Del 189 i serien Cambridge Studies in Advanced Mathematics

Higher Index Theory

Inbunden, Engelska, 2020

1 289 kr

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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.

Produktinformation

Utgivningsdatum2020-07-02
Mått158 x 234 x 36 mm
Vikt940 g
FormatInbunden
SpråkEngelska
SerieCambridge Studies in Advanced Mathematics
Antal sidor592
FörlagCambridge University Press
ISBN9781108491068

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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.

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.