Emmanuel Breuillard

Approximate lattices and quasi-crystals

Aperiodic tilings of the plane such as the Penrose tiling are instances of discrete approximate subgroups of R^2. These Euclidean quasi-crystals were studied by Yves Meyer in the seventies and proved to arise from periodic tilings of a bigger space by the familiar cut-and-project construction. In this talk I will discuss tilings of other non-Euclidean geometries, especially those arising from symmetric spaces of non-compact type. I will survey recent advances establishing super-rigidity and arithmeticity theorems for approximate lattices that directly generalize the Mostow-Margulis theorems.