## Overview of the p-Adic Local Langlands Correspondence for $GL(2,Q_p)$

(Introduction to the Lecture Series and and overview for those unable to attend the whole Lecture Series)

Pierre Colmez

National Center for Scientific Research

October 7, 2010

(Introduction to the Lecture Series and and overview for those unable to attend the whole Lecture Series)

Frank Calegari

Northwestern University; Member, School of Mathematics

October 6, 2010

Shachar Lovett

Institute for Advanced Study

October 5, 2010

October 5, 2010

Matthew Kahle

Institute for Advanced Study

October 5, 2010

In this talk I will overview two very different kinds of random simplicial complex, both of which could be considered higher-dimensional generalizations of the Erdos-Renyi random graph, and discuss what is known and not known about the expected topology of each. Some of this is joint work with Eric Babson and Chris Hoffman.

Richard Taylor

Institute for Advanced Study

October 4, 2010

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).

Institute for Advanced Study

October 4, 2010

Jozsef Beck

Institute for Advanced Study

October 4, 2010

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.

Srikanth Srinivasan

Institute for Advanced Study

September 30, 2010

Nikhil Srivastava

Institute for Advanced Study

September 30, 2010

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.