Spectral triples, introduced by A. Connes in 1985, have emerged as the preferred generalization of Riemannian geometry to the noncommutative realm, and provide a new formalism for the study of problems form quantum physics to fractal geometry. Many problems in physics and geometry suggests that a quantification on "how close" spectral triples can be from each others would prove very helpful; however, only heuristics can be found in the literature. Examples of problems where such a notion could prove useful include the convergence of matrix models in mathematical physics, or the geometry of fractal sets constructed, in a natural manner, as limits of simpler spaces.
In this talk, we will survey our construction of an actual metric, up to unitary equivalence, on a large class of spectral triples, which indeed provides a framework to discuss such problems . Our construction starts with our work in noncommutative metric geometry, spurred by the work of Rieffel, and our construction of a noncommutative analogue of the Gromov-Hausdorff distance on the class of quantum compact metric spaces. From this initial step, we will see how to enhance our construction to define convergence for Hilbert modules over quantum compact metric spaces, convergence of certain dynamics over quantum compact metric spaces and their modules, and, as a synthesis of these various tools, how to define our metric on spectral triples.
A geometry for the space of metric spectral triples Sponsored by the Simons Foundation