# School of Mathematics

## The Plancherel formula for L^2(GL_n(F)\GL_n(E)) and applications to the Ichino-Ikeda and formal degree conjectures for unitary groups

## Restriction problem for non-generic representation of Arthur type

## Relative character asymptotics and applications

## Representations of p-adic groups

Abstract: The building blocks for complex representations of p-adic groups are called supercuspidal representations. I will survey what is known about the construction of supercuspidal representations, mention questions that remain mysterious until today, and explain some recent developments.

## Supercuspidal L-packets

## Wave-front set of some representations of unipotent reduction of the group SO(2n+1)

## Boolean function analysis: beyond the Boolean cube.

Boolean function analysis traditionally studies Boolean functions on the Boolean cube, using Fourier analysis on the group Z_2^n. Other domains of interest include the biased Boolean cube, other abelian groups, and Gaussian space. In all cases, the focus is on results which are independent of the dimension.

## Local eigenvalue statistics of random band matrices

Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY) and transfer matrix approach to the analysis of local spectral characteristics of some specific types of RBM.

## Local to global relations of periods (continued)

Rankin-Selberg integrals provide factorization of certain period integrals into local counterparts. Other, more elusive, periods can be studied in principle by the relative trace formula and other methods.

Following Waldspurger, Ichino-Ikeda formulated a local to global conjecture about the Gross-Prasad periods. A more general setup was subsequently considered by Sakellaridis-Venkatesh.

I will discuss some of these principles as well as a result on Whittaker coefficients joint with Zhengyu Mao.