Homotopical tools for analysing approximate symmetry
In the 1980s Dan Shechtman discovered quasicrystals, by producing metal alloys whose electron diffraction patterns exhibited a great deal of structural order but also 5-fold rotational symmetry, precluding the periodically repeating arrangement of a regular crystal. Before this discovery, mathematically idealised infinite, aperiodic patterns had already been considered. For example, in the 60s the first aperiodic tile set was found by Berger in his resolution of Wang’s Domino Problem, and in the 70s Penrose discovered his famous aperiodic tilings. In this talk I will introduce the field of Aperiodic Order: the study of infinite tilings or point sets of Euclidean space which have a rich structure of approximate symmetries but few, if any, global symmetries. The classical study of periodic patterns is conducted through consideration of their space groups of global symmetries. Since aperiodic patterns lack global symmetry, new mathematical tools need to be introduced. I will explain how one defines associated moduli spaces of patterns, whose topological invariants turn out to be natural and important invariants of the original patterns. I will explain some key aspects of recent joint work with John Hunton, which incorporates their rotational and translational structure into this topological analysis.