Can you find Assaf in this picture? (Picture: Guy Kindler.)
In my post about ICM 2010 and India I hardly mentioned any mathematics. So here are a couple of mathematical snapshots from India. Not so much from the lectures themselves but from accidental meetings with people. (Tim Gowers extensively live-blogged from ICM10.) First, the two big problems in analysis according to Assaf Naor as told at the Bangalore airport waiting for a flight to Hyderabad. Next, a lecture on “proofs from the book” by Günter Ziegler. Then, some interesting things I was told on the bus to my hotel from the Hyderabad airport by François Loeser, and finally what goes even beyond q-analogs (answer: eliptic analogs) as explained by Eric Rains. (I completed this post more than two years after it was drafted and made major compromises on the the quality of my understanding of the things I tell about. Also, I cannot be responsible today for the 2-year old attempts at humor.)
The two big problems in analysis according to Assaf, and one bonus problem
The day before the ICM both Assaf Naor and I visited Microsoft Research at Bangalore hosted by Ravi Kannan (who had to fly early to make it to the grand rehearsal of the ICM opening ceremonies,) and Nisheeth Vishnoi. In the evening while we both waited for the flight to Hyderabad, Assaf told me what he regards as the two big problems in analysis.
1) The two-dimensional Carleson problem
Carleson’s famous theorem asserts that the Fourier expansion of a function in converges to the function almost everywhere. High dimensional analogs of the question itself or of some basic ingredients in the argument are very important and very hard. See here.
2) Quantitative versions of Hardy-Littlewood maximal principle
The story here starts with the Hardy-Littlewood maximal principle. Well, while I had heard about the maximal principle I really did not know what it was. The formulation is very simple. Start with a real function f in . So the expected value of |f(x)| is bounded. Then define g(x) to be the supremum over radius r of the expectation of f(x) over a ball of radius r around x.
As it turns out g is not necessarily in but it comes close. Let me say what “comes close” means here.
A functions f in has the property that the measure of the points x where |f(x)|>t is at most C/t. (Where C is the norm of f.) This is Markov’s inequality which is extremely useful in probability and it is extremely important to find better inequalities or much better inequalities for various specific cases. (The importance of going beyond Markov’s inequality is something I was interested in a while, so this aspect of the story appealed to me.)
Markov’s inequality is a consequence of being in , but it is a somewhat weaker condition.
Hardy and Littlewood proved that the maximum function g satisfies Markov’s inequality!
3) Bonus question: Pisier’s dichotomy conjecture.
A few days later I had a little chat with Assaf and he told me yet another very interesting problem about polytopes and about projections. This is Pisier’s dichotomy conjecture. (Gilles Pisier himself was also in Hyderabad). Let me simply quote Assaf on this one:
Pisier’s dichotomy problem is: If K is a centrally symmetric polytope in , with faces, then for every ε>0 there is a linear subspace E of , of dimension k that tends to infty with n (depending on the o(n) in the bound on the number of faces), and a parallelepiped , such that the section contains P, and is contained in (1+ε)P. Replacing ε by a universal constant (independent of dimension), and weakening the requirement that P is an image of a cube by just asking that it has polynomially (in k) many faces, is already interesting.
Update: Half a year later I met Leonard Schulman in Caltech and he told me about a project he is involved with and a question of isoperimetric type regarding Boolean functions. Having the conversation with Assaf fresh in my memory I noticed that the question (without any modification) is a about maximal principle for the Boolean cube, which turned out to be a useful “nudge”. See this paper by Aram Harrow, Alexandra Kolla and Leonard Schulman which is also featured in this post on GLL.
Proofs from the book lecture by Günter.
