I’ve entitled this post “fourth day” in an attempt to encourage myself to write less and get this account finished: with each passing day I find that more has slipped out of my mind (for instance, there are several hours of this day that I no longer remember anything about), and in any case the fourth day of a nine-day conference that ended last week is hardly hot news any more. Having said that, I have tried the trick with several previous posts in this sequence and been forced to change their titles.
Yet again the organizers gave the first slot of the day to a speaker I couldn’t bear to miss — David Aldous, one of the world’s very top probabilists. So yet again I arrived exhausted at the convention centre. Incidentally, here is a photo (from the second day, as it happens) that shows what arriving at the convention centre looked like. If you look closely you’ll see that there is a dramatic gender imbalance: that is because the “ladies” had been told to go to a different queue. At first I was extremely surprised by this, but there was a simple reason for the segregation: the male queue had a male frisker and the female queue had a female frisker. You can also just make out the airport-like metal-detecting cuboid skeletons we had to walk through on entering the building.
David Aldous did not disappoint me — he was definitely worth getting up for. (I think it was on this day that I finally overslept for the first time. That meant that I got up half an hour after my alarm had gone off, and had to get dressed in a hurry, miss breakfast, etc. But it might have been the day before.) There were three things that I wrote down, and one other that I remember. What I remember but did not write down was that he brought up a subject that I have encountered twice before: mathematicians who are not probabilists don’t know how to think about random variables. What did he mean by this? Well, we mathematicians like to define them as measurable functions on probability spaces — that makes us feel comfortable because that sort of reduces the probability to a nice simple piece of pure mathematics. But for some strange reason probabilists don’t want to get rid of the probability.
The first time I became aware that there was a problem here was when I was editing an article by James Norris, a colleague of mine at Cambridge, on probability distributions for the Princeton Companion to Mathematics. I think I did some rewriting that introduced the wrong way of thinking into his article, and had real difficulty in understanding what it was that he objected to. But eventually I sort of got there, and here’s one way of looking at it (related to what he said to me during our conversation, but if anyone objects to anything I say then it’s probably my fault for not doing justice to James’s arguments).
Suppose that is a random variable that is uniformly distributed on What, mathematically speaking, is ? Well, a simple answer would be to say that we can put Lebesgue measure on and that is just the function defined on the probability space that takes to So far so good. Now let be another random variable that is uniformly distributed on What is Well, surely whatever worked for must work for so surely is also the function on that takes to
But hang on, we don’t want to say that any two random variables that are uniformly distributed on are equal do we? Something has gone wrong. So let’s try again. Perhaps we ought to say that and are any two functions that are defined on and have the property that the set of values such that the function belongs to a measurable set always has the same measure as
OK, so let’s take an important case, where and are independent. What are the functions then? It turns out to be possible to define two functions such that for any measurable sets the probability that and is the product of the measures of and But it is not that easy, and it is not what we would ordinarily want to do. It is much easier to take the probability space to be and to define to be and to be
But in that case, if someone says, “Let be a random variable that is uniformly distributed on ,” what are we to think? One possibility is to say that is a function from some probability space that we don’t actually know until we’ve seen the entire problem, and in particular have found out whether it involves other random variables. But a much better approach is just not to be hung up on the sample space and think only about how the random variable itself is distributed.
So what is a random variable, I can hear some people still asking. It might be possible to define it as some sort of equivalence class of functions on probability spaces, but you won’t make any probabilist friends if you go down that road. I think it is better to treat random variables as we treat things like real numbers or functions: we know that there are definitions around such as Dedekind cuts or subsets of Cartesian products, but in practice we do not use these definitions and instead concentrate on various properties that are used over and over again. That is, we use an abstract approach rather than a concrete one. When I lectured a first-year course in probability (and the probabilists in Cambridge were slightly worried that I’d give the unfortunate pure-mathematics perspective) I found that it was indeed almost always possible to talk about random variables without mentioning sample spaces. The one place I found I couldn’t do it was when proving linearity of expectation, which is trivial if you use the functions-on-probability-spaces definition. But if you take that as an axiom then the rest follows.
