## Potential Automorphy for Compatible Systems of l-Adic Galois Representations

David Geraghty
Princeton University; Member, School of Mathematics
November 18, 2010

I will describe a joint work with Barnet-Lamb, Gee and Taylor where we establish a potential automorphy result for compatible systems of Galois representations over totally real and CM fields. This is deduced from a potential automorphy result for single l-adic Galois representations satisfying a `diagonalizability' condition at the places dividing l.

## Lecture 6

Pierre Colmez
National Center for Scientific Research; Member, School of Mathematics
November 18, 2010

## A New Approach to the Local Langlands Correspondence for $GL_n$ Over p-Adic Fields

Peter Scholze
University of Bonn
November 17, 2010

## Configuration Spaces of Hard Discs in a Box

Matthew Kahle
November 15, 2010

The "hard discs" model of matter has been studied intensely in statistical mechanics and theoretical chemistry for decades. From computer simulations it appears that there is a solid--liquid phase transition once the relative area of the discs is about 0.71, but little seems known mathematically. Indeed, Gian-Carlo Rota suggested that if we knew the total measure of the underlying configuration space, "we would know, for example, why water boils at 100 degrees on the basis of purely atomic calculations."

## Fractional Perfect Matchings in Hypergraphs

Andrzej Rucinski
Adam Mickiewicz University in Polznan, Poland; Emory University
November 15, 2010

A perfect matching in a k-uniform hypergraph H = (V, E) on n vertices
is a set of n/k disjoint edges of H, while a fractional perfect matching
in H is a function w : E → [0, 1] such that for each v ∈ V we have
e∋v w(e) = 1. Given n ≥ 3 and 3 ≤ k ≤ n, let m be the smallest
integer such that whenever the minimum vertex degree in H satisfies
δ(H) ≥ m then H contains a perfect matching, and let m∗ be defined
analogously with respect to fractional perfect matchings. Clearly, m∗ ≤
m.

## Counting Pattern Avoiding Permutations Via Integral Operators

Richard Ehrenborg
University of Kentucky; Institute for Advanced Study
November 23, 2010

## Modularity of Galois Representations

Chandrashekhar Khare
University of California, Los Angeles
November 22, 2010

In this expository talk, I will outline a plausible story of how the study of congruences between modular forms of Serre and Swinnerton-Dyer, which was inspired by Ramanujan's celebrated congruences for his tau-function, led to the formulation of Serre's modularity conjecture. I will give some hints of the ideas used in its proof given in joint work with J-P. Wintenberger. I will end by pointing out just one of the many interesting obstructions to generalising the strategy of the proof to get modularity results in more general situations.

## Historical Studies and Social Science: An Illustrated History

George Dyson
November 12, 2010

This lecture was part of the Institute for Advanced Study’s celebration of its eightieth anniversary, and took place during the events related to the Schools of Historical Studies and Social Science.