## 45 years of Deligne’s Fourier transform

It is exactly 45 years ago today that Deligne, in a letter to David Kazhdan (available here) introduced the $\ell$-adic version of the classical Fourier transform…

This construction, which operates on sheaves (or better complexes, or objects of the derived category, or…) on the additive group (or more generally a commutative unipotent algebraic group) has a definition that may look utterly bewildering at first for an analyst or an arithmetician, something like

$FT(M)=R\pi_{1,!}(p_2^*M\otimes \mathcal{L}_{\psi(xy)}).$

However, it has the key property that if $t_M$ is the trace function of the input object $M$, then the trace function of Deligne’s Fourier transform object $FT(M)$ is the discrete Fourier transform of $t_M$ (after fixing a suitable additive character). And this opens the door to the study of these discrete Fourier transforms using all the tools of algebraic geometry, and especially the Riemann Hypothesis of Deligne himself…

## Hédi Daboussi, in memoriam

I was very sad to learn today from R. de la Bretèche and É. Fouvry that Hédi Daboussi passed away yesterday. Hédi played an important part in my life; he was the first actual analytic number theorist that I met, one day at IHP in 1987, at the beginning of my first bachelor-thesis style project. (This was before internet was widely available.) Fouvry and him advised me on this project, which was devoted to the large sieve, especially the proof of Selberg based on the Beurling functions. They also introduced me to Henryk Iwaniec, who was visiting Orsay at the time (in fact, the meeting at IHP was organized to coincide with a talk of Iwaniec).

Daboussi is probably best known outside the French analytic number theory community for two things: his elegant elementary proof of the Prime Number Theorem, found in 1983, which does not use Selberg’s identity, and which is explained in the nice book of Mendès-France and Tenenbaum, and the “Rencontres de théorie élémentaire et analytique des nombres”, which he organized for a long time as a weekly seminar in Paris, before they were transformed, after his retirement, into (roughly) monthly meetings, which are still known as the “Journées Daboussi”, and are organized by Régis de la Bretèche. The first of these were two days of a meeting in 2006 in honor of Hédi.

For me, the original Monday seminar organized by Daboussi was especially memorable, both because I gave my first “real” mathematics lecture there (I think that it was about my bachelor project), and also because on another occasion (either the same time or close to that), I first met Philippe Michel in Hédi’s seminar. It is very obvious to me that, without him, my life would have been very different, and I will always remember him because of that.

## A bat in the office

If you have visited the ETH main building, you might have noticed birds that are sometimes lost within its great halls. Although it is not so obvious, bats are also often found there; so many, in fact, that the Stiftung Fledermausschutz has special contact people to intervene there when a lost bat is found.

This is what happened to me this morning.

The bat is an adult female of the Pipistrellus nathusii species; she was apparently dehydrated but is now being taken care of in the bat emergency room of the Zürich Zoo until it is warm enough for her to go back on her journeys…

(P.S. This is my last blog post; in addition to all other things taking too much of my time, the last WordPress update just makes the whole experience not worth going through; for instance, there should be a picture of the bat, but it doesn’t show for reasons that I’ve already lost too much time trying to understand; anybody interested in seeing it can email me.)

## Clean-up before closure

I’ve cleaned-up my list of publications and unpublished notes, moving in the second page some preprints that are not going to be submitted for publication (as well as two notes that I had not yet put online).

On February 1st, I’ll start a period of being vice-chairman for two years and a half, then chairman of the department for two years, so that my other activities are unlikely to flourish for a long time, and similarly this blog will be updated even less regularly than before.