In the representation theory of reductive p-adic groups, a lot of general structure was developed by Bernstein: supercuspidal representations, cuspidal support, Bernstein components and so on.

Title: Applications of the Ky Fan inequality to random (and almost periodic) operators
Abstract: We shall discuss the Ky Fan inequality for the eigenvalues of the sum of two Hermitian matrices. As an application, we shall derive a sharp version of a recent result of Hislop and Marx pertaining to the dependence of the integrated density of states of random Schroedinger operators on the distribution of the potential. Time permitting, we shall also discuss an application to quasiperiodic operators.

Let G be the group SL(2) over a finite extension F of Q_p, p odd. For a fixed r ≥ 0, we identify the elements of the Bernstein center of G supported in the Moy-Prasad G-domain G_{r^+}, by characterizing them spectrally.
We study the behavior of convolution with such elements on orbital integrals of functions in C^∞_c(G(F)), proving results in the spirit of semisimple descent.
These are ‘depth r versions’ of results proved for general reductive groups by J.-F. Dat, R. Bezrukavnikov, A. Braverman and D. Kazhdan.

Title:
Curves with boundary: Can you count them? Can you really?
Abstract:

Title: Systems of points with Coulomb interactions
Abstract:  Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability.