GANT beginnings

The special semester on Group Actions in Number Theory, organized by Ph. Michel and myself, started last week with our Winter School.

We were lucky to have J.-P. Serre give a short course on equidistribution, with emphasis on questions related to the Sato-Tate conjecture and its variants. Here are a few things I’ve learnt (from the lectures and discussions afterwards):

(1) Bourbaki writes
\mathbf{S}_1,\quad \mathbf{P}_2,\quad \mathbf{A}^3,
for the unit circle, the projective plane and the affine 3-space respectively, because only for the third it is true that
\mathbf{A}^m=(\mathbf{A}^1)^m\ldots

(2) Contrary to popular (e.g., mine…) belief, there is one canonical finite field besides the fields of prime order. It is \mathbf{F}_4, which is canonical because there is a unique irreducible quadratic polynomial over \mathbf{F}_2, so that
\mathbf{F}_4\simeq \mathbf{F}_2[X]/(X^2+X+1).

(3) For the same reason, Bourbaki regrets the notation
\mathbf{F}_q
for all finite fields, because it was their tradition to use bold fonts exclusively for objects which are completely canonical. Serre gave an example of a statement in his paper on Propriétés galoisiennes des points de torsion des courbes elliptiques which, if read too quickly, could give the impression of leading to a contradiction or a mistake — but only if one believes that \mathbf{F}_{p^2} is an unambiguous notation…

And finally — any links to mathematical brilliance will be left for the reader to contemplate — I learnt that around the 1940’s in Southern France, wine was the usual drink in middle and high-school lunches and dinners.

Disjointed thoughts on the joint meeting

The title is a pretty facile pun, I admit, but I probably can’t manage much wittier after coming back from the AMS-MAA Joint Mathematical Meeting through a four hour delay in Memphis snow and another hour in Amsterdam exchanging an apparently broken 737 for a sounder one.

I had been invited by A. Salehi Golsefidy and A. Lubotzky to participate in their special session on expander graphs (slides to be found here), and since I had never attended such a big meeting, the occasion seemed as good as it could get — other tempting events happening then included Lubotzky’s Colloquium Lectures on expanders and the invited addresses of K. Soundararajan and A. Venkatesh.

Jordan Ellenberg also gave a talk at the expander session on our joint work with C. Hall, already discussed previously. This meant that I could rely on his gastronomical acumen for my first meals in New Orleans — a topic not to be tossed aside lightly (nor thrown away with great force, either). His heuristic technique was perfectly successful, and besides a fair amount of charcuterie, our dinner at Cochon included quite tasty fried alligator.

Random thoughts follow:

  • Lubotzky’s lecture notes are excellent; especially fascinating is the mention that the first sighting of expanders, earlier typically attributed to Pinsker in 1973, has been found in a paper of Barzdin and Kolmogorov (Problemy Kibernetiki 1967, 19, 261–268), adding one more item to the long list of discoveries connected with Kolmogorov’s name. I hope to write a more complete post on this paper, which is very interesting (there is an English translation in Kolmogorov’s selected works).
  • A mathematical meeting is even better when there is a good second-hand bookstore nearby, in this case, “Crescent City Books”; readers looking for a particular volume of the translations of the Proceedings of the Steklov Institute have a good chance of finding it there. For my own part, I bought a used copy of C. Spurgeon’s book on Shakespeare’s imagery, which is somewhat bardolatric, but fascinating anyway. (Quick do-you-think-like-Shakespeare-quizz: what is the first, or most vivid, image that comes to your mind about snails?)
  • My last meal in New Orleans was the least satisfactory; however, a better one would probably not have left me enough time to wander around and stumble upon the excellent Idea Factory store, from which I left with some nice three-dimensional animal puzzles for my family…
  • Coincidentally, my advisor, H. Iwaniec, received the Steele Prize for Mathematical Exposition during the meeting, especially for his graduate textbooks on automorphic forms — as I was one among many who learnt a lot from them, this was a great occasion to celebrate!
  • Ancient astronomy was a bit on my mind, as I spent a lot of my time waiting in- or out-side airplanes listening to P. Glass’s opera Kepler, and reading a recent lively biography of Galileo. If you are doing the same, you may be interested in a new website of digitized books of old astronomy; I am planning to have a long look at some of the books where friends and foes of Galileo traded insults, insights, theoretical absurdities and experimental masterpieces…

Utilizing makes master

French people, as some of you may have noticed, can be a bit fractious when it comes to language use. I usually don’t care myself about real or perceived anglicisms (saying “conférence” instead of “colloque“, oh the horror!) but I’ve recently become susceptible to one special case, which gives me an idea of why people may take this kind of things so seriously.

