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.

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.

Semiclassical Eigenfunction Estimates

Melissa Tacy
Institute for Advanced Study
October 29, 2010

ANALYSIS/MATHEMATICAL PHYSICS SEMINAR

Concentration phenomena for Laplacian eigenfunctions can be studied by obtaining estimates for their $L^{p}$ growth. By considering eigenfunctions as quasimodes (approximate eigenfunctions) within the semiclassical framework we can extend such estimates to a more general class of semiclassical operators. This talk will focus on $L^{p}$ estimates for quasimodes restricted to hypersurfaces and the links between such estimates and properties of classical flow.