I mentioned in an earlier post that Aldous had some good lines. One of them, which I like very much despite still not completely understanding it, was, “There’s a difference between a recipe for a cake and a cake.” He explained that probability measures were analogous to recipes and random variables to actual cakes. I sort of see what he means — a function on a probability space is a way of realizing a given random variable — but … actually, perhaps I do finally understand it. Going back to my and example, perhaps you could say that the function on that takes to is a recipe for producing and but it cannot be or because they are distinct things that happen to be given by the same recipe. So you have to take your function and sort of detach it from itself until it becomes an autonomous object. (Perhaps another analogy would be that there is a difference between a musical score and a performance of the piece.) I wonder what two independent cakes would be like — I have a strange vision of their intersecting but not completely.
Aldous told us, or in some cases reminded us, that any random variable (probably subject to certain conditions that I have forgotten) can be realized as a function defined on This produced the second of his nice quotations: “It’s a non-obvious theorem that has to be true or life wouldn’t make sense whatsoever.” Why not? Because it would tell us that there were some random variables that could not be simulated.
The one other thing I remember particularly from the talk was that he mentioned some work of Tim Austin, who was one of Cambridge’s recent undergraduate superstars and has recently finished as a PhD student of Terence Tao. He told us that this was a name we should remember, and predicted that it would feature increasingly at future ICMs. From what I know of Tim, I wouldn’t want to bet against that.
After Aldous’s lecture I made another of my looking-after-myself decisions to skip a talk, and the third slot of the morning was free because Ngo Bao Chau had been due to give a plenary lecture then but would now be giving a lecture in the 1.45-2.45 slot as a new Fields medallist. So quite what I did between 10.00 and 12.30 I am not sure, though I am pretty sure it involved a reasonable amount of blogging.
At 12.30 I was invited to lunch in the restaurant of the conference hotel by some of the organizers of the Klein project, which aims to present research mathematics to schoolteachers in the spirit of the book, Elementary Mathematics from an Advanced Standpoint, by Felix Klein. I arrived at 12.35 and therefore (this being a lunch of mathematicians) last. I was asked what I wanted, the options being the full lunch buffet or a soup and salad option. Hint: everybody else had gone for soup and salad. I took my courage into both hands and went for the full lunch buffet (which turned out to be a good decision because going to the play in the evening left no opportunity for eating anything).
One of the things the Klein project wants to do is produce a book that covers mathematics area by area, with each chapter written by a team of authors. I was asked if I had any ideas about who could contribute. It’s a difficult one because although as a result of the Princeton Companion to Mathematics I now know a lot of people who can write general articles about mathematics, it may be that those very people are now less willing to write such articles, having already done so (not that the Klein project brief is the same as the Princeton Companion brief). But it occurs to me that publicizing the project here may achieve something. If you have any suggestions, then Bill Barton (firstname.lastname@example.org) would be delighted to hear from you. (He is looking for teams of people to write, in a way that would be interesting and comprehensible to schoolteachers, on such subjects as geometry, algebra, analysis, combinatorics, probability, etc.) If you follow the link above, you will find that there is also a wiki that they hope will develop into a useful resource.
When I got back to the main hall for Ngo’s lecture, I had to use the dare-to-sit-in-the-middle-of-the-fourth-row strategy again. That brought me near to an Indian woman who had introduced herself to me at some earlier point in the Congress — I no longer remember when. She handed me an envelope. I opened it and inside was a voucher that entitled me to a free set of proceedings (which at this ICM, unlike at the other ICMs I have been to, one had to pay extra for). That was very good of someone, I thought to myself, while not being quite sure who to be grateful to.
