Topological and Geometric Aspects of Tiling Dynamical Systems

University Of Texas At Austin, Austin TX

Investigators

Abstract

Dr. Lorenzo Sadun will analyze several problems on the topology and dynamics of tiling spaces. One problem concerns computing the Cech cohomology groups of tiling spaces. Another problem is to find criteria for when the natural translation action on a substitution tiling space (or a deformation of such an action) is topologically mixing. A third problem is to consider tilings of hyperbolic space, where the natural group action is nonabelian and in that the group is not amenable. These problems are motivated by quasicrystals, a class of solids that are highly ordered but are not crystals. Quasicrystals are modeled by aperiodic tilings, and ergodic theory relates the bulk properties of a quasicrystal (such as its ability to diffract light, its electrical conductance, and its rigidity) to certain averages taken over a space of possible quasicrystals, or equivalently a space of tilings. The better we understand the abstract mathematical properties of tiling spaces, the better we understand the down-to-earth physical properties of quasicrystals and of other materials.