Mathematics – Page 11 – E. Kowalski's blog

Who proved the Peter-Weyl theorem for compact groups?

Tamas Hausel just asked me (because of my previous post on the paper of Peter and Weyl) how could Peter and Weyl have proved the “Peter-Weyl Theorem” for compact groups in 1926, not having Haar measure at their disposal? Indeed, Haar’s work is from 1933! The answer is easy to find, although I had completely overlooked the point when reading the paper: Peter and Weyl assume that their compact group is a compact Lie group, which allows them to discuss Haar measure using differential forms!

So the question is: who first proved the full “Peter-Weyl” Theorem for all compact groups? Pontryaguin, in 1936, certainly does, without remarking that Peter-Weyl didn’t, possibly because it was clear to anyone that the argument would work as soon as an invariant measure was known to exist. But since there are “easier” proofs of the existence of Haar measure for compact groups than the general one for all locally-compact groups (using some kind of fixed-point argument), it is not inconceivable that someone (e.g., von Neumann) might have made the connection before.

In fact, there is an amusing mystery in connection with Pontryaguin’s paper and von Neumann: concerning Haar measure, he refers to a paper of von Neumann entitled Zum Haarschen Mass in topologischen Gruppen, and gives the helpful reference Compositio Math., Vol I, 1934. So we should be able to read this paper on Numdam? But no! The first volume of Compositio Mathematica there is from 1935; it is identified as Volume I, and there is no paper of von Neumann to be found…

[Update: as many people pointed out, the paper of von Neumann is indeed on Numdam, but appeared in 1935; I was tricked by the absence of 1934 on the Compositio archive and the author’s name being written J.V. Neumann (I had searched Numdam with “von Neumann” as author…)]

Radio

For those readers who understand spoken French (or simply appreciate the musicality of the language) and are interested in the history of mathematics, I warmly recommend listening to the recording of a recent programme of Radio France Internationale entitled “Pourquoi Bourbaki ?” In addition to the dialogue of Sophie Joubert with Michèle Audin and Antoine Chambert-Loir, one can hear some extracts of older émissions with L. Schwartz, A. Weil, H. Cartan, J. Dieudonné, for instance.

Аналитическая теория чисел

Thanks to the recent Russian translation of my book with Henryk Iwaniec, I can now at least read my own last name in Cyrillic; I wonder what the two extra letters really mean…

More conferences

It seems that most of my posts these days are devoted to announcing conferences in which I am involved as organizer… Indeed, there are two coming up this year (actually three, if I count the MSRI summer school):

(1) May 14 and 15, we will have the Number Theory Days 2015 at EPF Lausanne; the speakers are Gaetan Chenevier, Henryk Iwaniec, Alena Pirutka, Chris Skinner and Zhiwei Yun; this is co-organized by Ph. Michel and myself.

(2) Immediately afterward, from May 18 to 22, comes a conference at FIM, co-organized by H. Iwaniec, Ph. Michel and myself, with the title of “Analytic Aspects of Number Theory”; the current list of speakers is to be found on the web page; here is the poster (which is based on a picture taken by Henryk around Zürich last Fall):

AANT Poster — Analytic Aspects of Number Theory

Most importantly, there is a certain amount of funding available for local expenses of your researchers (doctoral and postdoctoral students). Applications can be made here (before Feburuary 6; the form states January 28, but this is an error that will be corrected).

A (not so well-known) theorem of Fouvry, and a challenge

A few weeks ago, as already mentioned, I was in Oxford for the LMS-CMI summer school on bounded gaps between primes. My mini-course on this occasion was devoted to the ideas and results underlying Zhang’s original approach, based on expanding the exponent of distribution of primes in arithmetic progressions to large moduli.

In the first lecture, I mentioned a result of Fouvry as a motivation behind the study of other arithmetic functions in arithmetic progressions: roughly speaking, if one can prove that the exponent of distribution of the divisor functions $d_1$ ,…, $d_6$ is strictly larger than $1/2$ , then the same holds for the primes in arithmetic progressions.

This statement (which I will make more precise below, since there are issues of detail, including what type of distribution is implied) is very nice. But it turned out that quite a few people at the school were not aware of it before. The reason is probably to a large extent that, as of today (and as far as I know…), it has not been possible to use this mechanism to prove unconditional results about primes: the problem is that one does not know how to handle divisor functions beyond $d_3$ … One could in fact interpret this as saying that higher divisor functions are basically as hard as the von Mangoldt function when it comes to such questions.

The precise statement of Fouvry is Theorem 3 in his paper “Autour du théorème de Bombieri-Vinogradov” (Acta Mathematica, 1984). The notion of exponent of distribution of a function $f(n)$ concerns a fixed residue class $a$ , and the average over moduli $q\leq x^{\theta}$ (with $q$ coprime to $a$ ) for some $\theta>1/2$ of the usual discrepancy
$\sum_{q\leq x^{\theta}} \Bigl|\sum_{n\equiv a\text{mod } q}f(n)-\frac{1}{\varphi(q)}\sum_{n}f(n)\Bigr|.$

The actual assumptions concerning $d_i$ , $1\leq i\leq 6$ , is a bit more than having this exponent of distribution $>1/2$ : this must be true also for all convolutions
$d_i\star \lambda$
where $\lambda(n)$ is an arbitrary essentially bounded arithmetic function supported on a very short range $1\leq n\leq x^{\delta_i}$ for some $\delta_i>0$ .

This extra assumption is reasonable because since $\delta_i$ can be arbitrarily small, certainly all known methods to prove exponents of distribution larger than $1/2$ would accommodate this tweak.

As far as the proof is concerned, this Theorem 3 is actually rather “simple”: using the Heath-Brown identity, all the hard work is moved to the proof of an exponent of distribution beyond $1/2$ for the characteristic function of integers $n$ having no prime factors $\leq z$ for $n\leq x$ and $z\leq x^{1/6-\varepsilon}$ . This is much deeper, and involves all the machinery of dispersion and Kloostermania…

In addition, Fouvry mentioned to me the following facts, which I didn’t know, and which are very interesting from a historical point of view. First, this theorem of Fouvry is a strengthened version of the results of Chapter III of his Thèse de Doctorat d’État (Bordeaux, September 1981, supervised by J-M. Deshouillers and H. Iwaniec). At that time, Kloostermania was under construction and Fouvry had only Weil’s classical bound for Kloosterman sums at his disposal, and this original version required an exponent of distribution beyond $1/2$ for the functions $d_1, d_2, \ldots, d_{12}$ . This illustrates the strength of Kloostermania!

Moreover, in this thesis, Fouvry used an iteration of Vaughan’s identity, instead of Heath-Brown’s identity, which only apparead in 1982. However, although this was less elegant, this iteration had the same property to transform a sum over primes into multilinear sums where all non smooth variables have small support near the origin.

Fouvry also suggests the following inverse challenge for aficionados: assuming an exponent of distribution $\theta>1/2$ for the sequence of primes, can one prove a similar exponent of distribution for all the divisor functions $d_k$ ?