I had asked Jim Arthur three days earlier whether Ngo gave good talks. Yes, he assured me, though he followed it up by saying that they were very clear and well-organized. That got me slightly worried, so I asked whether they were understandable if you didn’t know what an automorphic form was. “That might be a problem,” he confessed.
In the event, I didn’t get much out of Ngo’s talk that I had not already got out of Arthur’s laudatio, apart from this photo, which I think captures quite well the experience of being alone while surrounded by people.
The fact that I didn’t learn much from the talk was at least as much my fault as Ngo’s — I made a conscious decision not to concentrate very hard, because by this time I was feeling so unpleasantly tired that I thought I would get a lot more out of sleeping a little than I would out of the talk itself. But my plans were thwarted by my intolerance of certain sorts of noises, and a habit that my next-door neighbour had of building up pressure behind a part of the body I don’t know the name for, but it’s a sort of door thing about an inch up your nose that you can use to block the air flow, and suddenly releasing it. I found these little exhalations with their preliminary kicks maddening enough to have to abandon all thoughts of sleeping. They didn’t help me concentrate on Ngo either. As usual, the Gowers glare had no effect at all.
After the talk, I hastened to the stall in the exhibition area where I could pick up my ICM proceedings. But while I was still in the main hall, someone stopped me and asked if they could be photographed with me. When I got to the stall, I couldn’t find the envelope any more. They weren’t prepared to give me the proceedings without the voucher, so I was forced to think how it could possibly have disappeared. I realized it must have been when I transferred my belongings from one hand to the other to look better in the photograph. I was in a hurry because there were only 15 minutes between the end of Ngo’s talk and the beginning of Assaf Naor’s, so I ran back to the main hall and found that the envelope was on the floor exactly where my theory predicted it should be. I then ran back to the exhibition area, filled out a form, and was handed a canvas bag with volumes 2-4 of the proceedings in it.
The good news: I was now the owner of volumes 2-4 of the proceedings of ICM2010. The bad news: the bag weighed a ton. It wasn’t just that the weight of my luggage would now go up by a significant fraction (which wasn’t really a problem as Emirates has a generous baggage allowance). I also had to lug this bag around for the rest of the day, which would include going to the play later on.
Talking of additions to luggage, I meant to say that after each talk the speaker would be presented with two gifts. That included panellists, so I got to find out what these two gifts were. The first will be very easy to guess by any mathematician who has reached a certain level of experience. There is a bizarre tradition amongst mathematicians for distributing large ugly mugs of the kind that are designed for large milky cups of coffee. The trouble is, I like my coffee short, strong and black, and I have plenty of quite carefully chosen mugs and cups for that purpose. So over the years I have built up a collection of huge mathematical mugs that I never use. My wife, whose aesthetic sensibilities are more keenly felt than mine, can hardly bear even to look at them: when I proudly showed her my new mug that said, “Prof. Timothy Gowers, Panelist,” on it, she instructed me to find a place other than our house for it to live. Come to think of it, you can enjoy it for yourself: let me take a photo of it and post the photo.
When I got to Assaf’s talk, I stupidly went up and told him that there was a chance that my eyes would close from time to time, and that this would not be a reflection on his talk but just on my state of tiredness. “Why are you tired?” he asked. “Because I switched my light off late for no good reason” didn’t feel like an answer he would understand, so I ducked out of the question somehow and went to sit down. (Just to prove that I mean what I was saying, I even found it hard to keep my eyes open during part of Spielman’s talk, and that definitely had nothing to do with the talk itself.)
In the end, although there were one or two moments during Assaf’s talk where my eyelids felt a bit heavy, it was mostly fine. Assaf is another superb speaker: he often starts with problems that are easy to understand, and then takes you on a journey, several times transforming the problems into other equivalent ones until one leaves the original problem a long way behind and is bringing in tools from areas that one would not have expected to be relevant. For example, deep results in the geometry of Banach spaces turn out to be closely connected with approximation algorithms. To get a quick impression of what I am talking about, I would recommend glancing through his paper in the proceedings.
