[Update: this post is now complete.]
On my way to the ICM I bought my first ever digital camera. From the quality of the photo below, you may not be surprised to hear that it is my first, though actually I have taken some good photos with my wife’s — I just couldn’t seem to get mine to take decent photos in the windowless main hall of the convention centre, which was not very light but had screens that were lit in a way that made the rest of one’s photos come out dark. Also, I took this photo from a distance that meant that even with the zoom on full I had to crop it quite a bit to get what you see below. But that’s enough excuses — I also want to celebrate the first ever illustrated post on this blog. The picture shows Smirnov and Kesten just before the first of five talks given by a new Fields medallist or Nevanlinna prizewinner: Smirnov to give the talk and Kesten to introduce it.
My small experience of Smirnov’s talks has been very positive, and this was another good talk. But by this stage of the congress I was no longer trying nearly so hard to take enough notes to be able to remember the talks afterwards. He opened by saying that he would be talking about discrete complex analysis, which, if I remember correctly, was also part of his title. He mentioned the well-known theorem of Brooks, Smith, Stone and Tutte, and more particularly its beautiful proof using electrical networks (BSS&T were undergraduates when they found it), that a square can be decomposed into smaller squares, no two of which have the same side length. In general, he discussed how one might discretize the notion of analyticity so that it could be applied to functions on planar graphs, and drew attention to a very important criterion: one would like one’s definition to have a scaling limit, which means, roughly speaking, that if you let the mesh tend to zero then your function will tend, in some appropriate sense, to an analytic function on the plane. The climax of the talk was Smirnov’s remarkable result that the Ising model has a conformally invariant scaling limit and that it is universal. Again I didn’t catch the precise universality statement, but you don’t have to to know that it is a big deal.
I was slightly nervous during the talk because it was scheduled to end at 2.45 and the panel discussion I was chairing was scheduled to begin at 3.00. That in itself wasn’t a problem, but I had met only one of my co-panellists, Bill McCallum, whom I first met in 1994 when we were both at IHES for long visits. I had bumped into Bill a couple of times here in Hyderabad, which took the edge off my nerves somewhat. Incidentally, since I have just discussed Smirnov’s talk, perhaps it is a good moment to discuss a stochastic process that you will know well if you have ever been to an ICM: the process of meeting people you know. I myself, and I expect this behaviour is typical, made very few formal rendezvous (what on earth is the plural of rendezvous? — I mean it to be pronounced RONDAYVOOZ), and instead relied on chance meetings. These ought to be something like a Poisson process, and a nice aspect of this ICM was that the mean seemed to be fairly small. A partial list, off the top of my head, of people I knew (not necessarily well) and regularly bumped into is Bill Barton, Alex Bellos, Marianna Csornyei, Irit Dinur, Marianne Freiberger, Martin Grötschel, Roger Heath-Brown, Elon Lindenstrauss, Laci Lovász, Bill McCallum, Rob Morris, Assaf Naor, Oliver Riordan, Stas Smirnov, Benny Sudakov, Anatoly Vershik — I’m sure I’m missing some obvious people. But the point is that everyone would have had their own neighbourhood in the I-know-you graph (or is it a directed graph?) and the 3000 people at the congress were bound together by these chance meetings.
The reason I was nervous about the panel was twofold. The panellists were instructed to give short presentations of about ten minutes each. In my position as chairman, I had decided that the best way to fill two hours without the audience falling asleep would be to divide them into three portions, each with two ten-minute presentations and twenty minutes of discussion and questions. But I had no idea whether this format would work or whether I or any of the other panellists had anything worth saying. (In general, when people asked me about the panel discussion I advised them to go to the real mathematical talks that were going on at the same time.) My other source of anxiety was that there might be technical problems: I had uploaded my slides on to the ICM website, but if anything had gone wrong then I didn’t have a back-up plan (apart from a stick in my pocket that the organizers probably wouldn’t allow me to use in case of viruses).
In the event, my worries were unfounded. In particular, I found my co-panellists’ contributions much more interesting than they might have been, and everybody was good about sticking to time (another worry that I had as chairman). Here are a few highlights.
The first to talk was Carlos Bosch, who told us of a very amusing and interesting experiment they had done on mathematics teachers in Latin America. They asked the following question. Pablo buys a horse for $10,000, sells it again for $12,000, buys it again for $14,000, and finally sells it again for $16,000. Did Pablo make a profit or a loss, and if so, how much of one?
Apparently, only a very small percentage of answers were correct. Two common wrong answers were as follows.
1. Pablo bought the horse for $10,000 and sold it for $12,000, making a profit of $2,000. But then he bought it again for $14,000, so he made a loss of $2,000. Finally, he sold it for $16,000, making a profit of $2,000. Adding up the profits and loss you get $2,000.
2. Pablo started off buying the horse for $10,000 and ended up selling it for $16,000 so at the end of the whole process he had made a profit of $6,000.
But that wasn’t the end of the story. In an effort to get the point across to the schoolteachers, the experimenters said, “Ah, but imagine that Pablo had bought two horses, one for $10,000 and one for $14,000, and had sold them again for $12,000 and $16,000, respectively.” To this apparently excellent explanation, a typical response was, “You’ve changed the question! Now you’re asking about two horses whereas last time you asked about only one horse.”
