A metric in the space of spectral triples

Abstract

In 1996, Alain Connes introduced the spectral triple, which encodes the information of a spin manifold in a way that allows for a noncommutative generalization. Before that, in 1981, Misha Gromov introduced a metric on the space of compact metric spaces modulo isometry, called the Gromov-Hausdorff distance. With these two ideas in mind, Gunther Cornelissen and Bram Mesland are working on a metric space of spectral triples. They use the morphism between spectral triples introduced in 2009 by Bram Mesland, called a correspondence, and define the length of a correspondence. The distance between two spectral triples is then obtained as the infimum of the lengths of the correspondences between them. In my thesis, I review the theory needed to understand this metric, and give a concrete example of a correspondence.