## 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

Richard Ehrenborg

University of Kentucky; Member, School of Mathematics

November 2, 2010

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

Matthew Emerton

Northwestern University

November 2, 2010

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.

Elad Haramaty

Technion

November 1, 2010

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.

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.

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.

Enrico Bombieri

Institute for Advanced Study

October 29, 2010

In this lecture, Enrico Bombieri, IBM von Neumann Professor in the School of

Mathematics, attempts to give an idea of the numerous different notions of truth in mathematics. Using accessible examples, he explains the difference between truth, proof, and verification. Bombieri, one of the world’s leading authorities on number theory and analysis, was awarded the Fields Medal in 1974 for his work on the large sieve and its application to the distribution of prime numbers. Some of his work has potential practical applications to cryptography and security of data transmission and identification.Florian Herzig

Institute for Advanced Study

October 28, 2010

We will discuss a generalisation of Serre's conjecture on the possible weights of modular mod p Galois representations for a broad class of reductive groups. In good cases (essentially when the Galois representation is tamely ramified at p) the predicted weight set can be made explicit and compared to previous conjectures. This is joint work with Toby Gee

and David Savitt.

Jared Weinstein

Institute for Advanced Study

October 27, 2010

The usual Katz-Mazur model for the modular curve $X(p^n)$ has horribly singular reduction. For large n there isn't any model of $X(p^n)$ which has good reduction, but after extending the base one can at least find a semistable model, which means that the special fiber only has normal crossings as singularities. We will reveal a new picture of the special fiber of a semistable model of the entire tower of modular curves. We will also indicate why this problem is important from the point of view of the local Langlands correspondence for $GL(2)$ .

Margaret Readdy

University of Kentucky; Member, School of Mathematics

October 26, 2010

M. Sodin

Tel Aviv University

October 26, 2010

In the talk, I will describe recent attempts to understand the mysterious and beautiful geometry of nodal lines of random spherical harmonics and of random plane waves. If time permits, I will also discuss asymptotic statistical topology of other natural polynomial-like ensembles of random functions. The talk is based on a joint work with Fedja Nazarov.