Finite Summability in Noncommutative Topology and Geometry

The question of finite summability is one of importance for both K-homology and Spectral Triples. If a C*-algebra has uniformly finitely summable K-homology, we can compute the index pairing using Connes' Chern Character. In noncommutative geometry, a spectral triple represents a noncommutative manifold, and the summability of the triple provides a notion of dimension. The two notions are unified by the fact that the K-homology class associated to a finitely summable spectral triple has the same degree of summability as the spectral triple. In this expository talk, we define finite summability, discuss currently known examples, and outline potential avenues for research. No knowledge of K-homology or spectral triples will be assumed.