# School of Mathematics

## Members Seminar

## Potential Automorphy

I will introduce l-adic representations and what it means for them to be automorphic, talk about potential automorphy as an alternative to automorphy, explain what can currently be proved (but not how) and discuss what seem to me the important open problems. This should serve as an introduction to half the special year for non-number theorists. The other major theme will likely be the `p-adic Langlands program', which I will not address (but perhaps someone else will).

## Super-uniformity of the typical billiard path (proof included)

I will describe the proof of the following surprising result: the typical billiard paths form the family of the most uniformly distributed curves in the unit square. I will justify this vague claim with a precise statement. As a byproduct, we obtain the counter-intuitive fact that the complexity of the test set is almost irrelevant. The error term is shockingly small, and it does not matter that we test uniformity with a nice set (like a circle or a square), or with an arbitrarily ugly Lebesgue measurable subset of the unit square.

## STPM - L^p Concentration of Semiclassical Quasimodes

## STPM - Torsion-Freeness of Certain Cohomology Groups of PEL-Type Shimura Varieties

## STPM - Sparce Approximation of PSD Matrices

I will discuss the problem of approximating a given positive semidefinite matrix A , written as a sum of outer products $vv^T$ , by a much shorter weighted sum in the same outer products. I will then mention an application to sparsification of finite undirected graphs.