# School of Mathematics

## 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.

## On the Communication Complexity of Classification Problems

We will discuss a model of distributed learning in the spirit of Yao's communication complexity model. We consider a two-party setting, where each of the players gets a list of labelled examples and they communicate in order to jointly perform a learning task. For example, consider the following problem of Convex Set Disjointness: In this instance Alice and Bob each receive a set of examples in Euclidean space and they need to decide if there exists a hyper-plane that separate the sets.

## A Tight Bound for Hypergraph Regularity

The hypergraph regularity lemma — the extension of Szemeredi's graph regularity lemma to the setting of k-graphs — is one of the most celebrated combinatorial results obtained in the past decade. By now there are various (very different) proofs of this lemma, obtained by Gowers, Rodl et al. and Tao. Unfortunately, what all these proofs have in common is that they yield partitions whose order is given by the k-th Ackermann function.

## Representations of p-adic groups

I will survey what is known about the construction of (the building blocks of) representations of p-adic groups, mention recent developments, and explain some of the concepts underlying all constructions. In particular, I will introduce filtrations of p-adic groups and indicate some of their remarkable properties.

## Some closure results for polynomial factorization

In a sequence of extremely fundamental results in the 80's, Kaltofen showed that any factor of n-variate polynomial with degree and arithmetic circuit size poly(n) has an arithmetic circuit of size poly(n). In other words, the complexity class VP is closed under taking factors.

A very basic question in this context is to understand if other natural classes of multivariate polynomials, for instance, arithmetic formulas, algebraic branching programs, bounded depth arithmetic circuits or the class VNP, are closed under taking factors.

## Local to global relations of periods

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.

## Diophantine approximation with arithmetically small points

## On the long-term dynamics of nonlinear dispersive evolution equations

We will give an overview of some of the developments in recent years dealing with the description of asymptotic states of solutions to semilinear evolution equations ("soliton resolution conjecture").

New results will be presented on damped subcritical Klein-Gordon equations, joint with Nicolas Burq and Genvieve Raugel.