ICM2014 — Jim Arthur plenary lecture

The main other thing I did on day two of the congress was go to a reception in the evening hosted by the French Embassy. It was less formal than that makes it sound, and as I circulated I met a number of people I hadn’t seen for quite a while, as well as others I had got to know at the congress itself. The French ambassador, who was disconcertingly young looking, gave a speech, as did Artur Avila (as you know, Avila, like Ngo four years ago, is French), and one other person, whose name I’ve annoyingly forgotten. One interesting nugget of information to come out of those speeches was that Paris is planning to bid for the 2022 ICM. If that bid is successful, then Avila will have two successive ICMs in his home country. There was plenty of food at the reception, so, as I had hoped, I didn’t need to think about finding supper. When we arrived, we were asked for our business cards. In common with approximately 99.9% of mathematicians, I don’t have a business card, but for cardless people it was sufficient to write our names on little bits of paper. This, it turned out, was to be entered in a draw for a bottle of wine. When the time came, it was Avila who drew out the pieces of paper. Apparently, this is a Korean custom. There were in fact two bottles going, so two chances to win, which sort of became three chances when the person who should have won the first bottle wasn’t there to claim it. And so, for the first time in my life … not really. I have never won anything in a raffle and that puzzling sequence continued.

The next morning kicked off (after breakfast at the place on the corner opposite my hotel, which served decent espressos) with Jim Arthur, who gave a talk about the Langlands programme and his role in it. He told us at the beginning that he was under strict instructions to make his talk comprehensible — which is what you are supposed to do as a plenary lecturer, but this time it was taken more seriously, which resulted in a higher than average standard. Ingrid Daubechies deserves a lot of credit for that. He explained that in response to that instruction, he was going to spend about two thirds of his lecture giving a gentle introduction to the Langlands programme and about one third talking about his own work. In the event he messed up the timing and left only about five minutes for his contribution, but for everybody except him that was just fine: we all knew he was there because he had done wonderful work, and most of us stood to learn a lot more from hearing about the background than from hearing about the work itself.

ICM2014 — Barak, Guralnick, Brown

Here’s a little puzzle to get this post started. Of the fourteen 21st-century Fields medallists (if you count Perelman), seven — Lafforgue, Voevodsky, Tao, Werner, Smirnov, Avila and Mirzakhani — have something interesting in common that the others lack. What is this property?

That’s a fairly easy question, so let’s follow it up with another one: how surprised should we be about this? Is there unconscious bias towards mathematicians with this property? Of this year’s 21 plenary lecturers, the only one with the property was Mirzakhani, and out of the 20 plenary lecturers in 2010, the only one with the property was Avila. What is going on?

On to more serious matters. After Candès’s lecture I had a solitary lunch in the subterranean mall (Korean food of some description, but I’ve forgotten exactly what) and went to hear Martin Hairer deliver his Fields medal lecture, which I’m not going to report on because I don’t have much more to say about his work than I’ve already said.

By and large, the organization of the congress was notably good — for example, I almost never had to queue for anything, and never for any length of time — but there was a little lapse this afternoon, in that Hairer’s lecture was scheduled to finish at 3pm, exactly the time that the afternoon’s parallel sessions started. In some places that might have been OK, but not in the vast COEX Seoul conference centre. I had to get from the main hall to a room at the other end of the centre where theoretical computer science talks were taking place, which was probably about as far as walking from my house in Cambridge to the railway station. (OK, I live close to the station, but even so.)

ICM2014 — Emmanuel Candès plenary lecture

If you are a mathematical researcher, do you ever stop to ask yourself what the point is of all your research? Do you worry that the world could get along just fine without it?

One person who doesn’t lose any sleep over doubts like this is Emmanuel Candès, who gave the second plenary lecture I went to. He began by talking a little about the motivation for the kinds of problems he was going to discuss, which one could summarize as follows: his research is worthwhile because it helps save the lives of children. More precisely, it used to be the case that if a child had an illness that was sufficiently serious to warrant an MRI scan, then doctors faced the following dilemma. In order for the image to be useful, the child would have to keep completely still for two minutes. The only way to achieve that was to stop the child’s breathing for those two minutes. But depriving a child’s brain (or indeed any brain, I’d imagine) of oxygen for two minutes is not without risk, to put it mildly.

