Approximate lattices are discrete subsets of locally compact groups that are an aperiodic generalisation of lattices. They are defined as approximate subgroups (i.e. subsets that are closed under multiplication up to a finite multiplicative error) that are discrete and have finite co-volume. They were first studied by Yves Meyer who classified them in locally compact abelian groups by means of the so-called "cut-and-project schemes". Approximate lattices were subsequently used to model a diversity of objects such as aperiodic tilings (Penrose and the "hat"), Pisot numbers, and quasi-crystals.
In non-abelian groups, however, their structure remained mysterious. I will explain how the structure of approximate lattices in linear algebraic groups can be understood thanks to a notion of cohomology that sits halfway between bounded cohomology and the usual cohomology, thus generalising Meyer's theorem. Along the way, we will talk about Pisot numbers, extending a theorem of Lubotzky, Mozes and Raghunathan, amenability and (some) model theory.