Abstract: We study the classification of Hilbert space contractions belonging to the class $C_{0}$ via their relation with their Jordan models.This classification is carried out in two different settings: up to similarity and up to unitary equivalence, both of which are stronger than the usual quasisimilarity relation that is known to always hold between a $C_{0}$ contraction and its model.

We obtain positive results under a variety of assumptions, ranging from function theoretic (the Vasyunin approach using the Carleson condition for sequences of inner functions) to operator algebraic (the Arveson approach using boundary representations of operator algebras).

Abstract: The general aim of Alain Connes' noncommutative geometry is to develop selected geometric ideas using techniques from Hilbert space theory, with the goal of then transporting these concepts to new and unfamiliar contexts. But in this lecture I'll stay mostly with the familiar. I'll start with Hermann Weyl's theorem on eigenvalue asymptotics, examine it from the noncommutative-geometric point of view, and explain how a tentative idea of what a noncommutative geometric space might actually be starts to emerge in the process.