Analysis Seminar: "Convergence of Spectral Triples"

Abstract: Spectral triples, introduced by A. Connes, have emerged as the preferred tool to encode spectral geometric information over noncommutative algebras. They have found important roles in index theory and mathematical physics. In this presentation, we discuss the construction of a distance on a large class of spectral triple, inspired from the Gromov-Hausdorff distance. With such a distance at hand, we can then discuss issues such as finite dimensional approximations of spaces, continuity of quantum spaces obtained as deformations of classical spaces, and more. We will see that when spectral triples converge for our distance, their spectra converge as well, as does, in an appropriate sense, the associated bounded continuous functional calculi.

Host: Xiang Tang