Let X be a n by n unitary matrix, drawn at random according to the Haar measure on U_n, and let m be a natural number. What can be said about the distribution of X^m and its eigenvalues? 

The density of the distribution \tau_m of X^m can be written as a linear combination of irreducible characters of U_n, where the coefficients are the Fourier coefficients of \tau_m. In their seminal work, Diaconis and Shahshahani have shown that for any fixed m, the sequence (tr(X),tr(X^2),...,tr(X^m)) converges, as n goes to infinity, to m independent complex normal random variables (suitably normalized). This can be seen as a statement about the low-dimensional Fourier coefficients of \tau_m. 

In this talk, I will focus on high-dimensional spectral information about \tau_m. For example: 

(a) Can one give sharp estimates on the rate of decay of its Fourier coefficients?

(b) For which values of p, is the density of \tau_m  L^p-integrable? 

Using works of Rains about the distribution of X^m, we will see how Item (a) is equivalent to a branching problem in the representation theory of certain compact homogeneous spaces, and how (b) is equivalent to a geometric problem about the singularities of certain varieties called (Weyl) hyperplane arrangements.

Based on joint works with Julia Gordon and Yotam Hendel and with Nir Avni and Michael Larsen.

How to uniformly, or at least almost uniformly, choose an element from a finite group G? When G is too large to enumerate all its elements, direct (pseudo)random selection is impossible. However, if we have an explicit set of generators of G (e.g., as in the Rubik's cube group), several methods are available. This talk will focus on one such method based on the well-known product replacement algorithm. I will discuss how recent results on property (T) by Kaluba, Kielak, Nowak and Ozawa partially explain the surprisingly good performance of this algorithm.