Now, thanks to the famous work of Candès and others on compressed sensing, one can reconstruct the image using many fewer samples, which reduces the time the child must keep still to 15 seconds. Depriving the brain of oxygen for 15 seconds is not risky at all. Candès told us about a specific boy who had something seriously wrong with his liver (I’ve forgotten the details) who benefited from this. If you want a ready answer for when people ask you about the point of doing maths, and if you’re sick of the Hardy-said-number-theory-useless-ha-ha-but-what-about-public-key-cryptography-internet-security-blah-blah example, then I recommend watching at least some of Candès’s lecture, which is available here, and using that instead. Then you’ll really have seized the moral high ground.

Actually, I recommend watching it anyway, because it was a fascinating lecture from start to finish. In that case, you may like to regard this post as something like a film review with spoilers: if you mind spoilers, then you’d better stop reading here.

ICM2014 — Ian Agol plenary lecture

On the second day of the congress I hauled myself out of bed in time, I hoped, to have a shower and find some breakfast before the first plenary lecture of the congress started at 9am. The previous day in the evening I had chanced upon a large underground shopping mall directly underneath the conference centre, so I thought I’d see if I could find some kind of café there. However, at 8:30 in the morning it was more or less deserted, and I found myself wandering down very long empty passages, constantly looking at my watch and worrying that I wouldn’t have time to retrace my steps, find somewhere I could have breakfast, have breakfast, and walk the surprisingly long distance it would be to the main hall, all by 9am.

Eventually I just made it, by going back to a place that was semi-above ground (meaning that it was below ground but you entered it a sunken area that was not covered by a roof) that I had earlier rejected on the grounds that it didn’t have a satisfactory food option, and just had an espresso. Thus fortified, I made my way to the talk and arrived just in time, which didn’t stop me getting a seat near the front. That was to be the case at all talks — if I marched to the front, I could get a seat. I think part of the reason was that there were “Reserved” stickers on several seats, which had been there for the opening ceremony and not been removed. But maybe it was also because some people like to sit some way back so that they can zone out of the talk if they want to, maybe even getting out their laptops. (However, although wireless was in theory available throughout the conference centre, in practice it was very hard to connect.)

ICM2014 — Khot laudatio

After McMullen’s laudatio on Mirzakhani, it was time for Sanjeev Arora to talk about the work of the Nevanlinna prize winner Subhash Khot. It was also the time that a significant proportion of the audience decided that enough was enough and left the room. The same thing happened in Hyderabad four years ago, and on both occasions I was fairly shocked: I think it shows a striking disrespect, not so much for the speaker and prizewinner, though there is that aspect too, as for theoretical computer science in general. It seems to say, “Right, that’s the interesting prizes over — now we’re on to the ones that don’t really matter.” Because I have always been interested in computational complexity and related areas, my interest in the Nevanlinna prize is comparable to my interest in the Fields medals — indeed, in some ways it is greater because there is more chance that I will properly appreciate the achievements of the winner. And the list of past winners is incredible and includes some of my absolute mathematical heroes.

When the announcement was made a few hours earlier, my knowledge of Subhash Khot could be summarized as follows.

1. He’s the person who formulated the unique games conjecture.
2. I’ve been to a few talks on that in the past, including at least one by him, and there have been times in my life when I have briefly understood what it says.
3. It’s a hardness conjecture that is a lot stronger than the assertion that PNP, and therefore a lot less obviously true.

What I hoped to get out of the laudatio was a return to the position of understanding what it says, and also some appreciation of what was so good about Khot’s work. Anybody can make a conjecture, but one doesn’t usually win a major prize for it. But sometimes a conjecture is so far from obvious, or requires such insight to formulate, or has such an importance on a field, that it is at least as big an achievement as proving a major theorem: the Birch–Swinnerton-Dyer conjecture and the various conjectures of Langlands are two obvious examples.

ICM2014 — Mirzakhani laudatio

I’m going to try the same exercise with Curt McMullen’s talk about Mirzakhani’s work that I did with Ofer Zeitouni’s about Hairer: that is, I’ll begin by seeing what I can remember if I don’t look at my notes. However, I remember disoncertingly little, and what I do remember is somewhat impressionistic.

