it is non-trivial information. (What are they good for? There I cannot say anything that’s worth saying.)
Anyhow, the Langlands programme is about connecting automorphic forms with representation theory and looking at objects called automorphic representations. The difficulty of the area is this. One has some nice concrete operators (the Hecke operators mentioned above) and would like to know about their eigenvalues. However, just because an operator is concrete, it doesn’t mean you can write down its eigenvalues, and in this case you can’t. However, what you can hope to do is relate automorphic representations for different groups to each other, and this, if you manage it, gives you very deep reciprocity laws.
There’s something else called the principle of functoriality, which I won’t attempt to describe here even vaguely. Jim Arthur said, “The principle of functoriality awaits the efforts of future Fields medallists.”
The slides were getting more and more difficult to get anything out of by this stage, and I think I won’t say very much more. But Arthur gave us some idea of why the Fundamental Lemma has so many interesting consequences, and then started to explain what was so remarkable about Ngô’s proof.
If you want to impress your friends, here’s how to pretend you understand the proof in detail. If someone asks what his main idea was, you can reply, “Well, his deepest insight was to show that the Hitchin fibration of the anisotropic part of the trace formula is a Deligne-Mumford stack.” If that doesn’t do the job, then try to drop the phrase “perverse sheaves” into the conversation — they are relevant apparently. If you want to show that you have a broad view, then you could also say that Ngô very remarkably used global methods such as Hitchin fibrations to prove a local theorem. If you’re looking for a single amazing idea, then probably the use of Hitchin fibrations was it. Finally, here’s a list of names to splash about: Goresky, Hales, Kottwitz, Langlands, Laumon, MacPherson, Shelstad and Waldspurger. (This is apparently a far from complete list of the people on whose work Ngo builds.) A final summary from Jim Arthur: Ngô’s work opens up automorphic forms to some wonderful applications.