Abstract: 

Large-scale geometry is the study of metric spaces up to equivalence by "zooming out"; for instance, the integer line and the real line (as metric spaces) are "large-scale equivalent. Many topological (i.e. "small-scale") notions have analogues in large-scale geometry; in particular, the notion of topological dimension has a large-scale analogue due to Gromov, called asymptotic dimension. In the past five years, asymptotic dimension has become one of the main tools in the descriptive set theory of actions of countable groups. We will talk about this connection, and mention the state of the art in the area, including joint work with Iyer, Grebík, Marks and Rozho.