How close are two noncommutative spaces? This question is natural for an analyst, and an answer would provide a formal framework to various statements in mathematical physics, where one may wish to approximate a (quantum) space-time using finite dimensional approximations, or one may wish to study perturbations of a given metric or geometry. Ideally, we thus seek a distance between noncommutative spaces.

A common approach to endow a noncommutative algebra acting on a Hilbert space with a form of quantum Riemannian geometry is given by spectral triples, introduced by Connes. Under reasonable assumptions, such spectral triples give rise to quantum metric spaces. Thus, a natural starting point to answer our initial question is to generalize the Gromov-Hausdorff distance to noncommutative geometry. This is a nontrivial task, and in particular, it must be done in a manner which makes it natural to modify the construction to work with more structures. For instance, a spectral triple includes a module --- its defining Hilbert space --- and a (usually unbounded) self-adjoint operator.

The spectral propinquity is indeed a metric on the class of metric spectral triples, up to the natural conjugacy equivalence. It is built from the Gromov-Hausdorff propinquity, our analogue of Gromov's distance for compact quantum metric spaces based on C-algebras, and utilize many of our constructions over the past few years: our modular Gromov-Hausdorff propinquity and our covariant versions.

This talk present an informal review of the construction of the spectral propinquity, introducing all necessary ideas on the way, and putting together the picture of convergence for quantum geometries which has motivated our research for the past few years.