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.

Speaker: Andrew Sale (Cornell U.)

Title: On the outer automorphism groups of right-angled Artin and Coxeter groups

Abstract: In geometric group theory, a fundamental, and broad, question to answer is that of understanding the world of finitely presented groups. Two of the simplest examples are free groups Fn and free abelian groups Z^n. With Fn and Z^n being their extreme examples, right-angled Artin groups (RAAGs) give us some idea of what happens “between” these groups. RAAGs are an important class of groups which appear in diverse situations, perhaps most significantly in Agol’s proof of the virtual Haken conjecture.

In studying their outer automorphism groups, we are looking at a class of groups that again interpolates between two classically important families of groups: Out(Fn), the outer automorphism group of Fn, and GL(n,Z). While there are numerous similarities between these families, they also differ in some important contexts. One such context concerns the nature of quotients that they have, and I will describe a couple of properties that make rigorous the notion of “having many quotients”. I will explain what happens for outer automorphism groups of RAAGs, and also the closely related family of right-angled Coxeter groups, and the consequences this has for Kazhdan’s Property (T).

Speaker: Sam Nariman (Northwestern U.)

Title: On the homology of diffeomorphism groups made discrete.

Abstract: Let $G$ be a finite dimensional Lie group and $G^{delta}$ be the same group with the discrete topology. The classifying space $BG$ classifies principal $G$-bundles and the classifying space $BG^{delta}$ classifies flat principal $G$-bundles (i.e. those bundles that admit a connection whose curvature vanishes). The natural homomorphism from $G^{delta}$ to $G$ induces a continuous map from $BG^{delta}$ to $BG$. Milnor conjectured that this map induces an equivalence after the profinite completion. In this talk, we discuss the same map for infinite dimensional Lie groups, in particular for diffeomorphism groups and symplectomorphisms. In these cases, we use techniques from homotopy theory to show that the map from $BG^{delta}$ to $BG$ induces a split surjection on cohomology with finite coefficients in the stable range. If time permits, I will discuss applications of these results in foliation theory, in particular, characteristic classes of flat surface bundles.