In the next slot, I should have gone to hear Manfred Einsiedler (of Einsiedler, Katok and Lindenstrauss fame) but opted instead for a blog break. That meant that my next, and for this year final, ICM talk was Benny Sudakov, speaking about Ramsey and Turán type problems. Like Irit Dinur’s talk, this one contained quite a lot that I knew (things like statements of the Erdős-Stone theorem) but that a typical ICM delegate could not be expected to know and that were therefore essential to include. But to my slight surprise there were also a number of very interesting results mentioned that I had not yet heard about, including some that were co-authored by Jacob Fox and David Conlon — the latter being a collaborator and former research student of mine.
One of these I found particularly memorable — I took no notes and can still remember it. (I hope I’ll still say that when I’ve tried to explain it.) An old problem in Ramsey theory is to understand the rough form of the functions : that is the number of points you need in a set such that if you colour all the subsets of of size with two colours, then you can find a subset of size such that all its subsets of size have the same colour. (In the language of hypergraphs, you colour the edges of a -uniform hypergraph and you want a monochromatic complete subhypergraph with vertices.) When one has the usual finite Ramsey theorem for graphs, and although getting good asymptotics for is a major open problem in combinatorics, at least the general type of function is known: the growth is exponential in
When the general type of the function is no longer known. What is known is that it is at least exponential in and is at most doubly exponential in It is also known that beyond 3, each time you add 1 to the lower bound goes up by an exponential. What this basically means is that the bound for is either a -fold exponential or a -fold exponential. So sorting out the problem for will sort it out for general
It might seem rather strange that the function should go up by one exponential each time, but only from onwards. I don’t myself know why the proof works only from onwards, so I can’t give any insight into that. But I wanted to mention a few interesting facts connected with the problem when First of all, Erdős and Hajnal showed that if you allow four colours, then you do obtain a doubly exponential lower bound. Again, I do not know, so cannot give you, any hint about why having more colours helps so much. (It obviously helps, but why it should help enough to change the form of the function is what I don’t know.) They (or at least I think it was they) also introduced a variant of the problem where you try to find a set of vertices such that almost all its subsets of size have the same colour.
It is on this problem, amongst others, that Conlon, Fox and Sudakov have a very interesting result. They showed that for every the size of the set you need in order to guarantee the existence of a set of vertices such that at least of its subsets of size 3 have the same colour is exponential rather than doubly exponential. Since in the case of graphs (that is, the case ) there is very little difference to the bounds when you try to make almost all the edges the same colour rather than all of them the same colour, this looks like impressive evidence that the smaller of the two bounds is the right one.
But not so fast, Benny explained: their proof worked just as well with four colours as it did with two, and that, in view of the lower bound of Erdős and Hajnal for four colours, shows that the problem is genuinely different from the everything problem. I very much enjoy the feeling that you can get from certain talks, of which this was one, where you think you know where the speaker is going, but as you lean back smugly in your seat the mathematics jumps up and bites you.
Benny pitched his talk at exactly the right level. I got plenty out of it, but I’m pretty sure that people who knew less about combinatorics would have as well. (Another good aspect of the talk was that it was in a proper-sized room — the size Lurie’s would have been if Richard Thomas had got his way.) Quite a lot of what he mentioned, but by no means all, can be found in this paper.
After another hour of hanging about, it was time to catch a bus to A Disappearing Number. I said a rather sheepish hello to Roger Heath-Brown, who had also been persuaded by the speeches at the previous evening’s reception that he wanted to go. (I said that I had been intending to anyway, which is true, but the reception turned that into an even more definite decision.)