That gave us some idea of the challenges facing us mathematicians if we want the whole world to see the light.
But the positive side of the story was that in Mexico there has been an initiative where professional mathematicians get together with mathematics teachers and show them how they think, by discussing the solutions to various problems, and more importantly the processes by which they reach these solutions. It seems that even after a fairly small number of sessions, the results of those taught by the teachers improve dramatically — not just in mathematics but in Spanish as well.
Next was Ivan Yashchenko, who told us how a mathematics education in Russia was much more problem-oriented than in most countries. The syllabus content took a back seat. He was fairly pessimistic about the quality of Russian mathematics education, but his pessimism was directed at the education of the masses rather than of the elite — the very best mathematicians in Russia, it seems, are still receiving an education that is as good as it has always been. He told us a nice problem that has been used in mathematics circles in Russia: you take a square grid — say ten by ten. Then round the boundary you put an arrow in each square, going in the direction that corresponds to going clockwise round the boundary. (At the corners it doesn’t matter which of two possibilities you choose.) Now the object is to fill in all the remaining squares with horizontal or vertical arrows in such a way that no two neighbouring squares — including diagonal neighbours — have oppositely pointing arrows. Or rather, the object is to prove that this cannot be done.
This post isn’t finished, but since I’m not going to be able to finish it for over 24 hours I think I’ll post an incomplete version for now and finish it later.
Later. During the over 24 hours, as it turns out, since I broke off from writing I have indeed thought of several more mathematicians who were part of my acquaintance neighbourhood at the ICM (and I still don’t expect it to be a complete list): John Ball, John Coates, Ingrid Daubechies, Gil Kalai, Gyula Katona, Peter Markovic, Jaroslav Nesetril, Gilles Pisier, Richard Thomas, Ulrike Tillmann, John Toland and Cédric Villani. One thing I noticed was that the vast majority of the people I knew had to be there for one reason or another — either they were speaking or they were high up in the IMU. Since they were also a very small sample of the mathematicians that I know comparably well, I want to ask why so few people decide to go to the ICM just for the fun of it. Of course, the number of delegates is large, but the proportion of mathematicians from faculties at major universities round the world, say, seems to be pretty small. I think quite a lot of people just don’t really like ICMs all that much, and others were perhaps reluctant to go all the way to Hyderabad, organizing flights, visa, hotel, etc. But perhaps there are further reasons.
Going back to the panel discussion, the next person to speak was Heinz Steinbring, who I soon realized was a man after my own heart when he advocated giving examples first. He also recommended another principle, for which I do not have a good slogan but which I strongly believe in, which is to teach theory having first established the need for it by means of a problem. An example I have in mind for this is matrix multiplication. If you have got to the point where somebody understands how matrices related to linear maps — in two dimensions, say — then it is not a hugely difficult exercise to work out the matrix that gives you the composition of the maps defined by two other matrices. And if you’ve been through that exercise, then matrix multiplication seems much less bizarre and arbitrary.
Next was Ramanujam. (He liked to be known just by that one name.) The thing I remember best from his presentation was a beautiful list of thought processes that all research mathematicians would agree are essential to mathematics — things like generalizing a problem to make it simpler — and that he claimed, I think rightly, were almost entirely absent from school mathematics.
It occurs to me that I may not have mentioned that the title of the discussion was “Relation between the discipline and school mathematics.” “The discipline” referred to research mathematics. I also haven’t said much about the questions from the floor. Some of these were quite hard to understand (in which case I would look expectantly at my co-panellists in the hope that one of them would have a go at an answer), but several of them were interesting. One that I remember best was a suggestion that professional mathematicians should write school textbooks, which is something I’ve vaguely (VERY vaguely) thought of doing at some point.
Bill McCallum gave a very interesting and entertaining presentation about quadratic equations, a mixture of history (the discovery of the solution, the difficulties that were involved, and so on) and current affairs (a debate in the British parliament about whether it was worth teaching children how to solve them). And I gave the last presentation, in which I argued that university-style analysis with epsilons and deltas is not as far removed from school-style analysis as is often supposed.
Once the final question was over, I found myself whisked to some chairs at the back of the hall, where I was interviewed, mostly about the Polymath project. That lasted for nearly an hour, and in particular ruled out going to an invited talk between 5 and 5.45.
From 6 to 7.30 was a performance of classical Indian dance in the main hall. I decided I would go to it but not necessarily stay for the whole thing. The criterion I applied was that I would stay until the information content was close to zero. By that I mean that at the beginning I didn’t have all that much idea what to expect, but as the performance went on it all started to seem fairly similar (which was of course a reflection of my ignorance of what was going on, but given that ignorance the rate of extra information I was receiving was small). Anyhow, I stayed for half an hour, and then came back for the final quarter of an hour, having worked on a blog post in between. The performance began with a man who had a kind of tragic clown face on his own on the stage. I should perhaps say, after the rather dry words about information content, that I enjoyed what I saw — the very characteristically Indian shapes that the dancers made with their bodies, which again reminded me of Indian sculptures. I have another photo, again not a good one but it gives some impression of what it was like.
