Title: The minimal subdynamics problem
Abstract: Topological dynamics is the study of continuous group actions. The classical and best understood case is when the acting group is either $\mathbb{Z}$ or $\mathbb{R}$, while the general theory remains far less clear. However, recent years saw the emergence of powerful methods that often allow one to work without any assumptions on the group at all. Instead of group theory, these methods are based on combinatorics and descriptive set theory. In this talk, I will illustrate this confluence of ideas by addressing the following very basic problem: If $\Delta$ is a subgroup of $\Gamma$, does $\Gamma$ have a free continuous action on a compact space without any nontrivial closed $\Delta$-invariant subsets? This talk is based on joint work with Joshua Frisch.