Topological invariants for aperiodic patterns

Abstract

A periodic pattern of Euclidean space, such as an infinite chequerboard decoration in the plane, is essentially determined by its space group of global symmetries. Quite recently there has been a surge of interest in aperiodically ordered patterns, which never precisely repeat themselves but frequently ‘almost repeat’. Such objects are used as models for quasicrystals, and arise as natural objects associated to certain dynamical and number theoretic constructions. Being notably short of global symmetries, these objects require new mathematical tools to study them. The approximate symmetry of these patterns is effectively captured by the topology of associated moduli spaces of patterns. In this talk I will introduce the field of Aperiodic Order and how one may use Algebraic Topology to define natural invariants for aperiodic patterns. I will discuss recent joint work with John Hunton, which introduces new techniques for analysing both their translational and rotational structure. Applied in the periodic setting, one may recover the classical space group of global symmetries. For special ‘model sets’, used as mathematically idealised quasicrystals, we obtain an invariant which has a natural quotient isomorphic to the crystallographers’ aperiodic space group.