Speakers and abstracts

We consider the problem of constructing finite-to-one equivariant(continuous) maps from a topological dynamical system $(X,T)$ into the cubical shift $([0,1]^k)^{\mathbb{Z}}$, $k\in \mathbb{N}$. The question of determining the minimal value of $k$ for which a one-to-one equivariant map (i.e., an embedding) exists, in terms of the mean dimension of (X,T) ,has been actively investigated over the past 25 years. Extending this line of work, we show that if a system (X,T) has the marker property and mean dimension $m<k$, then it admits anequivariant map into $([0,1]^k)^{\mathbb{Z}}$ whose fibers contain at most $\left\lfloor \frac{k}{k-m} \right\rfloor \frac{k}{k-m}$ points. In particular, by choosing $k>2m$, one recovers the optimal embedding results previously obtained by Gutman and Tsukamoto (2015), as well as by Gutman, Qiao, and Tsukamoto (2019), for $\mathbb{Z}$-actions. Unlike these earlier works, our proof relies on classical topological techniques originating in the work of Ostrand (1965), Kolmogorov (1957), and Arnold (1957).

Based on  a joint work with Michael Levin.

Coboundary expansion of a simplicial complex is a high-dimensional analogue of the Cheeger constant. In contrast to the graph-theoretic setting, there are very few known families of 2-dimensional simplicial complexes that simultaneously have uniformly bounded degree and uniformly positive coboundary expansion.

In my talk, I will introduce the notion of coboundary expansion and then present a construction - arising from group theory, specifically from the theory of Kac-Moody-Steinberg groups - of a family of 2-dimensional simplicial complexes with bounded degree and uniformly positive coboundary expansion.

This talk is based on a joint work with Inga  Valentiner-Branth.

Given an automorphism of a compact abelian group, we investigate whether the finitely supported invariant probability measures are dense in the space of all invariant probability measures. We illustrate this question through several examples and discuss its implications for Hilbert–Schmidt stability of groups and for Livšic-type theorems.