Aperiodic order: C*-algebras and tiling spaces

Abstract

A tiling or point set of Euclidean space is aperiodically ordered if it frequently almost repeats but never precisely repeats. That is, there is no infinite order isometry preserving the pattern, but any given finite sub-pattern may be found within some globally bounded distance from anywhere in the pattern. Periodic patterns, those with a full rank subgroup of translational symmetries, are classically classified by their space groups of global symmetries. Since aperiodic patterns, by their very nature, are lacking in global symmetry, new tools are needed to capture their internal structure. In this talk I will introduce the field of Aperiodic Order and some of the mathematical machinery which has been developed to study it. I will show how C*-algebras are defined for aperiodic tilings and what is known about their K-theory. I will also explain the closely related approach of studying these patterns topologically, via associated moduli spaces of patterns, sometimes called tiling spaces. I will briefly summarise some new directions in this area from joint work with John Hunton, in which we identify and compute invariants for aperiodically ordered patterns which keep track of both rotational and translational structure.