Today’s Google doodle celebrates Emmy Noether.

Algebra would look very different without her (“successive sets of symbols with the same second suffix“).

Comments on mathematics, mostly.

23.03.2015
### Noether on Google

Today’s Google doodle celebrates Emmy Noether.

Algebra would look very different without her (“successive sets of symbols with the same second suffix“).

16.03.2015
### A parity lemma of A. Irving

In his recent work on the divisor function in arithmetic progressions to smooth moduli, A. Irving proves the following rather amusing lemma (see Lemma 4.5 in his paper):

LemmaLet be an odd prime number, let be an integer and let be a -tuple of elements of . For any subset of , denote

and for any , let

denote the multiplicity of among the .

Then if none of the is zero, there exists some for which isodd.

I will explain two proofs of this result, first Irving’s, and then one that I came up with. I’m tempted to guess that there is also a proof using some graph theory, but I didn’t succeed in crafting one yet.

**Irving’s proof.** This is very elegant. Let be a primitive -th root of unity. We proceed by contraposition, hence assume that all multiplicities are even. Now consider the element

of the cyclotomic field . By expanding and using the assumption we see that

In particular, the norm (from to ) of is an even integer, but because is odd, the norm of is known to be odd for all . Hence some factor must have , as desired.

**A second proof.** When I heard of Irving’s Lemma, I didn’t have his paper at hand (or internet), so I tried to come up with a proof. Here’s the one I found, which is a bit longer but maybe easier to find by trial and error.

First we note that

is even. In particular, since is odd, there is at least some with *even*.

Now we argue by induction on . For , the result is immediate: there are two potential sums and , and so if , there is some odd multiplicity.

Now assume that and that the result holds for all -tuples. Let be a -tuple, with no equal to zero, and which has all multiplicities even. We wish to derive a contradiction. For this, let . For any , we have

by counting separately those with sum which contain or not.

Now take such that is odd, which exists by induction. Our assumptions imply that is also odd. Then, iterating, we deduce that is odd for all integers . But the map is surjective onto , since is non-zero. Hence our assumption would imply that all multiplicities are *odd*, which we have seen is not the case… Hence we have a contradiction.

15.03.2015
### 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…)]

27.02.2015
### 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.

09.02.2015
### Аналитическая теория чисел

« Previous Page — Next Page »
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…

© 2015 E. Kowalski's blog Hosted by uzh|ethz Blogs