Tiling spaces and tiling C*-algebras

Abstract

Through examples I will introduce the field of aperiodic order. This is the study of infinite, idealised patterns of space which, despite having no translational symmetry, nevertheless are highly regulated, in the sense that finite portions of the pattern reappear frequently. I will explain how one associates moduli spaces to these objects, “tiling spaces”, and thus how one studies aperiodic patterns via topological invariants. A different route to these invariants is to define associated C* -algebras. This has been an effective tool in establishing links from the mathematical theory of aperiodic patterns to the properties of quasicrystals modelled by them. I will give a brief overview of the construction of these C*-algebras and discuss the current state of the art as to computation of their invariants.