Aperiodic order, number theory and topology

Abstract

In this talk I shall introduce the field of aperiodic order, a branch of geometry which aims to study infinite patterns which display a great amount of structural order, despite lacking translational symmetry, i.e., without “repeating themselves”. These intriguing patterns make for instructive models of “quasicrystals”, amongst other physical applications, but I shall show how they also arise in areas of pure mathematics. In particular, the properties of a special class of these patterns, called cut-and-project sets, are intimately linked with questions in Diophantine approximation, the area of number theory which investigates quantitatively how rational numbers approximate real numbers. As time permits, I shall also explain the approach to studying aperiodic patterns through topology, via associated moduli spaces of patterns.