Title: Distribution of powers of random unitary matrices through singularities of hyperplane arrangements

Abstract: 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? 
In the first part of the talk, I will answer (a) and (b). In addition, using works of Rains about the distribution of X^m, I will show how Items (a) and (b) are closely related (and that (b) is in fact equivalent) to a geometric problem about the singularities of certain varieties called (Weyl) hyperplane arrangements.
In the second part of the talk, I will sketch the proof of this geometric problem, building on a known algorithm for resolution of singularities of hyperplane arrangements.

Based on joint works with Julia Gordon and Yotam Hendel ([GGH]), and with Nir Avni and Michael Larsen ([AGL]).