of the building as it looked in 1870. The red square indicates where my office is located…

]]>(1) My lecture notes on representation theory (expanded) will appear in September, published in the Graduate Studies in Mathematics series; the preview material contains Chapter 1, and a fair bit of Chapter 2; the index is also available, and perusing it will give an idea of the range of topics mentioned.

(2) Henryk Iwaniec has also a new book coming, in October, containing his lectures notes on the Riemann zeta function. I haven’t seen it yet, but he told me that the highlight, in his opinion, is the second part which contains his personal treatment of the Levinson method for finding critical zeros of zeta. This should be quite interesting to read…

**Update** (August 29): the AMS web site confirms that my book is already available!

Did I like the book? This is probably far from the right question to ask in any case, but it’s therefore worth investigating in a post-modern spirit. Since I finished the whole of the seven volumes in about seven months, during which time I had to take care of many other activities, and also started (and often stopped) reading a fair number of other books, I was certainly finding something in Proust that kept me engaged in his work.

One technical aspect of Proust’s style that struck me was how he manages to capture the way that crucial events or characters will first appear informally and casually in life. Here’s for example the first time the narrator meets Gilberte while playing at the Champs-Élysées:

Retournerait-elle seulement aux Champs-Élysées? Le lendemain elle n’y était pas; mais je l’y vis, les jours suivants; je tournais tout le temps autour de l’endroit où elle jouait avec ses amies, si bien qu’une fois où elles ne se trouvèrent pas en nombre pour leur partie de barres, elle me fit demander si je voulais compléter leur camp, et je jouai désormais avec elle chaque fois qu’elle était là.

And here is the first appearance of the *jeunes filles*, among whome is Albertine:

J’aurais osé entrer dans la salle de balle, si Saint-Loup avait été avec moi. Seul je restai simplement devant le Grand-Hôtel à attendre le moment d’aller retrouver ma grand-mère, quand, presque encore à l’extrémité de la digue où elles faisaient mouvoir une tâche singulière, je vis s’avancer cinq ou six fillettes, aussi différentes, par l’aspect et par les façons, de toutes les personnes auxquelles on était accoutumé à Balbec, qu’aurait pu l’être, débarquée on ne sait d’où, une bande de mouettes qui exécute à pas comptés sur la plage — les retardataires rattrapant les autres en voletant — une promenade dont le but semble aussi obscur aux baigneurs qu’elles ne paraissent pas voir, que clairement déterminé dans leur esprit d’oiseaux.

This comes with no warning or no articifial build-up of *something is going to happen, drumroll, drumroll*.

I was also very touched by the last pages, which certainly affected my overall impression and reaction in a way that I’ve only felt before when finishing Faulkner’s *Absalom, Absalom* (after which I went into a rather intense faulknerian phase during my PhD years). It seems that to understand Proust (as much as I can…), I would have to re-read the whole text, in the light of these last pages. Is that the expected reaction? It could well be… Will I do it? Who knows…

On a different note, I was amused to see that while the first Pléiade edition is a rather straightforward edition of the novel with short notes and biographical information (rather like the Library of America editions of Faulkner, for instance), the second edition succombs

to editorial inflation on a magnificent scale: the variants, *esquisses*, notes and notes on the *esquisses*, take up more space than the actual text!

Here is the first volume of the old edition:

compared with the second of the new edition:

This can of course be helpful, as are certainly useful the 125 pages of *Liste des personnages cités* which allow you to quickly locate all the places where Rembrandt, or the Marquis de Norpois, or Saint Simon, or any other character, real or imagined, makes an appearance in the whole text.

(There is a similar list for names of places and names of works of arts, again real or imagined).

Another thing I noticed is that the first edition doesn’t use accented letters as capitals at the beginning of sentences, while the second does:

compared with

]]>I certainly learnt things myself, especially in the course of K. Soundararajan, who discussed (among other things) some recent works of his with M. Radziwiƚƚ that I had intended to read, without finding the time…

My own lectures were on trace functions over finite fields. It was the first occasion I’ve had to give more than one talk on this topic, and I used the opportunity to see which ideas could work for a good presentation of the basic ideas to analytic number theorists. I’m quite happy with the outcome, and this will be very useful since I will give courses on the subject during the next two semesters, which should provide at least some amount of notes and drafts for the book that É. Fouvry, Ph. Michel and mysefl are hoping to write.

For those who could not attend the event (which is probably a fair number of people, in view of the fact that an unfortunate independent-scheduling event led to it being simultanenous with the ENFANT/ELEFANT conference organized by L. Pierce and D. Schindler in Bonn), it is quite nice that the whole programme was filmed and is now available on Youtube on the IHÉS channel! (The ordering of the videos is a bit strange, but it is easy to use the descriptions to watch, for instance, all four of Sound’s lectures one after the other).

