Higher Index Theory

Rufus Willett (University of Hawaii, Manoa), Guoliang Yu (Texas A & M University)

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English

Cambridge University Press
02 July 2020

Calculus & mathematical analysis; Algebraic topology

Series: Cambridge Studies in Advanced Mathematics

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.

By: Rufus Willett (University of Hawaii Manoa), Guoliang Yu (Texas A & M University)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Dimensions: Height: 234mm, Width: 158mm, Spine: 36mm
Weight: 940g
ISBN: 9781108491068
ISBN 10: 1108491065
Series: Cambridge Studies in Advanced Mathematics
Pages: 592
Publication Date: 02 July 2020
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Introduction; Part I. Background: 1. C*-algebras; 2. K-theory for C*-algebras; 3. Motivation: positive scalar curvature on tori; Part II. Roe Algebras, Localisation Algebras, and Assembly: 4. Geometric modules; 5. Roe algebras; 6. Localisation algebras and K-homology; 7. Assembly maps and the Baum–Connes conjecture; Part III. Differential Operators: 8. Elliptic operators and K-homology; 9. Products and Poincaré duality; 10. Applications to algebra, geometry, and topology; Part IV. Higher Index Theory and Assembly: 11. Almost constant bundles; 12. Higher index theory for coarsely embeddable spaces; 13. Counterexamples; Appendix A. Topological spaces, group actions, and coarse geometry; Appendix B. Categories of topological spaces and homology theories; Appendix C. Unitary representations; Appendix D. Unbounded operators; Appendix E. Gradings; References; Index of symbols; Subject index.

Rufus Willett is Professor of Mathematics at the University of Hawaii, Manoa. He has interdisciplinary research interests across large-scale geometry, K-theory, index theory, manifold topology and geometry, and operator algebras. Guoliang Yu is the Powell Chair in Mathematics and University Distinguished Professor at Texas A & M University. He was an invited speaker at the International Congress of Mathematicians in 2006, is a Fellow of the American Mathematical Society and a Simons Fellow in Mathematics. His research interests include large-scale geometry, K-theory, index theory, manifold topology and geometry, and operator algebras.

Reviews for Higher Index Theory

'This book is an exceptional blend of clear, concise, delightfully written exposition and thorough scholarship. The book also has much to offer more experienced researchers.' Peter Haskell, European Mathematical Society