## The most valuable mathematical restaurant cards in the world!

Now that Akshay Venkatesh has (deservedly) received the Fields Medal, I find myself the owner of some priceless items of mathematical history: the four restaurant cards on which, some time in (probably) 2005, Akshay sketched the argument (based on Ratner theory) that proves that the Fourier coefficients of a cusp form at $n$ and at (say) $2n$, for a non-arithmetic group, do not correlate. In other words, if we normalize the coefficients (say $a(n)$) so that the mean-square is $1$, then we have
$\lim_{X\to +\infty} \frac{1}{X}\sum_{n\leq X} a(n)\overline{a(2n)}=0.$

(Incidentally, the great persifleur of the world was also present that week in Bristol, if I remember correctly).

The story of these cards actually starts the year before in Montréal, where I participated in May in a workshop on Spectral Theory and Automorphic Forms, organized by D. Jakobson and Y. Petridis (which, incidentally, remains one of the very best, if not the best, conference that I ever attended, as the programme can suggest). There, Akshay talked about his beautiful proof (with Lindenstrauss) of the existence of cusp forms, and I remember that a few other speakers mentioned some of his ideas (one was A. Booker).

In any case, during my own lecture, I mentioned the question. The motivation is an undeservedly little known gem of analytic number theory: Duke and Iwaniec proved in 1990 that a similar non-correlation holds for Fourier coefficients of half-integral weight modular forms, a fact that is of course related to the non-existence of Hecke operators in that context. Since it is known that this non-existence is also a property of non-arithmetic groups (in fact, a characteristic one, by the arithmeticity theorem of Margulis), one should expect the non-correlation to hold also for that case. This is what Akshay told me during a later coffee break. But only during our next meeting in Bristol did he explain to me how it worked.

Note that this doesn’t quite give as much as Duke-Iwaniec: because the ergodic method only gives the existence of the limit, and no decay rate, we cannot currently (for instance) deduce a power-saving estimate for the sum of $a(p)$ over primes (which is what Duke and Iwaniec deduced from their own, quantitative, bounds; the point is that a similar estimate, for a Hecke form, would imply a zero-free strip for its $L$-function).

For a detailed write-up of Akshay’s argument, see this short note; if you want to go to the historic restaurant where the cards were written, here is the reverse of one of them:

If you want to make an offer for these invaluable objects, please refer to my lawyer.

## Coworkers of the world, unite! (or: “They who must not be named”)

If you have not perused it yet, I encourage you to read carefully the press release announcing the arrival of A. Venkatesh at the Institute for Advanced Study. Once you have done so, let’s try to answer the trick question: Who has collaborated with Venkatesh?

In this masterpiece of american ingenuity, we both learn that Venkatesh is great in part because of his ability to work with many people, but on the other hand, none of his “coworkers” deserve to be named. Bergeron, Calegari, Darmon, Einsiedler, Ellenberg, Harris, Helfgott, Galatius, Lindenstrauss, Margulis, Michel, Nelson, Prasanna, Sakellaridis, Westerland, who they? (I probably forget some of them, for which I apologize). In fact, the only mathematicians named are (1) past professors of IAS; (2) current professors of IAS; (3) Wiles.

It’s interesting to muse on what drives such obscene writing. My current theory is that the audience of a press release like this consists of zillionaire donors (past, present, and especially future), and that the press office thinks that the little brains of zillionaires (liberal, yes, but nevertheless zillionaires) should not be taxed too much with information of a certain kind.

(Disclaimer: I have the utmost admiration for A. Venkatesh and his work.)

[Update (August 2): the leopard doesn’t change its spots…]

## Horizon

The Swiss Science Foundation (SNF in German, FNS in French) publishes a regular magazine that surveys current topics concerning science and research in Switzerland. It often highlights subjects that are unexpected and interesting in all areas of science, from the humanities to forestry, the deterioration of prussian blue in paintings and so on, and does so in its three parallel editions, in English, French and German.

The last issue has a special focus on Open Science in its various forms. For some reason, although there is no discussion of Polymath per se, the editors decided to have a picture of a mathematician working on Polymath as an illustration, and they asked me if they could make such a picture with me, and in fact two of them (the photographer is Valérie Chételat) appear in the magazine. Readers may find it amusing to identify which particular comment of the Polymath 8 blog I am feigning to be studying in those pictures…

Besides (and of greater import than) this, I recommend looking at the illustration pages 6 and 7,

which is a remarkably precise computer representation of the 44000 trees in a forest near Baden, each identified and color-coded according to its species… (This is done by the team of M. Schaepman at the University of Zürich).

## L-functions database!

People studying automorphic forms, automorphic representations, number fields, diophantine equations, function fields, algebraic curves, equidistribution and many other arithmetic objects (j’en passe, et des meilleurs), often end up with some “L-function” to deal with — indeed, probably equally often, with a whole family of them, sometimes not so well-behaved… These objects are fascinating, mystifying, exhilarating, random and possibly spooky. Where they really come from is still a mystery, even with buzzwords aplenty ringing around our ears. But one remarkable thing was already known to Euler and to Riemann: one can compute with L-functions. One impressive research project has been building, for quite a few years, a very sophisticated website presenting enormous amounts of data about L-functions of many kinds. The L-functions Database is now out of its beta status: go see it, and have a look at the list of editors to see who should be thanked for this amazing work!