Topological methods in aperiodicity (mini-course)

Abstract

7th June 11.30 until 13.30 (2 hours): Aperiodic order

The idea of aperiodic order, introduced through examples and applications. The two main approaches to constructing aperiodic patterns: substitution and the cut and project method. Interactions of aperiodic order and number theory through the cut and project method, and selected review of some more general directions in the field of aperiodic order.

8th June 11.30 until 12.30 (1 hour): Pattern spaces

How one associates to a pattern a moduli space of associated patterns: the translational hull of an FLC pattern. Basic topological and dynamical properties of the translational hull. Presentations as inverse limits of approximants.

9th June 11.30 until 13.30 (2 hours): Pattern cohomology

Elementary introduction to cellular cohomology and Čech cohomology. Example computations of Čech cohomology groups of some pattern spaces, along with discussion of more general approaches. Visualising pattern cohomology through pattern equivariance. Applications of pattern cohomology and discussion on future directions and open problems.