Noncommutative Geometry Seminar

Date: January 22, 2021

Time: 1:00PM - 2:00PM

Location: Zoom 951 5490 42

Speaker: Emil Prodan, Yeshiva University

Title: Non-Commutative Geometry and Materials Science

Abstract: Starting with the pioneering works of Jean Bellissard in the 1980’s, Non-Commutative Geometry has emerged as one of sharpest tools in the arsenal of a theoretical materials scientist. Perhaps for this audience, the most important questions are why Non-Commutative Geometry and how much of it? Using well understood examples, where specially designed materials display extraordinary behaviours in extreme conditions, I will try to convince the audience that one has to walk the entire sequence: algebra of observables → K-theory →pairing with cyclic cohomology → local index formula, plus one additional step which I call “pushing into the Sobolev.” Furthermore, among such extraordinary behaviours is a certain relation between the dynamics of degrees of freedominside the bulk and at the boundary of a sample, dubbed the bulk-boundary correspondence principle. It is captured by a certain extension of C*-algebras and, as such, KK-theory offers a natural framework and supplies the necessary tools to investigate such phenomena. If the time permits, I will also discuss recent efforts trying to steer these tools from their usual use of explaining observed behaviours towards the discovery of new dynamical behaviours in materials science.

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