I am teaching a class on expander graphs this semester, and as usual there is a web page about it. For the moment, do not be fooled by the link to “lecture notes”, since I haven’t put up anything yet (I am trying to learn the easiest way to incorporate nice pictures of graphs in LaTeX; tkz-graph and tkz-berge seem to be the best way to do that…)
The course meets only 2 hours per week (and those are ETH hours, which means 1h30 really), so it will necessarily be very restricted in what I can cover. I have decided on the content of the first two parts: a general discussion of elementary material (of course), and then a full account of the combination of the results of Helfgott and Bourgain–Gamburd, which together imply that the family of Cayley graphs of , with respect to the projections modulo primes of a subset which generates a Zariski-dense subgroup of , is an expander family.
If there is time after this, which I hope, I haven’t quite decided what to do. One possibility would be an account of the “expander philosophy” discussed at the end of this post, another would be to discuss some applications to theoretical computer science (admittedly, this would be because I would learn about them much more than I know at the current time…)
After a long hiatus, I have just put up online an updated version of my lecture notes on representation theory. The delay was psychologically interesting: after a long period where I added material more or less in the order I wanted it to appear in the text, I started in June to proceed in much more chaotic (or random?) manner, with an explanation of the Larsen alternative for unitary groups coming before the Peter-Weyl theorem, and so son. Inly in the last few days did the text regain at least some coherence. (In particular, it took me a long time to finally sit up and write an account of the Peter-Weyl theorem that I felt to be at least somewhat motivated.)
There are still things missing before the notes contain all that I’d like, in particular at least a few pages of survey concerning the representation theory of some locally compact, non compact, groups. There are still a few weeks before the beginning of the new semester, however, and hopefully I will have time to do some work on this part in the coming weeks…
As usual, any remarks or corrections will be very appreciated!
The fundamental fact about the representation theory of a compact topological group is the Peter-Weyl Theorem, which can be described as follows: the regular representation of on (defined using the probability Haar-measure on ) decomposes as an orthogonal Hilbert direct sum of the spaces of matrix coefficients of the finite-dimensional irreducible representations of . (Books, for instance the one of Knapp on semisimple Lie groups, often include more in what is called Peter-Weyl Theory, but this is the statement that is proved in the original paper.)
As I am currently preparing to write down a proof of the Peter-Weyl theorem for my notes on representation theory, I had a look at this paper. Although I probably won’t follow it quite completely, I found it very interesting — it is quite subtly different from all modern treatments I have seen, in an interesting way, and without being much more complicated than what is found, e.g., in Knapp’s book. (For a short masterful online account, this post of Terry Tao is very good; from a search in the same Göttingen archive web site, it seems that — maybe — the modern treatment dates back to Pontryagin, in 1936.)
In any case, the question for today is: Who was Peter? Only the initial “F.” and the affiliation “in Karlsruhe” identifies this coauthor on the original paper (even the “F.” is misread as “P.” on the PDF cover page…) It seem he was a student of Weyl, but note that there is no Peter on the math genealogy page for Weyl. (A joke here at ETH, is that when the lecture room known as the Hermann Weyl Zimmer is renovated this summer, some unfortunate skeletons will be found in the closets and under the floor, or behind the blackboards…)
The Spring semester at ETH is starting next week, and I will be teaching an introductory course on representation theory (for third-year and later students). I am looking forward to it, since this is the first time I teach a class identified as “algebra” (except for linear algebra, of course).
My lecture notes will be available as I prepare them (the first chapter will be ready soon, at least in draft form) and it will be seen that (partly because of my own bias) I think of a representation of a group as a homomorphism , and not as modules over the group algebra. I certainly intend to mention the latter approach at some point (indeed, I have in preparation a long “Variants” chapter, where I mention also, e.g., topological representations, unitary representations, permutation representations, projective representations, Lie algebra representations, etc, with basic examples), but I find it useful to go through the elementary theory of representations completely “by hand”, for a course at this level. In some sense, this is because this is what I can do best, with my own knowledge; but also, going through these basic facts purely with the tools and language of representations-as-homomorphisms does provide excellent opportunities to start seeing various ways of using representation theory (I will for instance prove Burnside’s irreducibility criterion and the linear independence of matrix coefficients using my understanding of Burnside’s original arguments). And I do intend to use this course to introduce some functorial language to the students, and I feel that abstract nonsense will be quite appealing after trying to make sense of the confusion that may arise from proving the transitivity formula for induction strictly using the “subset of functions on the group” model for induced representations…
Here’s a question about the module-over-the-group-algebra approach: what is the first (or simplest, or most fundamental) argument where it is better? The one I can think of for the moment — after going through the first chapter of my notes — is that it seems to give the simplest way to argue that a semisimple representation always contains an irreducible subrepresentation. (And of course this is only relevant for infinite-dimensional representations, since otherwise one can argue using dimension.)
[Update (23.2.2011): the first chapters are now online.]
Here’s an update on the front of exponential sums…
(1) The conference I co-organized at FIM on exponential sums over finite fields
ended about two weeks ago. It was quite nice (at least from my point of view…). As usual here, the excellent organization made it possible to enjoy the mathematics with no extra issues to take care of. Most talks were on blackboard but, with authorization, here are the files for those that were on beamer (I think I may miss one or two, which I will add later):
(2) The second part of my class on exponential sums (cohomological methods) continues. I only barely started the lecture notes I was planning to write, the first few weeks of the semester having been simply too busy. As a result, although they are available and will be updated regularly from now on, they are quite incomplete. Most damagingly, the beginning material (introduction and motivation for going towards describing algebraic exponential sums using traces of Frobenius on suitable “Galois” representations of fundamental groups) is mostly missing for the moment. Still, the end parts of Chapter III are written and the contents, from Chapter IV onwards, will hopefully be kept in sync with the class, and no further gaps will appear. Since Chapter IV will start introducing the Grothendieck-Lefschetz trace formula, the étale cohomology groups, and then go on towards the formalism of weights and Deligne’s general version of the Riemann Hypothesis, this might still be of interest even before I find the time (next semester, probably) for filling up the gaps.
My plan next year, when (and if) this second part is done nicely enough, will be to put it together with the first part of the class (on elementary methods) to produce a proper book on exponential sums over finite fields. I’ll be happy to receive even before any feedback on the way the text shapes up, especially from the point of view of people in analytic number theory or in other fields where exponential sums are applied.