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Hilbert-space based analysis of differential operators, with the goal of using elliptic differential operators to build K-homology classes. Particular focus on geometric aspects related to propagation speed. Also some more precise Schatten-class theory.
Introduction to assembly as a 'forget control' map from K-homology to K-theory of Roe algebras. Introduction of the Baum-Connes and coarse Baum-Connes conjectures as universal assembly maps. Concrete models for the Baum-Connes conjecture for special cases, and proof of the coarse Baum-Connes conjecture for Euclidean space.
Introduction of localisation algebras, and use of these to build an analytic model for K-homology, including a proof that this model is a generalised homology theory. Equivariant versions, and relationships to other models of K-homology.
General pairings, and cap and slant products between K-theory and K-homology, including the use of representable K-homology where needed. Spin-c manifolds and using Dirac operators to prove Bott periodicity and Poincaré duality.
Applications of assembly maps to the Kadison-Kaplansky conjecture on the existence of idempotents in group algebras, to the existence and study of positive scalar curvature metrics, and to the Novikov conjecture in manifold topology. Historical motivation and some overview of the literature.