]]>I am waiting impatiently for a more refined approach that will include meta-statistics:

Did you know? France has never lost a game where a single US-based newspaper presented more than three human-interest stories concerning the opposing team players in the three days before the game.

Did you know? It is the first time since the invention of the personal computer that the BBC has listed more than 243 statistical facts about a game.

Did you know? Three out of four statistical facts about the Italy-Switzerland game have involved numbers larger than thirteen.

These will be the days…

]]>Of course, even without being present, the written text of the seminar is always available later to learn what the talks were about. But, especially when the subject is not close to something I know, it’s often much better to have seen first a one-hour presentation which distills the most important information, which may well be a bit hidden in the written text for non-specialists. So it is rather wonderful to see that, since last March, the lectures of the Bourbaki seminar are recorded and made available on Youtube on the channel of the Institut Henri Poincaré.

Here is the playlist for the March seminars, with lectures by Golse (on Bolztmann’s equation), Bolthausen (spin glasses), Benoist (curves on K3 surfaces) and myself (guess…), and here that of last Saturday, with lectures by A. Valette (the Kadison-Singer problem), Smulevici (general relativity), Hales (formal proofs) and Coquand (dependent type theory and univalent foundations; with Voevodsky, who was present, explaining at the end his choice of the word “univalent”).

The links to the videos (as well as links to the texts) can also be found on the web page of Bourbaki.

(There was also mention that the seminar was streamed as it happened, but I don’t know if that’s really the case; if yes, will fashionable bars in Manhattan and Princeton start opening at 4 A.M. on selected Saturdays to offer croissants, coffee, cognac, and the Bourbaki talks?)

]]>But no! The actual question is to compute times ! We must correct this! But it’s just as easy without starting from scratch: we turn the “plus” cross a quarter turn on the left-hand side:

and then switch the digits on the right-hand side:

This is a fun little random fact about integers and decimal expansions, certainly.

But there’s a bit more to it than that: it is in fact independent of the choice of base , in the sense that if we pick any other integer , and consider base expansions, then we also have

as well as

(where we underline individual digits in base expansion.)

At this point it is natural to ask if there are any other Léo-pairs to base , i.e., pairs of digits in base such that the base expansions of the sum and the product of and are related by switching the two digits (where we always get two digits in the result by viewing a one-digit result as ).

It turns out that, whatever the base , the only such pairs are and the “degenerate” case .

To see this, there are two cases: either the addition leads to a carry, or not.

If it does, this means that where . The sum is then

So this is a Léo-pair if and only if

This equation, in terms of and , becomes

which holds if and only if . Since the factors are integers and non-negative, this is only possible if , which means , the solution found by Léo.

Now suppose there is no carry. This means that we have and . Then

and we have a Léo-pair if and only if

i.e., if and only if .

This is not an uninteresting little equation! For a fixed (which could now be any non-zero rational), this defines a simple quadratic curve. Without the restrictions on the size of the solution , there is always a point on this curve, namely

This does not fit our conditions, of course. But we can use it to find all other integral solutions, as usual for quadratic curves. First, any line through intersects the curve in a a second point, which has rational coordinates if the line is also defined by rational coefficients, and conversely.

Doing this, some re-arranging and checking leads to the parameterization

of the rational solutions to , where is an arbitrary non-zero rational number. In this case, this can also be found more easily by simply writing the equation in the form

Now assume that is an integer, and we want to be integers. This holds if and only if is an integer such that .

Such solutions certainly exist, but do they satisfy the digit condition? The answer is yes if and only if , which means , giving the expected degenerate pair. Indeed, to have , the parameter must be a negative divisor of . We write with positive. Then to have non-negative digits, we must have

the first one of these inequalities means , while the second means that …

where the Royal Society has a conference center. This was quite a fun occasion, and not only because the curtains had one the most interesting decorative pattern I have seen during years of extensive study:

The talks were only 30 minutes long; because of this and the absence of blackboards, they were all beamer talks, and I’ve uploaded my slides here. They might be useful even for participants in the meeting, since I only had time to cover the first 60% or so of what I had planned, which was a survey of my most recent work with É. Fouvry and Ph. Michel (available here, and which I hope to discuss in more detail in a later post)…

]]>Can anybody make a guess of what is the first line? It is

What is the **** historical perspective?

but I can’t read the missing fourth word!

And what is the Kierkergaardian idea, really?

]]>