Shimura Varieties, Local Models and Geometric Realizations of Langlands Correspondences

Elena Mantovan
California Institute of Technology; Member, School of Mathematics
November 1, 2010

I will introduce Shimura varieties and discuss the role they play in the conjectural relashionship between Galois representations and automorphic forms. I will explain what is meant by a geometric realization of Langlands correspondences, and how the geometry of Shimura varieties and their local models conjecturally explains many aspects of these correspondences. This talk is intended as an introduction for non-number theorists to an approach to Langlands conjectures via arithmetic algebraic geometry.

Fourier Spectrum of Polynomials Over Finite Fields

Shachar Lovett
Institute for Advanced Study
November 2, 2010

Let $f(x_1,...,x_n)$ be a low degree polynomial over $F_p$. I will prove that there always exists a small set $S$ of variables, such that `most` Fourier coefficients of $f$ contain some variable from the set $S$. As an application, we will get a derandomized sampling of elements in $F_p^n$ which `look uniform` to $f$.

The talk will be self contained, even though in spirit it is a continuation of my previous talk on pseudorandom generators for $CC0[p]$. Based on joint work with Amir Shpilka and Partha Mukhopadhyay.

Ground States of the 2D Edwards-Anderson Spin Glass

Michael Damron
Princeton University
November 5, 2010

I will discuss the problem of determining the number of infinite-volume ground states in the Edwards-Anderson (nearest neighbor) spin glass model on $Z^D$ for $D \geq 2$. There are no complete results for this problem even in $D=2$. I will focus on this case and explain recent results which go some way toward proving that (with zero external field, so that ground states come in pairs, related by a global spin flip) there is only a single ground state pair (GSP).