Cambridge University Press, 2020. — 597 p. — (Cambridge Studies in Advanced Mathematics 189). — ISBN 9781108867351.Index theory studies the solutions to differential equations on geometric spaces, their relation to the underlying geometry and topology, and applications to physics. If the space of solutions is infinite dimensional, it becomes necessary to generalise the classical Fredholm index using tools from the K-theory of operator algebras. This leads to higher index theory, a rapidly developing subject with connections to noncommutative geometry, large-scale geometry, manifold topology and geometry, and operator algebras. Aimed at geometers, topologists and operator algebraists, this book takes a friendly and concrete approach to this exciting theory, focusing on the main conjectures in the area and their applications outside of it. A well-balanced combination of detailed introductory material (with exercises), cutting-edge developments and references to the wider literature make this a valuable guide to this active area for graduate students and experts alike. Contents Introduction Background C∗-algebras K-theory for C∗-algebras Motivation: Positive Scalar Curvature on Tori Roe Algebras, Localisation Algebras and Assembly Geometric Modules Roe Algebras Localisation Algebras and K-homology Assembly Maps and the Baum–Connes Conjecture Differential Operators Elliptic Operators and K-homology Products and Poincar´e Duality Applications to Algebra, Geometry and Topology Higher Index Theory and Assembly Almost Constant Bundles Higher Index Theory for Coarsely Embeddable Spaces Counterexamples Appendices Topological Spaces, Group Actions and Coarse Geometry Categories of Topological Spaces and Homology Theories Unitary Representations Unbounded Operators Gradings
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