# School of Mathematics

## p-adic Hodge-theoretic properties of etale cohomology with mod p coefficients, and the cohomology of Shimura varieties - 2

## Local-global compatibility at primes dividing l

## Error-Correcting Codes

## Perfectoid spaces

## A semistable model for the tower of modular curves

## Workshop on Galois Representations and Automorphic Forms

March 21-25,2011

## On the Fourier Spectrum of Symmetric Boolean Functions

It is well-known that any Boolean function f:{-1,+1}^n \to {-1,+1} can be written uniquely as a polynomial f(x) = \sum_{S subset [n]} f_s \prod_{i in S} x_i. The collection of coefficients (f_S's) this expression are referred to (with good reason) as the Fourier spectrum of f. The Fourier spectrum has played a central role in modern computer science by converting combinatorial and algorithmic questions about f into algebraic or analytic questions about the spectrum.

## Analytic Geometry Over F_1

I'll talk on work in progress on algebraic and analytic geometry over the field of one element F_1. This work originates in non-Archimedean analytic geometry as a result of a search for appropriate framework for so called skeletons of analytic spaces and formal schemes, and is related to logarithmic and tropical geometry. I'll explain what analytic spaces over F_1 are, and will describe non-Archimedean and complex analytic spaces which are obtained from them.

## Galois Representations Associated to Holomorphic Limits of Discrete Series

We attach Galois representations to automorphic representations on unitary groups whose weight (=component at infinity) is a holomorphic limit of discrete series. The main innovation is a new construction of congruences, using the Hasse Invariant, which avoids q-expansions and so is applicable in much greater generality than previous methods. Our result is a natural generalization of the classical Deligne-Serre Theorem on weight one modular forms and work of Taylor on GSp(4).