Speaker: **Frederic Latremoliere** (University of Denver)

Title: The Gromov-Hausdorff Propinquity

Time/Date: 5:15-6:45pm, Wednesday, December 5, 2018.

Room: 126

Abstract: The founding allegory of noncommutative metric geometry is to study certain noncommutative algebras, which we call quantum compact metric spaces, as generalizations of the algebras of Lipschitz functions over metric spaces. Motivated by metric geometry for manifolds and by mathematical physics, we have developed a noncommutative analogue of the Gromov-Hausdorff distance on the class of quantum compact metric spaces, called the Gromov-Hausdorff propinquity. This metric is complete, and induces the same topology on classical compact metric spaces, while allowing us to prove, for instance, that quantum tori are limits of certain natural finite dimensional algebras. We then extended our construction to modules over quantum compact metric spaces, to define a complete metric on such modules, allowing for instance to discuss the continuity of the Heisenberg modules over quantum tori. In this talk, we will present a survey of our work and touch on how it can be used to discuss the convergence of such structures as spectral triples.