I enjoyed the bus journey because it was in a different direction from all the bus journeys I had gone on previously, though after a while we found ourselves at a junction that I remembered from my very first journey — the one from the airport to the convention centre. It was there that the taxi I had been in had turned left, done a U-turn, and turned left again. Now, approaching from the other direction, we turned left, did a U-turn, and went straight on. This was because there was some half-built flyover that made it impossible to turn directly right. (Another explanation might have been that turning right would be difficult when it meant crossing the lanes that were going from right to left. But that did not in general seem to be a problem in Hyderabad: at a certain point a bus would just cross the lanes and the cars had to stop for it.)
Soon we were at the Global Peace Auditorium, and I went in to see if I could get hold of the ticket that Simon McBurney had said would be waiting for me. As an insurance, I had bought a ticket that morning, since it was only 100 rupees (about £1.50), but I then saw that there were many different ticket prices, so if I had to use my 100-rupee ticket I would probably be in an awful seat.
There wasn’t an obvious box office, but there were several tables with people behind them. I was by no means the only person vying for the attention of those people, but in due course I managed to get it. At first they appeared to have no knowledge of any ticket being left for me, but after a bit they gave me one and I went into the auditorium to sit down. I was in the third row but right round to the side.
Next to me was a man who said he was a journalist and asked whether I was a mathematician. When I said I was he said he would like to interview me briefly after the play about whether I had liked it. I said I’d be prepared to do that. We chatted for a bit and then fell silent.
Then, to my not quite total surprise, a woman arrived and asked me what seat number I had. I told her C41, and she told me that she was also in C41. I showed her my ticket, but also admitted to her that “C41” had been written on it within the last few minutes, so she was probably more entitled to sit there than I was. She said, “It’s OK,” and went off to sort things out. Not long after that she was back with one of the people from the theatre who asked me to follow him. I felt slightly as though I had done something naughty, but in the end all I had to do was walk up the rows of the by now pretty full auditorium, go back into the foyer, get my “real” ticket, give back the wrong one, and go back and sit in a place (E40 or something) that was very close to where I had been before but actually slightly better because the angle was less extreme.
I’m not sure how much it is worth doing this, but here are two photos, one of the exterior of the theatre (if you look very carefully with your screen at maximum brightness, you may be able to make out two sculpted elephants guarding the entrance) and one of the auditorium, looking towards the stage, which is set for the first scene, which takes place at the front of a lecture theatre.
And what about the play itself? A more basic question might be how one can base a play around the Hardy-Ramanujan story when so much of the fascination of that story depends on the actual mathematics. Perhaps it will give you some idea if I say that the play began with Saskia Reeves talking about infinite series. She started with some simple ones, like but soon moved to the assertion that which she proceeded to justify by means of the functional equation for the Riemann zeta function (which allowed her to think of the series as and calculate it by relating it to ). The amusing thing about the justification was that it was correct, and that, presumably, Saskia Reeves didn’t understand what she was saying — although she was of course a good enough actress to sound completely convincing. During some of this introduction she spoke only quietly and one of the other actors addressed the audience about the nature of mathematics.
The rest of the play was a mixture of talking about mathematics, showing some famous episodes from the Ramanujan story (the discovery of the approximation to the partition function, the 1729 episode, the election of Ramanujan as a fellow of Trinity against the wishes of several fellows, his death at the age of 32) but not in a straight chronological sequence, and interspersing all that with a present-day story about Saskia Reeves’s character and a young businessman who falls in love with her. The whole thing works very well — especially the interaction between the two main stories — and you should see it if you get the chance. If I had to make a small negative comment, it would be that the characters tell us too often that mathematics is beautiful, which came across to me as a bit preachy (especially as one knew that the actors saying it had almost certainly not experienced that beauty for themselves to any great extent). But I’m not sure whether I’d feel the same way if I were a non-mathematician: perhaps then I would have had a little thrill each time the delicious paradox that mathematics can be beautiful was mentioned. And, just as a plenary lecture should be aimed at non-experts, so this play is principally aimed at non-mathematicians.
