Mushfeq Khan will speak on amenability and symbolic dynamics.

As usual the seminar is in Keller 314.

A Talk Story in Number Theory.

There is a childish misconception that the occupation of a professional mathematicians

is to operate with very big numbers. That is presumably primarily applicable to those who

do Number Theory. In this talk, I will show that this sometimes may be not too far from truth.

The talk is supposed to be entertaining and is directed to grad students willing to get a rough idea

about what it takes (and what it may give) to choose Number Theory as a research speciality.

Speaker: Claus Sorensen (UCSD)

Title: Local Langlands in rigid families

Abstract: The local Langlands correspondence attaches a representation of GL(n,F) to an n-dimensional representation of the Galois group of F (a local field). In the talk I will report on joint work with Johansson and Newton, in which we interpolate the correspondence in a family across eigenvarieties for definite unitary groups U(n). The latter are certain rigid analytic varieties parametrizing Hecke eigensystems appearing in spaces of p-adic modular forms. These varieties carry a natural coherent sheaf and we show that its dual fibers are built from the local Langlands correspondence by taking successive extensions; even at the non-classical points. Our proof employs certain elements of the Bernstein center which occur in Scholze’s trace identity. The first half of the talk is intended for a general audience with a limited background in number theory.

Continuing the theme of symbolic dynamics, I will demonstrate a proof of Simpson’s result that “Entropy = Dimension” for N^d and Z^d, and discuss some of Adam Day’s work generalizing these results to amenable groups.

This week Umar Gaffar will give Shelah’s proof of the following result:

Let $\lambda$ be the cardinality of an ultraproduct of finite sets. If $\lambda$ is infinite then $\lambda=\lambda^{\aleph_0}$.

Speaker: Rohit Nagpal

Title: Stability in the high dimensional cohomology of certain arithmetic groups

Abstract: Borel-Serre duality relates high dimensional cohomology of arithmetic groups to the low dimensional homology of these groups with coefficients in the Steinberg representation. We recall Bykovskii’s presentation for the Steinberg representation and explain its connection to modular symbols. Next, we describe the Steinberg representation as an object in a symmetric monoidal category, and use its presentation to describe an action of the free skew commutative algebra. Finally, we perform a Gröbner-theoretic analysis of this action to obtain new information on the homology of certain arithmetic groups with coefficients in the Steinberg representation. For example, we show that the sequence of homology groups H_1(Gamma_n(3), St_n) exhibit representation stability. This is an ongoing project with Jeremy Miller and Peter Patzt.

Speaker: Khrystyna Serhiyenko (UC Berkeley)

Title: Frieze patterns

Abstract: Frieze is a lattice of positive integers satisfying certain rules. Friezes were first studied by Conway and Coxeter in 1970′s, but they gained fresh interest in the last decade in relation to cluster algebras. In particular, there exists a bijection between friezes and cluster algebras of type A. Moreover, the categorification of cluster algebras developed in 2006 yields a new realization of friezes in terms of representation theory of Jacobian algebras. In this talk, we will discuss the beautiful connections between all these objects.

An operation called mutation is the key notion in cluster algebras. We will introduce a compatible notion of mutation for friezes and describe the resulting entries using combinatorics of quiver representations. We will also mention an important generalization of the classical friezes, called sl_k friezes and their connections to Grassmannians Gr(k,n).

Speaker: Kenshi Miyabe (Meiji University)

Title: Continuity of limit computable functions

Abstract:

The Darboux-Froda theorem says that, for every non-decreasing function,

the set of the points of non-continuity is at most countable.

This is probably the simplest case saying that every function

well-behaves at almost every point.

I introduce some similar theorems, and their computable versions.

Then, we discuss the relation with randomized algorithm.