Zeta function rigidity : a view from noncommutative geometry

Abstract

In this thesis we study the zeta function formalism of finitely summable spectral triples in noncommutative geometry introduced by Alain Connes in 1995. In particular we study how these zeta functions, that naturally come in different classes, can be used to classify objects. The thesis starts with the study of finite, connected, unoriented graphs with Betti number at least 2 and valencies at least 3. We start by constructing a finitely summable spectral triple for these and prove that the first class of zeta functions determines the graph. Next closed smooth Riemannian manifolds are studied. The ideas of finitely summable spectral triples are applied to the Laplacian. We prove that the first class together with the diagonal of the second class of zeta functions determines the Riemannian manifold. This is in contrast with the usual zeta function where the phenomenon of isospectrality occurs: different objects can have the same spectrum (which is equivalent of having the same usual zeta function). Finally, the last result is formalized by introducing the concept of length categories and distances. We construct the length of a smooth map between smooth Riemannian manifolds and this in turn induces a distance between them. We conclude by proving that this distance induces the topology of uniform convergence of smooth Riemannian manifolds.