I speak here of what seems to be a new rash of use of the verb “utilize” in English. Every time I read it, I shudder from head to foot — why not use the simpler, rounder and altogether nicer “use“? Psychologically, this is somewhat amusing because, after all, “utiliser” is the French version of the word, and also I don’t mind “utility” at all. I have the impression that for some reason I get annoyed because I don’t really know how to parse or say the word internally (where is the accent?), and this might be just because I have never (that I remember) heard this word used in a way which would make it sound good.

The other question is whether it is really something new, or happening more often, or if — somehow — I just managed to miss it before? (According to Google, it seems “utilize” is rather decreasing in use at the moment; but maybe it is rising in certain places, e.g. on the internet, and not so much in books?)

On Coscinomancy

Thanks to a query of A. Venkatesh, I have just been looking at the tangled and instructive etymology of the word sieve…

To the Greeks, a sieve is a koskinon (κóσκινον), whether it be of use as a cooking utensil, or as devised by that clever fellow, Eratosthenes, to find prime numbers. But the English word has a completely different etymology, of Germanic origin if the OED is to be believed.

From the same source, I’ve learnt that a coarse-meshed sieve can also properly be called a ridder or a riddle (the latter being an alteration of the former). In this form, it is said to be related to the Indo-European stem kreit-, which is also, as it turns out, (carrying a meaning of to separate, to judge) at the root of many fine words, such as crisis, critic, criterion; through the Latin cribrum, it leads to the French word for sieve, namely crible. (Though, as far as cooking is concerned, the instrument is rather called a tamis in French cuisine; this last word, although considered obsolete, does exist in English, as does the variant temse…)

As for the Greek word, it has left no trace in French, and (apparently) remains in English only under the guise of a delightful word, coscinomancy, which I regret not having known about at the time of deciding on the title of my inaugural lecture:

coscinomancy, n.
Pronunciation: /ˈkɒsɪnəʊˌmænsɪ/

Etymology:
< medieval Latin coscinomantia, < Greek κοσκινόμαντις, < κόσκινον sieve.

Divination by the turning of a sieve (held on a pair of shears, etc.).

1603 C. HEYDON Def. Iudiciall Astrol. xvii. 356 Comparing Astrologie with Aruspicie, Hydromancie, Chiromancie, Choschinomancie, and such like.
1653 H. MORE Antidote Atheism (1712) III. ii. 89 Coskinomancy, or finding who stole or spoiled this or that thing by the Sieve and Shears.
1777 J. BRAND Observ. Pop. Antiq. (1870) III. 301–2.
1871 E. B. TYLOR Primitive Culture I. 116 The so-called coscinomancy, or, as it is described in Hudibras, ‘th' oracle of sieve and shears’.

(To quote the OED again; note the amusing oscillations of the popularity of this word…)

GANT Winter School draft schedule

The first draft of a schedule for the GANT Winter School, co-organized by Ph. Michel and myself from Jan. 18 to 28, 2011, at the Bernoulli Center of EPF Lausanne (or, as we like to say here in Zürich, ETH Lausanne) is now available. For the moment, it is found on Google Calendar to accomodate possible tweaks, but a downloadable PDF will also be produced soon for offline convenience. The names of the courses can easily be correlated with those on the updated poster; for convenience, here is the list (in chronological order of appearance during the school):

  • “Automorphic forms” — “Modular forms, automorphics forms and automorphic representations”, E. Lapid and Ph. Michel
  • “Sieve methods” — “Sieve methods”, É. Fouvry and myself
  • “Growth and expansion in groups” — “Sum-product phenomena, expansion in groups and applications”, H. Helfgott and P. Varjú
  • “Sato-Tate”: “La conjecture de Sato-Tate” — J.-P. Serre
  • “Spectral and ergodic methods” — “Point counting on homogeneous varieties: spectral and ergodic-theoretic methods”, M. Einsiedler and S. Mozes
  • “Point counting on algebraic varieties” — “Point counting on algebraic varieties and analytic number theory”, T. Browning and R. de la Bretèche

We will also provide abstracts of the courses fairly soon…

(By the way, does anyone know which — if any — URL incantation can force the calendar link to open on the correct date in 5-day-week-display mode?)