ICM2010 — Lindenstrauss laudatio

After the unceremonial closing of the opening ceremony it was lunch time. I soon found myself in the first of what would turn out to be many queues of the kind where the optimal strategy is far from obvious but obviously not very good. One of the army of volunteers (the males of whom were wearing smart purple Indian shirts — collarless and going down to the thigh) told me to go up a floor, so I did, and I found a number of parallel queues. There was also a borderline-legitimate queuelet round to the side, and I joined that, hoping that it was legitimate enough not to annoy people and not too far to the side to get the attention of the people serving lunch. Lunch was basically an Indian takeaway — a choice of various Indian dishes served in an aluminium tray and a white cardboard lid that was rectangular apart from being curved off at the vertices, and held in place by a fold at the top of the aluminium. (I’m not certain it was aluminium but that’s my best guess.) After about 15 minutes of anxious waiting I was in a position to ask for lamb biryani, which I had also had for breakfast. It cost me two coupons from my kit bag, and came with a small pot of yoghurt, and the option of a fierce looking chutney, which I took. (I was about to say “took with relish” but then realized that to some people that would be ambiguous.) I was told that if I wanted to sit down I could go into the gallery of the main hall, so I did. A table would have been nice, but I could manage without.

The lamb biryani was slightly disappointing in that it had just one piece of lamb which, though large, was about 80% bone. That left quite a lot of rice to get through, and I was glad of the opportunity to spice it up with the chutney. For what it’s worth, we were provided with small wooden disposable spoons and forks to eat with. The chutney passed a basic test: it made my nose run.

After I’d finished that I wandered about and accidentally found myself in a room where there was a press conference with the new prizewinners. Nobody seemed to mind my standing there at the back of the room (and I wasn’t the only one, though it wasn’t clear whether anyone else was there quite as randomly as I was). Spielman was in the middle of a spiel about codes and smooth analysis, which I’ll discuss a bit later. He was obviously extremely articulate in a way that makes me look forward very much to his talk.

When he had finished, Alex Bellos, a British mathematics popularizer and journalist (author of the recent Alex’s Adventures in Numberland, which he was forced to call Here’s Looking at Euclid for the US edition) asked two questions: what did the prizewinners think about the 40 age limit for Fields medals and the Nevanlinna prize, and did they believe in the distinction between pure and applied mathematics?

The answers were interestingly uniform. Yves Meyer said flat out that there was no distinction any more between pure and applied mathematics, which for someone working in the areas he has worked in is a very natural view to hold. He said that in the 16th century Montaigne, aged 40, had described himself as entering old age, and then pointed to Nirenberg, still active at 85, as a demonstration that the world has changed a lot since then.

Spielman basically agreed with Meyer. Lindenstrauss came nearest to disputing the claim that there was no pure/applied distinction (but in a context of having said that there wasn’t one) when he drew attention to the point that mathematics should not be judged by its immediate applicability. To his great credit, he didn’t wheel out the example of Hardy and the RSA algorithm. He also mentioned that Marina Ratner had proved her big theorem aged over 50.

Smirnov agreed that the boundary between pure and applied was blurred. (I much prefer this way of stating it to saying that there is no distinction. I think there is a distinction, one that can sometimes even be useful, but obviously I agree that the boundary is blurred. In that respect it’s a bit like the distinction between left wing and right wing.) He also mentioned applications to biology.

Nirenberg marvelled at how different parts of mathematics influence each other. On the age question he said, “I’m still trying.”

Villani said he didn’t have much to add as he agreed with everyone else. And I couldn’t hear Ngo Bao Chau’s answer.

After that I wandered out of the press conference again.

I got to the laudationes a little bit early to be sure of a good seat. I have, annoyingly, reached the age where my eyesight, which was previously extremely good, has started to go. Even more annoyingly, this was predicted by my optometrist in my last eye test but one. She said, “You’ll need glasses next time,” and went on to explain that she was saying that purely on the basis of my age. I felt as I might have felt if I had been told, “You’ll find that although you continue to prove theorems, they won’t be as interesting any more.” But she was right. (I hope, however, that the analogy breaks down right there …) I now have a pair of reading glasses, but I’m not in the habit of using them.

I’ve got them in India, but I hadn’t taken them with me, and was to be punished for it, since what I’m bad at is changing rapidly from looking at things that are close to things that are at a distance, which I had to do in order to listen to the laudationes and take notes on them at the same time. But I’m getting ahead of myself here.

