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.

Akshay's cards
Akshay’s cards

(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:

Restaurant card
Restaurant card

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

The Magic Mountain

Once more, I have yielded to the arch-Tempter, the Book-Buying demon.

This time, it started when I bought (second-hand — I actually think today that only second-hand books are really authentic, unless of course the book is brand new) the translation by J. E. Woods of the novel “Joseph and his Brothers” by Thomas Mann. I expected that (like Joyce’s “Finnegans wake” and Faulkner’s “A Fable”, which I both own and in one case read) this was only a gesture of respect for the work of a writer that I admire. To my astonishment, I read this four-part fifteen-hundred page book (“The stories of Jacob”, “The young Joseph”, “Joseph in Egypt”, “Joseph the Provider”) in a few weeks, and found it too short, and realized that it was a masterpiece. The story of Joseph and Mut-em-Ênet in the third book is, indeed, an extraordinary act of literary empathy. And this story was written in exile by a conservative sixty-year old german, when most of everyone and everything he loved was either utterly betraying his culture or was being destroyed.

Well, so when I learnt (from a blog post of the ETH Bibliothek) that — after who knows how many years of work from the editors — the commented edition of this book was appearing in April this year (the Grosse kommentierte Frankfurter Ausgabe announced it in 2008 as “in plan, 2012”), I couldn’t resist and ordered it. I actually had already bought a German version of the book (“Die Geschichten Jaakobs”, “Der Junge Joseph”, “Joseph in Ägypter”, “Joseph der Ernährer”, to use the original titles), and since the available room in my apartment doesn’t really allow for more than one copy of thousand pages long German books, I donated these to my colleague Ian Petrow who had told me of his liking for the “Magic Mountain”.

But then, could I really keep my paperback German copies of “Der Zauberberg” and of “Doktor Faustus”, when both existed in the same amply commented edition? I couldn’t, donated the old ones (to the same colleague), and bought both. So here I am:

Thomas Mann
Thomas Mann

(on the left, the older (in)complete works of Shakespeare for scale).

The empty slot in the middle is that of the “Zauberberg”, which I am now trying to read in German, with much help from online dictionaries. And it reminds me that I started reading “The Magic Mountain” in Rutgers (and in translation, of course), when a friend there recommended it to me, especially because of the character of Lodovico Settembrini:

Auf dem Wege von links kam ein Fremder daher, ein zierlicher brünetter Herr mit schön gedrehtem schwarzen Schnurrbart und in hellkariertem Beinkleid, der, herangekommen, mit Joachim einen Morgengruss tauschte – der seine war präzis und wohllautend – und mit gekreuzten Füssen, auf seinen Stock gestützt, in anmutiger Haltung vor ihm stehen blieb.
GFKA, p. 88

For the Yiddish version, translated by Isaac Bashevis Singer, see here.

Like Joyce, Thomas Mann died in Zürich, and his grave can be found there.

Thomas Mann
Thomas Mann

Where will the Tempter bring me next? I believe that, most likely, it will be the Opere of Primo Levi, or those of Niccolò Machiavelli, although my Italian is now rather worse than my German.

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…]

BBD is dead, long live BBDG!

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…

Winter School at Monte Verità

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…