One other small negative aspect of the experience had nothing to do with the play, but with my second dose of maddening neighbour noise for the day. This time it was two young men who talked during the performance. They weren’t even whispering, and sometimes I missed lines of the play — though whether that was mainly due to the noise or mainly due to my being distracted by my annoyance at the noise is hard to say. I gave them several glares, once again with no result, and lacked the courage to ask them to shut up.
Afterwards, I loitered about a bit in case I met any of the cast and crew (which Simon McBurney had ordered me to do) but I didn’t, and I gave up fairly quickly because I was anxious not to miss the bus to my hotel. If I had, I would certainly have managed to get there somehow, but equally certainly I would have got there a lot later, and given that I had to pack, and also to get up early to catch a flight the next day, that was something I did not want to do. (An additional factor in my calculations was that the following day I was expecting to reach Cambridge by about 10pm and would have to get up in time for a taxi at 5am the following morning.)
When I got back to the hotel, I started to think that I couldn’t leave Hyderabad without experiencing very slightly more the city itself. Basically, all I had seen of it was from inside a bus, which was certainly interesting but it didn’t count as seeing it properly. Neither did what I did next, but it helped slightly: I went for a short walk. As I may have mentioned, my hotel had a quaint view over a flyover, so as soon as I could, I walked in the opposite direction from that. After I had walked for a while, I decided to turn back, but to walk back on the other side of the road, which was quite a main one. I chose some traffic lights to cross at, but they didn’t help much because first they took ages to go red, and then when they did go red there was no discernible difference in the behaviour of the constant stream of traffic. But I cast my mind back to Furstenberg’s words of wisdom: what can happen will happen. If I waited long enough, there would be a gap in the cars that was long enough for me to get across. (It’s possible that there was some convention I didn’t know, such as that if you walk across a busy road then the cars will not run you over. But I didn’t want to experiment.) I suppose, strictly speaking, I needed a quantitative form of this principle, but in any case eventually my gap appeared and I got across. I did not make any blogworthy observations about India during this walk, but I was glad to have done it.
It remained to pack. Although my suitcase had been underfull when I came out, packing turned out to be a serious challenge. And the main reason for that was the three volumes of ICM2010 proceedings, which I still had with me despite numerous opportunities to leave it in the bus, or the theatre, or the convention centre. I also kept thinking that I had finished and that I had managed the problem rather ingeniously, when I would see something else that I had forgotten to pack. A phase transition occurred when I spotted the mug and the other gift I had been given after the panel discussion, which I had not yet opened. I decided to open it then and there, even if it meant leaving the ICM2010 wrapping paper behind, and found a quite nice small chess set — one of those boxes that contains the pieces, which were wooden, and unfolds into a board. These occupied enough space to force a rethink — in the end I squeezed my hand-luggage bag into the slightly larger conference bag so that they would count as one item, with the plan that I would separate them again once I was through airport security. (I don’t suppose they would have been all that bothered if I’d taken them as two bags as, combined, they were much smaller than the hand luggage allowance.)
I could go on — as I write I realize that I haven’t mentioned ordering a taxi when I arrived at the hotel, and a telephone call to say that because of likely traffic on a Monday morning I should leave half an hour earlier than planned. In the end I left at 7.10am instead of the 7.00 that was now planned because the promised taxi wasn’t there. And we got onto a raised road that went for miles and miles with barely another car to be seen and I arrived at the airport in time to write almost an entire blog post before getting on to the plane.
I also feel as though some kind of summary is called for, but this post has reached over 5,500 words, the series of posts is now longer than Mathematics, A Very Short Introduction (yes, I actually checked), and in any case the congress was only at the half-way stage. I may at some point write a sort of summing-up post, but for now I must get back to some real work. If anyone wants to tell me how the rest of the congress went — perhaps even contribute a guest post — then I’d be very interested to know, since I’ve heard nothing about it at all. But the next post here will be EDP18.