I got my good seat, and then sat, and sat, and sat. The talks were due to start at 2. Somehow when nothing had happened by 2.05 I got a bad feeling about it, not so much because it was five minutes after the scheduled start time but because there didn’t seem to be any pre-talk activity — things like the president of the ICM up on the stage milling around talking with other very important people. At 2.20 there was an announcement: “Is Harry Kesten here please?” From where I was sitting, the answer appeared to be no. But even now I’m a little mystified because Kesten was on third, so his absence was not a good reason for delaying proceedings. Some canned music, which had stopped for the announcement, started up again. I had quite liked it at first — sort of Indian/jazz fusion — but it was on a cycle, and the third time round I was less keen.

Finally Jacob Palis, who was IMU president at the 1998 congress and was chairing this session, got up on stage, the music faded, there was a pause, and Palis said a few words … honour to chair session … writing names in history of mathematics … Ramanujan … and it was time for the first laudatio.

This was Furstenberg talking about Lindenstrauss. The full text can be found here so instead of trying to summarize the talk I’ll try to summarize my experience of the talk.

As Furstenberg climbed up on to the stage, there was a smattering of applause — why just a smattering I couldn’t understand. He fished out some papers, hunched a little over his laptop, looked puzzled, didn’t say anything for just long enough to get us worried, then started. He said that Lindenstrauss’s main achievements were in number theory and dynamical systems, in particular the progress on the Littlewood conjecture and on the solution of the quantum unique ergodicity conjecture of Rudnick and Sarnak. He then said, “I was expecting some of this to go on screen.” Eventually it did.

Furstenberg always speaks at a measured pace, and I was beginning to admire just how well-chosen his words were, but when his slides started working it was revealed that he was reading from them. This is perhaps a natural interpretation of the instructions to submit one’s talk in advance of the proceedings, and Furstenberg was by no means the only person to do it. But I think if I were organizing the next ICM I would instruct speakers to prepare slides that were distinct from, and much more schematic than, the full articles. However, if you want to, you can at least now find out exactly what Furstenberg said in his talk. One memorable quote was, “If it can happen, it will happen.” This is supposed to convey the philosophy of ergodic theory. (To be strictly accurate, he qualified it with the words “approximately, at least”.) As an example (this doesn’t come from Furstenberg’s talk), if you take a pair of real numbers and look at its multiples mod (equivalently, if you repeatedly translate the torus by ) then the closure of the orbit will consist either of a finite set of points (this happens if and only if some multiple of belongs to which happens if and only if both and are rational), or a closed curve in the torus (this happens, for instance, if and are irrational but their ratio is rational, which forces the multiples to lie on a line that goes through an integer point, and therefore forces their reductions mod 1 to lie on a curve that looks like a finite union of line segments if you draw it on the unit square), or the entire torus (this happens if and are irrational and independent over ). In each case, one can argue that there are certain obvious restrictions you can place on the orbit, which confine the orbit to some “nice” set, and then the orbit itself is equidistributed in that nice set.

This turns out to be a very simple example of a very general and important story. I am not competent to tell that story, but fortunately Terence Tao has blogged on it. (That is not the only post of Terry’s on that topic: another I would recommend is this one. More generally, you can click on promising tags such as “Ratner’s theorems” and get a list of relevant posts.)

I’ve got lots of notes on the talk, taken before I realized that the whole talk would be available online. But I think Furstenberg has done a beautiful job, so there really isn’t much point in my providing a different and worse version, especially as his full text is short. I’ve just noticed one thing that I can add to it. Towards the end, Furstenberg mentions that Hecke operators, which came into the work on the Littlewood conjecture, also come into Lindenstrauss’s solution of the QUE conjecture. In the written version Furstenberg just states this, but when he gave the talk he emphasized that it was very surprising that Lindenstrauss’s methods were applicable in this different-looking context. I feel this is an important point — one wants to hear words like “very surprising” or “technical tour de force” in a talk of this kind.

Perhaps I should also make clear that Lindenstrauss’s work is about groups that do not satisfy the conditions for Ratner’s theorem, and concerns hyperbolic systems, which Ratner’s theorem does not deal with. (I may have made that last statement too general, but in any case the area that Lindenstrauss works in is the area of groups where Ratner’s theorem breaks down but conclusions of a similar type still seem to be valid.)

Last night, I realized that my sleep strategy for the previous night had not been even close to optimal: I had had too much sleep on the previous night so had a bout of insomnia (which I normally never suffer from) last night. So now I’m back to feeling jet-lagged. I promised I’d be in bed by midnight and it’s now nearly quarter past, so the remaining laudationes will have to wait.