# Expanders course

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 $\mathrm{SL}_2(\mathbf{F}_{p})$, with respect to the projections modulo primes $p$ of a subset $S\subset \mathrm{SL}_2(\mathbf{Z})$ which generates a Zariski-dense subgroup of $\mathrm{SL}_2$, 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…)