I will start by defining the coarse equivalence of metric spaces. This is an equivalence relation meant to capture the large scale geometry of a given space.

To a coarse metric space one can associate a C*-algebra called uniform Roe algebra.

When does isomorphism of uniform Roe algebras associated imply coarse equivalence of the underlying coarse spaces ? A recent result of Spakula and Willett gives sufficient conditions in the case of coarse metric spaces. These conditions are uniform discreteness and property A (the ‘coarse’ variant of amenability). I will discuss a weakening of these conditions. Very recently, these rigidity results were extended to Roe coronas (quotients of uniform Roe algebras modulo the compact operators). No previous knowledge of coarse spaces, Roe algebras, or logic is required. (This is a joint work with Bruno De Mendoca Braga and Alessandro Vignati.)