The most concrete thing I remember (without being 100% sure I’ve got it right) is that one of Mirzakhani’s major results concerns counting closed geodesics in Riemann surfaces. A geodesic is roughly speaking a curve that feels like a straight line to an inhabitant of the surface. Another way of putting it is that if you take two points that are close together on a geodesic, then the part of the geodesic between those points is the shortest curve that joins those two points. (Hmm, on writing that I feel that I’ve made an elementary mistake of exposition, in that I have assumed that you know what a Riemann surface is, and then gone to a little trouble to say what a geodesic is, when not many people will know the former without also knowing the latter. To atone for that, let me add a link to the Wikipedia article on Riemann surfaces, though I’m afraid that article is not much good for the beginner. A beginner’s definition, not precise at all but perhaps adequate for the purposes of reading this post, is that a Riemann surface is a surface like a sphere or a torus, but with some very important extra structure that comes from the fact that each little patch of surface looks like a little patch of the complex plane.)

ICM2014 — Hairer laudatio

I haven’t kept up anything like the frequency of posts at this ICM that I managed at the last one. There are at least three reasons for this. One is that I was in the middle of writing up a result, so I devoted some of my rare free moments to that. Another is that the schedule was so full of good talks that I hardly skipped any sessions. And the third is that on the last day I was taken ill: I won’t go into too much detail, but let’s say that what I had sort of rhymed with “Korea”, but also left me feeling fairly terrible. So I didn’t much enjoy the conference banquet — at least from the food point of view — and then the next day, which I can’t quite believe was actually yesterday, when I got up at 5am in order to catch the bus from the hotel to the airport in time for my 9:30 flight back to Paris, I felt sufficiently terrible that I wasn’t quite sure how I would get through the 11-hour flight, four-hour stopover in Paris and four-and-a-half-hour train journey from Paris to Béziers.

I was rescued by an extraordinary piece of luck. When I got to the gate with my boarding card, the woman who took it from me tore it up and gave me another one, curtly informing me that I had been upgraded. I have no idea why. I wonder whether it had anything to do with the fact that in order to avoid standing any longer than necessary I waited until almost the end before boarding. But perhaps the decision had been made well before that: I have no idea how these things work. Anyhow, it meant that I could make my seat pretty well horizontal and I slept for quite a lot of the journey. Unfortunately, I wasn’t feeling well enough to make full use of all the perks, one of which was a bar where one could ask for single malt whisky. I didn’t have any alcohol or coffee and only picked at my food. I also didn’t watch a single film or do any work. If I’d been feeling OK, the day would have been very different. However, perhaps the fact that I wasn’t feeling OK meant that the difference it made to me to be in business class was actually greater than it would have been otherwise. I rather like that way of looking at it.

ICM2014 — Bhargava laudatio

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.

ICM2014 — Avila laudatio

As I said in my previous post, I don’t think I’m going to try all that hard to explain the work of the prizewinners, since it has been very well explained in other places (except that much more attention has gone to the Fields medallists than to the Nevanlinna prize winner — maybe I’ll try to redress the balance a little bit there). Instead, I’d just like to mention a few things that I found interesting or amusing during the laudationes.

The first one was an excellent talk by Etienne Ghys on the work of Artur Avila. (The only other talk I’ve heard by Ghys was his plenary lecture at the ICM in Madrid in 2006, which was also excellent.) It began particularly well, with a brief sketch of the important stages in the history of dynamics. These were as follows.

1. Associated with Newton is the idea that you are given a differential equation, and you try to find solutions. This has of course had a number of amazing successes.

2. However, after a while it became clear that the differential equations for which one could hope to find a solution were not typical. The next stage, initiated by Poincaré, was to aim for something less. One could summarize it by saying that now, given a differential equation, one tries merely to say something interesting about its solutions.

3. In the 1960s, Smale and Thom went a stage further, trying to take on board the realization that often physicists don’t actually know the equation that models the phenomenon they are looking at. As Ghys put it, the endeavour now can be summed up as follows: you are not given a differential equation and you want to say something interesting about its solutions.

ICM2014 — opening ceremony

I’d forgotten just how full the first day of an ICM is. First, you need to turn up early for the opening ceremony, so you end up sitting around for an hour and half or so before it even starts. Then there’s the ceremony itself, which lasts a couple of hours. Then in the afternoon you have talks about the four Fields Medallists and the Nevanlinna Prize winner, with virtually no breaks. Then after a massive ten minutes, the Nevanlinna Prize winner talks about his (in this case) own work, about which you have just heard, but in a bit more detail. That took us to 5:45pm. And just to round things off, Jim Simons is giving a public lecture at 8pm, which I suppose I could skip but I think I’m not going to. (The result is that most of this post will be written after it, but right at this very moment it is not yet 8pm.)