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Higher Index Theory

$84.99 (C)

Part of Cambridge Studies in Advanced Mathematics

  • Date Published: August 2020
  • availability: Available
  • format: Hardback
  • isbn: 9781108491068

$ 84.99 (C)

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About the Authors
  • 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.

    • Explains key techniques and motivating background material to introduce a vibrant and highly active area of modern mathematics
    • Written in a concrete, example-focused style, and gives new and simpler approaches in many cases
    • A self-contained reference for many facts that can be difficult to find in the existing literature
    Read more

    Reviews & endorsements

    ‘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

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    Product details

    • Date Published: August 2020
    • format: Hardback
    • isbn: 9781108491068
    • length: 592 pages
    • dimensions: 234 x 158 x 36 mm
    • weight: 0.94kg
    • contains: 1 b/w illus. 150 exercises
    • availability: Available
  • Table of Contents

    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
    Index of symbols
    Subject index.

  • Authors

    Rufus Willett, University of Hawaii, Manoa
    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, Texas A & M University
    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.

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