# Mathematics Topology Seminar - Claude Schochet

This event is in the past.

**Date:**September 10, 2019

**Time:**2:00 p.m. - 3:00 p.m.

**Category:**Seminar

Speaker: Claude Schochet, Technion

Title: Cohomology and K-Theory for Tilings

Abstract: An aperiodic tiling of R^d has a hull, which typically is a compact foliated space X. The hull comes equipped with an invariant transverse measure, corresponding to a Z^d-invariant measure on a totally disconnected transversal N. Mathematical physicists attach two types of invariants to this situation. On the one hand, the invariant transverse measure gives rise to a real-valued map on H^d(X; R) (Cech cohomology) which corresponds to the diffraction pattern aspect of tilings. On the other hand. it also gives rise to a real-valued map on K_0(C^*G(X)), the topological K-theory of the C^*-algebra of the holonomy groupoid of the tiling, which corresponds to the spectral (of Hamiltonians defined on such tilings) aspect of tilings. We use the Index Theorem for foliated spaces to prove that under some general assumptions that these points of view are equivalent. This is joint work with Eric Akkermans (physics, the Technion) and Jonathan Rosenberg (math, U. Maryland).