I was very saddened to read on T. Tao’s blog of the death of Elias Stein. Although I did not know him personally, or even worked in the same area of mathematics, I felt a great admiration and respect for him. I remember browsing intently through some of his books, especially at the beginning of my PhD thesis, but also still in recent times, and especially the one (partly) about quasi-orthogonality. I tried to apply one of the statements I found there (“Cotlar’s Lemma”) to some cases of large-sieve inequalities. It didn’t work, but I remember that during a party at Fine Hall some time later, my advisor H. Iwaniec introduced me to Stein (this was my only personal encounter with him!) by saying jokingly that “Your lemma is not strong enough”.

I have always felt a great attraction to the type of harmonic analysis that I read about in those books, even if I don’t understand it. Maybe, one day, I will be able to know more…

The registration for the Winter School on trace functions at Monte Verità can now begin. As explained on the web page, because the number of participants is limited, you should send an email to this address to indicate your interest. For the most part, the school is for PhD students, so please indicate who is your PhD advisor, and if you are a postdoc, your motivation for attending the school.
We will then send to the selected participants the link to the official registration page.

Like every year, the mathematics department of ETH offers some postdoc positions. This year, a slightly different organization has been chosen, combining some resources with the FIM. The positions are now called “Hermann Weyl Instructors”, and the main change (besides slightly earlier deadlines) is that the teaching duties are clearly stated upfront: the postdoc should teach 50% during two semesters of the three year position. (So if we compare with the Veblen Instructorships offered by Princeton and the IAS, for example, we request one year teaching/two year research, instead of two year teaching/one year research for the Veblen position).

The web page with information on the positions (including salary, which follows a standard ETH scale) is available on the FIM website. The deadline for application is November 1 (for full consideration — slightly later applications are permitted, but depending on the schedule of the committee meetings, they might be too late). Finally, the application can be done using this form.

Just a quick update to indicate that the official FIM pages for the conferences in honor of Alessandra Iozzi’sbirthday and Bill Duke’sbirthday (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.