## ICM2014 — Ian Agol plenary lecture

August 22, 2014

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.)
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## ICM2014 — Khot laudatio

August 20, 2014

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 P$\ne$NP, 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.
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## ICM2014 — Mirzakhani laudatio

August 19, 2014

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.)
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## ICM2014 — Hairer laudatio

August 18, 2014

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.
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## ICM2014 — Bhargava laudatio

August 15, 2014

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 $ax^2+bxy+cy^2$. One assumes that they are primitive (meaning that $a$, $b$ and $c$ don’t have some common factor). The discriminant of a binary quadratic form is the quantity $b^2-4ac$. The group SL$_2(\mathbb{Z})$ then acts on these by a change of basis. For example, if we take the matrix $\begin{pmatrix}2&1\\5&3\end{pmatrix}$, we’ll replace $(x,y)$ by $(2x+y, 5x+3y)$ and end up with the form $a(2x+y)^2+b(2x+y)(5x+3y)+c(5x+3y)^2$, which can be rearranged to
$(4a+10b+25c)x^2+(4a+11b+30c)xy+(a+3b+9c)y^2$
(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

August 14, 2014

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.
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## ICM2014 — opening ceremony

August 13, 2014

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.) Read the rest of this entry »

## ICM2014 — first impressions

August 12, 2014

I’m writing this at 6:22am in my hotel room in Seoul, which is in a hotel that is right next to the conference centre, to the point where you don’t have to go out of doors to get from one to the other. I’ve just had a good night’s sleep, even though in French time (which is what I was used to until the day before yesterday — if that concept still means anything) my entire night has been during the day time, and now is about the time I’d be thinking of going to bed. I feel a bit strange, and I may have trouble staying awake during an opening ceremony that lasts several hours and then a pretty full programme of talks later in the day. But during the latter I’ll be taking notes in order to be able to blog about them in reasonable detail, so at least I’ll have something to keep my brain from relaxing too much.
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## ICM2014 — introductory post

August 11, 2014

Four years ago I blogged from the ICM in Hyderabad. The posts are amongst the most popular I have written — my statistics show that some of them are still being read quite regularly even now. Right now I’m sitting in Charles de Gaulle airport waiting to board a plane to Seoul, where I will be attending this year’s ICM, or rather, as I did last time, attending the first half of it. I’m not sure I’ll have the time or energy to write quite as much about ICM2014 as I did about ICM2010, but I’ll do what I can. In particular, I’ll try to convey exactly what I manage to understand from some of the main talks — especially the talks about the work of the new Fields Medallists. Given all the rumours about the likelihood of one of them being female, I am particularly glad to be going to the opening ceremony to witness (I hope) an important moment in mathematical history.

Just as the last ICM was the first (and still only) time I had been to India, this one will be my first visit to Korea. I’m looking forward to that aspect too, though my hotel is right next to where the congress is taking place and the programme looks pretty packed, so I’m not sure I’ll see much of the country. Talking of the packedness of the programme, I can already see that there are going to be some agonising decisions. For example, Tom Sanders is giving an invited lecture at the same time as Ryan Williams, two speakers I very much want to listen to. I suppose I’ll just have to read the proceedings article of the one I don’t go to. Equally unfortunate is that Ben Green’s plenary lecture is not until next week, when I’ll have gone. But I hope that I’ll still be able to get some kind of feel for where mathematics is now, what people outside my area consider important, and so on, and that I’ll be able to convey some of that in the next few posts.

I’d better stop this now, since I’ll soon be getting on to an Airbus 380 — a monstrously large double-decker plane. One of my children is something of a transport enthusiast and told me in advance that this would be the case (he had looked it up on the internet). I had hoped to end up on the top floor, but that turns out to be for business class only. The flight is about 11 hours: it leaves at 9pm French time and arrives at around 2:30pm Korean time. The challenge will be not to be utterly exhausted by the time of the opening ceremony on Wednesday morning. My memory of Hyderabad is that by the end of the four days I was so tired that I was almost getting anxious about my health. I plan to look after myself a bit better this time, but it may be difficult.

## Mini-monomath

July 19, 2014

The title of this post is a nod to Terry Tao’s four mini-polymath discussions, in which IMO questions were solved collaboratively online. As the beginning of what I hope will be a long exercise in gathering data about how humans solve these kinds of problems, I decided to have a go at one of this year’s IMO problems, with the idea of writing down my thoughts as I went along. Because I was doing that (and doing it directly into a LaTeX file rather than using paper and pen), I took quite a long time to solve the problem: it was the first question, and therefore intended to be one of the easier ones, so in a competition one would hope to solve it quickly and move on to the more challenging questions 2 and 3 (particularly 3). You get an average of an hour and a half per question, and I think I took at least that, though I didn’t actually time myself.

What I wrote gives some kind of illustration of the twists and turns, many of them fruitless, that people typically take when solving a problem. If I were to draw a moral from it, it would be this: when trying to solve a problem, it is a mistake to expect to take a direct route to the solution. Instead, one formulates subquestions and gradually builds up a useful bank of observations until the direct route becomes clear. Given that we’ve just had the football world cup, I’ll draw an analogy that I find not too bad (though not perfect either): a team plays better if it patiently builds up to an attack on goal than if it hoofs the ball up the pitch or takes shots from a distance. Germany gave an extraordinary illustration of this in their 7-1 defeat of Brazil.

I imagine that the rest of this post will be much more interesting if you yourself solve the problem before reading what I did. I in turn would be interested in hearing about other people’s experiences with the problem: were they similar to mine, or quite different? I would very much like to get a feel for how varied people’s experiences are. If you’re a competitor who solved the problem, feel free to join the discussion!

If I find myself with some spare time, I might have a go at doing the same with some of the other questions.

What follows is exactly what I wrote (or rather typed), with no editing at all, apart from changing the LaTeX so that it compiles in WordPress and adding two comments that are clearly marked in red.

Problem Let $a_0 be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that

$a_n <\frac{a_0+a_1+\dots+a_n}n\leq a_{n+1}\ .$
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