After that was the conference banquet, something that happens at every ICM and has a different style each time. I think the most memorable was the one in Beijing, where they had a room that was large enough for everyone to sit down (at round tables). This was memorable for the logistics of getting three thousand people from the convention centre to the different place where the banquet was held, in buses that took about twenty people each. A huge mass of delegates assembled just outside a (large) side entrance to the building, and a sort of semi-orderly queue formed. We were told many times please to be patient, which by and large we were, but a little bit of impatience right at the end seemed to be necessary to get on a bus — those who were too well-behaved would simply stand and watch as thousands of other delegates swarmed past. I was one of the first to go, by which I mean one of the first two or three hundred to go, and to my surprise we were set down next to something that I had passed each day in the bus on the way to the convention centre and found interesting. I don’t know how to describe it but it had pink walls and the phrase that jumped into my mind was “Hindu park.” Anyway, it was a sort of park, and it had a slightly Hindu feel, I thought, quite possibly wrongly. We walked about two or three hundred yards through it until we got to a sort of marquee, which had a covered corridor leading to it. I was relieved to get there because it was starting to rain. (The weather forecast I looked at before leaving predicted heavy rain for the entire week, but although there were one or two very heavy showers they were also quite brief and I was lucky enough never to be caught by one.)
I’ll finish the description of the banquet tomorrow, and then I’ll have finished day two.
Banquet. Come to think of it, I can’t remember whether they actually called it a banquet. Perhaps it was just the Conference Dinner. Foodwise, it was pretty similar to what I had been getting used to from various receptions, breakfast, and so on. For instance, bits of chicken were handed round, and various familiar Indian were lined up in chafing dishes at tables round the edge of the marquee (which was open at the sides). However, there were so many people at this dinner that there were long queues for the real food.
After a while I decided to brave one of them, and when I got to the front about fifteen minutes later (and glared, with absolutely no effect, at people who just walked up to the front and queue barged), I found that I was in the queue for vegetarian food. I asked for a bit of that and then tried to sidle over to a meatier set of chafing dishes. But it didn’t work, with the result that I ended up with a half-full plate of vegetarian food and not much prospect of supplementing it. I went back to the people I had been chatting with, who were standing round a little heap of conference bags and discarded outer clothing — it was getting extremely hot with all the people there — and continued to chat until somebody said that there was a heavenly world just a few yards away from us. That is, there was another marquee, just like the one we were in but cooler, with fewer people, with an abundance of readily accessible food. It sounded too good to be true, but it was true. We went up another corridor (red carpet below, tented roof) to a higher level where to the left was another large marquee that was just as described. The one thing that kept it earthly was that the half plate I had eaten had taken the edge off my appetite. But it was worth it for the reasonable temperature alone.
Getting back to the hotel was another logistical miracle. At some point I sensed that it might be a good idea to make my way outside and try to find out how the system would be working. I asked someone who appeared to know, who told me to go a bit further and wait for somebody carrying a placard with the name of my hotel on it. I decided to take a little bit of initiative, and walked out of the park and into the street (which had, now as before, a rather sulphurous smell) where a mass of buses had congregated. I walked down, looking for one that had the name of my hotel in its window, and eventually found it a long way back. I got in, and was the first in, which enabled me to choose the one seat that had room for my legs: right at the back with the aisle in front of me rather than another seat.
The one other thing to report about the day was that after the rest of the bus had filled up apart from the rest of the back row, three people got on. There were four seats in the back row, and I was in the second from the left (as you looked forward). The bus had single seats on the left and pairs on the right. OK, so obviously I had to share the back row with those three people. But I made a mistake: I moved my legs to the left, thereby gently encouraging the first person, a young woman, towards my right. Except that, thinking about it now, it wasn’t really a mistake, because if I had sent her the other way it would have been forcing her to sit next to me when she had had the chance not to. If you don’t understand why I couldn’t have done that, then perhaps it’s a British thing.
Anyhow, it turned out the other two people were a couple, and they, quite reasonably, asked if I would move so that they could sit together. Had I, by getting there way before them, built up a right to leg room that exceeded their right to sit together? I don’t know. Again the most my Britishness would allow me was just the faintest hint of reluctance and the tiniest gesture of discomfort during the journey.
I find that one thing I lose at conferences is discipline about going to sleep. I often get very tired and think, “I must have a good night or I’m going to have some kind of collapse.” And I’m quite good at getting myself in a position where a good night is possible. And then I write one of these blog posts — which is OK, because there’s still a reasonable amount of night left — and then, just to help me get in a sleepy mood, I … er … switch on the TV and discover that there’s a not very well known film on starring someone like Mickey Rourke. As I watch further, I come to realize that it is fairly near the beginning …
That’s not exactly what happened this time, but I didn’t do a great job catching up on the sleep I badly needed to catch up on. And the next day I absolutely needed to be at the convention centre by 9am, because the first talk was to be given by Artur Avila and I didn’t want to miss that.