Connes-Chern Character for Manifolds with Boundary and Eta Cochains.

Intro -- Contents -- List of Figures -- Introduction -- Chapter 1. Preliminaries -- 1.1. The general setup -- 1.2. Relative cyclic cohomology -- 1.3. The Chern character -- 1.4. Dirac operators and -graded Clifford modules -- 1.5. The relative Connes-Chern character of a Dirac operator over a manifold with boundary -- 1.6. Exact b-metrics and b-functions on cylinders -- 1.7. Global symbol calculus for pseudodifferential operators -- 1.8. Classical b-pseudodifferential operators -- 1.9. Indicial family -- Chapter 2. The b-Analogue of the Entire Chern Character -- 2.1. The b-trace -- 2.2. The relative McKean-Singer formula and the APS Index Theorem -- 2.3. A formula for the b-trace -- 2.4. b-Clifford modules and b-Dirac operators -- 2.5. The b-JLO cochain -- 2.6. Cocycle and transgression formulæ for the even/odd b-Chern character (without Clifford covariance) -- 2.7. Sketch of Proof of Theorem 2.11 -- Chapter 3. Heat Kernel and Resolvent Estimates -- 3.1. Basic resolvent and heat kernel estimates on general manifolds -- 3.2. Comparison results -- 3.3. Trace class estimates for the model heat kernel -- 3.4. Trace class estimates for the JLO integrand on manifolds with cylindrical ends -- 3.5. Estimates for b-traces -- 3.6. Estimates for the components of the entire b-Chern character -- Chapter 4. The Main Results -- 4.1. Asymptotic b-heat expansions -- 4.2. The Connes-Chern character of the relative Dirac class -- 4.3. Relative pairing formulæ and geometric consequences -- 4.4. Relation with the generalized APS pairing -- Bibliography -- Subject Index -- Notation Index.

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