Moduli spaces of patterns and their cohomology

Abstract

Periodic patterns of Euclidean space are decorations by motifs, such as point patterns or tiles, which have full-rank global translational symmetry. This means that they can be described from just a fundamental domain and their symmetry group. An aperiodically ordered pattern is one which can frequently repeat itself on finite patches but without being globally periodic. These are far more complicated to analyse and a variety of abstract tools has been developed to understand them. In this talk I shall explain how one studies them topologically, via associated moduli spaces of locally indistinguishable patterns. Topological invariants are applied, such as K-theory or Cech cohomology. I shall briefly outline how one goes about computing these invariants and how one may visualise what they say about the original pattern. At present most attention is dedicated to studying these patterns translationally. Bringing in rotations introduces some interesting challenges; a 3-dimensional periodic pattern, for example, has associated translational moduli space simply the 3-torus, but the rotational version is a 6-manifold whose topology depends crucially on the rotational symmetries of the pattern. I shall explain some recent progress with John Hunton in computing topological invariants for these spaces.