Topological dynamics of tilings

University Of Texas At Austin, Austin TX

Investigators

Abstract

Dr. Lorenzo Sadun will investigate topological and dynamical properties of spaces of nonperiodic tilings. His emphasis will be on the structure of these spaces as inverse limits of finite CW complexes, and on their Cech cohomology. Although the cohomologies of many tiling spaces have been computed, little is known about the functorial properties of the Cech cohomology (e.g., what happens to the cohomology when the tiling space is changed in a prescribed way), and what the cohomology tells us about the underlying tilings. Along the way, he will study tilings with continuous rotational symmetry and tilings that lack finite local complexity. Neither class of tilings is currently understood, but the techniques that Dr. Sadun and his collaborators are developing should allow him to extend results about translationally finite tilings to these other categories. Nonperiodic tilings, such as the Penrose tiling, have been used to model physical materials such as quasicrystals. Abstract mathematical properties of a space of tilings are closely related to concrete physical properties of the material being modeled. These properties include the diffraction spectrum, the electrical conductivity, and the ability of the material to resist shears. Dr. Sadun's goal is to develop this correspondence further, both by calculating topological properties of tiling spaces that have previously defied understanding, and by tracking how the topology of a tiling space reflects changes to the underlying tilings.