Just a quick update to indicate that the official FIM pages for the conferences in honor of Alessandra Iozzi’s birthday and Bill Duke’s birthday (both at FIM next year) are now available. Most important are the forms for young researchers to request funding (local expenses) to attend the conferences, here and there. (I was almost going to say to be careful not to apply to the wrong conference, but both will be great, so it doesn’t really matter…)
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 and at (say) , for a non-arithmetic group, do not correlate. In other words, if we normalize the coefficients (say ) so that the mean-square is , then we have
(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 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 -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.
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…]
Aficionados of Perverse Sheaves will be happy to learn that the famed volume Astérisque 100, volume 1, known throughout the world as BBD, has finally been re-printed by the SMF. In fact, it now comes with errata and addenda, and most importantly O. Gabber has accepted to appear as a co-author. All references should therefore now be updated to BBDG…
Philippe Michel and myself are organizing a Winter School in January 2019 on the topic of trace functions and their applications to analytic number theory. There is a very basic web page for the moment. Most importantly, the application form, which will be setup by the conference center where the school will be held, is not yet available.
The setting is the CSF (Congressi Stefano Franscini), which is a conference center of ETH located in Ticino (so this will be a good occasion to practice italian). The school is intended essentially for current PhD students, together with a smaller number of recent PhDs ; the total number of participants should be around 50. There will be five minicourses, given by T. Browning, Ph. Michel, L. Pierce, W. Sawin and myself. A more detailed programme will appear in due course…