Title: Cohomology of smooth abelian group actions and applications of exotic topological groups

Speaker: Kurt Vinhage (University of Chicago)

Abstract: We will discuss the cohomology of homogeneous partially hyperbolic abelian group actions and recently (re)discovered tools which combine group homology, Lie criteria for topological groups, and some exotic topological groups, namely the free product of Lie groups. We will also discuss how these tools may be used to obtain smooth rigidity results, as well as other possible applications, including in the rank one setting.

Title: Khintchine recurrence for upper Banach density along filters

Speaker: John Johnson (Ohio State University)

Abstract: Khintchine’s recurrence theorem is a well-known extension of Poincaré’s recurrence theorem improving the “quality” of the recurrence in a measure-preserving transformation on a probability space: if $A$ is a subset of the probability space with positive measure, then the collection $\{ n \in \mathbb{N} : \mu(A \cap T^{-n}A) > 0 \}$ has bounded gaps. This recurrence theorem also has a combinatorial analog for sets with positive upper Banach density: if $A \subseteq \mathbb{N}$ has positive upper Banach density $d^*(A) > 0$, then the collection $\{ n \in \mathbb{N} : d^ *(A \cap -n + A) > 0 \}$ has bounded gaps. By iteration, this combinatorial theorem yields Szemerédi’s affine cube lemma as an easy corollary. (The affine cube lemma is one important component in the combinatorial proof of Roth’s theorem on three-term arithmetic progressions.)

In joint work with Florian Richter, we define a notion of upper Banach density, relative to filters on the positive integers, and, under some relatively mild conditions on the underlying filters, prove an analog of Khintchine’s recurrence theorem for this generalization of Banach density as well. As a combinatorial consequence, we can obtain a restricted version of Szemerédi’s affine cube lemma

Title: Disjointness properties of some parabolic flows

Speaker: Adam Kanigowski (Pennsylvania State University)

Abstract: We study disjointness properties of some parabolic flows such as smooth flows on surfaces, time changes of horocycle flows and Heisenberg nilflows. We show that in the above classes time p and q automorphisms are disjoint. This in particular implies Sarnak conjecture. The approach is based on a new version of Ratner’s property (non-uniform Ratner’s property).

Title: Divergent trajectories in arithmetic homogeneou spaces of rational rank two

Speaker: Nattalie Tamam (Tel Aviv University)

Abstract: In the theory of Diophantine approximations, singular points are ones for which Dirichlet’s theorem can be infinitely improved. It is easy to see that all rational points are singular. In the special case of dimension one, the only singular points are the rational ones. In higher dimensions, points lying on a rational hyperplane are also obviously singular. However, in this case there are additional singular points. In the dynamical setting the singular points are related to divergent trajectories. In the talk I will define obvious divergent trajectories and explain the relation to rational points. In addition, I will present the more general setting involving Q-algebraic groups. Lastly I will discuss results concerning classification of divergent trajectories in Q-algebraic groups.

Title: The dichotomy between structure and randomness in multiplicative number theory

Speaker: Florian Richter (Ohio State University)

Abstract: We will begin the talk by discussing a dichotomy theorem in multiplicative number theory which asserts that any multiplicative function (that satisfies certain minor regularity conditions) is either a (special kind of) almost periodic function or a pseudo-random function. Then we will explore how this phenomenon extends to other classical objects coming from multiplicative number theory. In particular, we will study the combinatorial and dynamical properties of level sets of multiplicative functions and I will present a structure theorem which says that for any level set E of an arbitrary multiplicative function there exists a highly structured superset R such that E is a pseudo-random subset of R.

Title: Distributional Lattices in Symmetric spaces.

Speaker: Elliot Paquette (Ohio State University)

Abstract: A Riemannian symmetric space X is a Riemannian manifold in which it is possible to reflect all geodesics through a point by an isometry of the space. A lattice in such a space can be considered as a discrete subgroup G of isometries so that a Borel fundamental domain of the quotient space G/X has finite Riemannian volume. Lattices mirror the structure of the ambient space in many ways: for example, X is amenable if and only if the the ambient space is amenable. We introduce the notion of a distributional lattice, generalizing the notion of lattice, by considering measures on discrete subsets of X having finite Voronoi cells and certain distributional invariance properties. Non-lattice distributional lattices exist in any Riemannian symmetric space: the Voronoi tessellation of a stationary Poisson point process is an example. With an appropriate notion of amenability, the amenability of a distributional lattice is equivalent to the amenability of the ambient space. We give some open problems related to these processes and some pretty pictures.

Title: Moebius disjointness for models of an ergodic system and beyond

Speaker: Mariusz Lemanczyk (Nicolaus Copernicus University, Toruń, Poland)

Abstract: In 2010, P. Sarnak formulated the following conjecture: 

For each zero entropy topological system (X,T), we have 

$$

\lim_{N\to\infty}\frac1N\sum_{n\leq N}f(T^nx)\mu(n)\to 0$$

for each $f\in C(X)$ and $x\in X$.  Here $\mu$ stands for the arithmetic

Moebius function. The talk will concentrate on motivations for Sarnak’s conjecture 

(relations with celebrated Chowla conjecture in number theory), role of ergodic theory in it and some recent progress.    

Here is the current schedule of talks for Fall semester 2017. All talks are on Thursdays in MW154 at 3.00pm-4.00pm (unless otherwise indicated).

September 7: Mariusz Lemanczyk (Nicolaus Copernicus University, Poland)

September 14: Elliot Paquette (Ohio State)

September 28: Florian Richter (Ohio State)

October 10 (TUESDAY): Nattalie Tamam (Tel Aviv University, Israel)

October 26: Adam Kanigowski (Penn State)

November 2: John Johnson (Ohio State)

November 16: Kurt Vinhage (Chicago)