## A Satake Isomorphism $mod.p$

## Local-Global Compatibility in the p-Adic Langlands Progra for GL(2) over Q

I will outline the proof of various cases of the local-global compatibility statement alluded to in the title, and also explain its applications to the Fontaine--Mazur conjecture, and to a conjecture of Kisin.

## Fourier Spectrum of Polynomials Over Finite Fields

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.

## 3/2 Firefighters Are Not Enough

## The Topology of Restricted Partition Posets

The d-divisible partition lattice is the collection of all partitions of an n-element set where each block size is divisible by d. Stanley showed that the Mobius

## Local-Global Compatibility in the p-Adic Langlands Program for GL(2) over Q

I will outline the proof of various cases of the local-global compatibility statement alluded to in the title, and also explain its applications to the Fontaine—Mazur conjecture, and to a conjecture of Kisin.

## On the Structure of Cubic and Quartic Polynomials

In our work we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results.

## Shimura Varieties, Local Models and Geometric Realizations of Langlands Correspondences

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

**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.