On the way from Hyderabad airport to our hotels I set next to an Indian number theorist, and Günter Ziegler and Assaf were sitting one row ahead. Günter showed us the slides for his wide-audience “proofs from the book” talk which contain quite a few beautiful proofs. The slides of this lecture are now available online here. (Update: link fixed)
Bercovich, Ngo and Model theory according to François
I met François Loeser a couple of times in New Haven and in Jerusalem, and after Assaf and Günter left the bus for their hotel, François and I discovered that we were heading to the same hotel. François is collaborating with Ehud Hrushovski in several projects that combine model theory and algebraic geometry. And François told me about a paper that they are going to arXive in just in a few days about Berkovich spaces.
Now, when I talked to François I really felt that I gained some understanding about this project. I remember that there was one aspect in which the new theory is a large extension of what Berkovich was doing while from another angle it is a special case. The theory is also related to real algebraic geometry (which is quite close to discrete geometry). But the truth of the matter is that I probably cannot tell you more about this paper and about the whole model theory – algebraic geometry endeavor than what you can get by looking at the paper. If there are good survey articles or popular accounts about this direction I will be happy to learn about them.
“So does this model theory connection have anything to do with… ehh… say… the fundamental lemma?” I tried my luck? “Yes”, said François, not so much the work with Ehud but rather another work that he had with some partners. It turns out that it was important to know how to transfer results from finite fields to p-adic fields and model theory can help. Although by now more precise proofs of the required results can be proved using “pure” algebraic geometry.
A couple of days later I met by chance (well, to meet another mathematician in an ICM is not a sheer coincidence) another famous figure in the connection between Model theory and algebra: Anand Pillay. Sixteen years earlier in Zurich ICM 1994 we were walking with several people and Lenore Blum said how wonderful Pillay’s talk was. I heard about his mathematics even before so I was quite curious to meet him then, and I even had a (bogus) mental image of how he looks but it waited 16 years.
Update: From what I hear Hrushovski and Loeser’s approach to Berkovich spaces is a great hit, and model-theoretic thinking in algebraic geometry breaks new grounds. Unrelatedly, the recent proposed proof of the ABC conjecture also has a dose of logic-theoretic aspects.
Elliptic analogs according to Eric Rains
q-Analogs are not the end of the road – there are also elliptic analogs!
We are familiar with q-analogs. The q analog of n! is . The q analogs of binomial coefficients are Gaussian coefficients (just replace the factorials by their q-analogs). Among the names associated with q-analogs, Euler, Ramanujan, Macdonald, Andrews (who was at Hyderabad) Zeilberger, Cherednick and many more. A famous younger guy is Eric Rains, and since I did not met Eric Rains before, I invented a mental image of how he looks which turns out to be completely wrong (but it was based on a wrong spelling “Reins”).
One evening, while shoving my way together with Synthia Dwork, Ursula Hamenstädt, and Salil Vadhan, to the bus from the convention center to our hotel, Eric, whom we just met, told us about even more general objects called “elliptic analogs”. The idea is very simple: you replace the parameter q with a parameter representing elliptic curves. This goes back to a 1997 paper by Frenkel and Turaev. Of course, as simple as it sounds it made no sense to me at all. But after listening a to Eric and looking a little at some papers I got a very vague picture about what it is about. So here are some references: 1, 2, 3, 4, 5.
Updates: When I was in Caltech half a year later I met Eric and he told me in greater detail about elliptic analogs, and I even took careful notes for the purpose of explaining the matter a little more professionally, but somehow I cannot find them now, but will welcome comments from people who can explain this further (and also the other stuff mentioned in this post). Cynthia Dwork gave a beautiful talk at the ICM about the mathematics and computer science of privacy. (Mathematical notions related to questions about privacy are connected to discrepancy questions, This connection played a role in the recent determination by Nikolov and Talwar of the hereditary discrepancy analog for Erdos’ discrepancy problem. See also this post in Gowers’s blog.)
Being at an ICM is a very special experience. There are many friends you want to catch up with and many people you know and would like to talk to, and things are too hectic for this. Here you see some well-known distinguished people who you always wanted to see, and here, lo and behold, a few people treat you this way. I like these random and short meetings that leaves you unsatisfied, but I know some mathematicians that cannot stand it. The next ICM is in Seoul 2014.