The topology of aperiodic order

Abstract

I will introduce the field of aperiodic order and how one begins to study it via topology. Aperiodic order concerns infinite, idealised patterns which have a rich system of partial symmetries between finite patches (that is, they are highly repetitive) but nonetheless lack global translational symmetry. One natural way of studying such a structure is to pass to a moduli space of associated patterns, often called the tiling space. I will explain how one defines this space, its basic properties and what one can typically say about its topological invariants, such as its Cech cohomology. I will also describe recent work with John Hunton (Durham University) which places more emphasis on rotational aspects, in addition to the more commonly studied translational structure. These techniques produce a new topological invariant for these patterns coming from shape theory, the counterpart to homotopy theory designed for dealing with pathological kinds of spaces. In the periodic setting this invariant corresponds to the classical space group of symmetries of the pattern. For certain classes of aperiodic patterns this invariant factors onto an aperiodic extension of the space group used by crystallographers, although it appears to contain further information.