# Quantum Chaos in Bordeaux

I was in Bordeaux this week until yesterday to attend (parts of) the conference Quantum Chaos 2009 organized by M-L. Chabanol, S. Nonenmacher and G. Ricotta. This was a very enjoyable stay, although it was threatened by natural chaos on the left due to a terrible storm that went over the South West part of France last Saturday, and by more social chaos on the right from a fairly general strike tomorrow, including that of the public transportation system and of the university personnel restaurant in Bordeaux.

I will not try to give a summary of what Quantum Chaos is, since I am not at all an expert in this field in general, and also because Arnd Bäcker, the first lecturer, gave a wonderful overall presentation from the point of view of mathematical physics (which is where the subject grew). The downloadable version does not include the very nice computer simulations that he presented, unfortunately.

I lectured myself on the part of the subject that I know (a bit) about: the Arithmetic Quantum Unique Ergodicity Conjecture of Rudnick and Sarnak [see this survey of J. Marklof for some background on this and other arithmetic aspects of Quantum Chaos], which claimed that the probability measures on the modular surface

$SL(2,\mathbf{Z})\backslash \mathbf{H}$

defined by

$\mu_f=|f(z)|^2 y^{-2}dxdy,$

for eigenfunctions f of the Laplace operator which are of norm 1 and are also eigenfunctions of all Hecke operators, should converge weakly to the Poincaré measure

$3/\pi y^{-2}dxdy$

on the modular surface. I said “claimed” because this is a conjecture no more: rather dramatically, G. Prakash pointed out at the end of my second lecture that K. Soundararajan had posted on arXiv the same morning a preprint containing a proof of the last missing step required to verify the correctness of the conjecture! This is based on the ground-breaking work of E. Lindenstrauss, who had used ergodic theoretic methods to prove that any weak limit had to be a multiple of the Poincaré measure, but without being able to avoid the possibility that some of the unit mass would be lost (“escape in the cusp”): Soundararajan manages to exclude this last possibility. (I find somewhat ironic that this happened the first day probably in months where I didn’t have time to check the arXiv posting in the morning…)

What I had lectured on was the recent work of R. Holowinsky, and of (another work of) Soundararajan. roughly speaking, Holowinsky had managed to prove the conjecture either “with very few exceptions” or under the Ramanujan-Petersson conjecture for Hecke eigenforms, using a very beautiful and clever sieve argument (the link between sieve methods and a conjecture motivated by mathematical physics is part of the beauty of mathematics…) Soundararajan, also with a remarkable argument of analytic number theory, had proved a fairly general “weak subconvexity” result for values of L-functions satisfying the Ramanujan-Petersson conjecture. This gave — among other things! — another proof of the Unique Ergodicity with very few exception (assuming the Ramanujan-Petersson) conjecture, because of earlier work, in particular, of Sarnak and a formula of Watson linking the desired equidistribution to some special values of triple product L-functions.

Although the (brand new) result of Soundararajan means that these two works are not needed any more to try to prove the original conjecture, their interest is not just as beautiful pieces of mathematics, as Holowinsky and Soundararajan show in another joint work: their methods work also for holomorphic cusp forms of large weight (still Hecke eigenfunctions). There, on the one hand, we have the Ramanujan-Petersson conjecture (by Deligne’s one-two knockout step: first linking it with the Riemann Hypothesis over finite fields, and then proving the latter). Also, on the other hand, a rather wonderful stroke of good fortune arises: the two “exceptional subsets”, where each method fails to give equidistribution, are disjoint! So Holowinsky and Soundararajan end up proving the holomorphic case of AQUE by a completely devious route (which recalls the first ineffective solution of the class number problem of Gauss, where either assuming that GRH holds, or that it doesn’t, led to the result!)

It is also interesting to note that this holomorphic case of AQUE is currently inaccessible to ergodic-theoretic methods, the reason being (apparently) that there is no “classical dynamics” behind it — the geodesic flow being the classical system underlying the original conjecture. (This was explained to me by M. Einsiedler, whose lectures in Bordeaux cover the ergodic-theoretic methods of Lindenstrauss).

Other lecturers in the programme are P. Kurlberg, describing the quantization of toral automorphisms and the problems that arise (where it is again very satisfying from the point of view of the unity of mathematics that the Riemann Hypothesis over finite fields turns out to play a big role), and J. Keating who will lecture Thursday and Friday on Quantum Graphs (which, unfortunately, I know nothing about, but I’ll try to find a good survey to link to).