C∗-algebras associated to dynamical systems

There is a long history of interaction between operator algebras and dynamical systems. At the core of this interaction are constructions of operator algebras which have that in common that they replace dynamical behavior by something static at the prize of non-commutativity. This interaction is asymmetric, but mutually beneficial. There is much left to learn about the universe of so-called C∗-algebras, and examples with an origin in dynamics have proven to be important and amenable test cases, as they often come with extra structure provided by our understanding of the underlying dynamical system. In the other direction, methods and strategies from C∗-algebras have been successfully translated to dynamical systems. One of the most prominent examples of this transport of ideas is the Bratteli-Vershik model for Cantor minimal systems. I shall try to give an overview here of an equally important area of contact, that of classification of dynamical systems up to various coarse equivalences.