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Topology and Analysis

The Atiyah-Singer Index Formula and Gauge-Theoretic Physics

  • Textbook
  • © 1985

Overview

Authors:
  1. B. Booss

    1. IMFUFA, Roskilde Universitetscenter, Roskilde, Denmark

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  2. D. D. Bleecker

    1. Department of Mathematics, University of Hawaii, Honolulu, USA

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    You can also search for this author in PubMed   Google Scholar

Part of the book series: Universitext (UTX)

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Table of contents (24 chapters)

  1. Front Matter

    Pages i-xvi
    Download chapter PDF

  2. Operators with Index

    1. Fredholm Operators

      • B. Booss, D. D. Bleecker
      Pages 1-5

    2. Algebraic Properties. Operators of Finite Rank

      • B. Booss, D. D. Bleecker
      Pages 5-12

    3. Analytic Methods. Compact Operators

      • B. Booss, D. D. Bleecker
      Pages 12-25

    4. The Fredholm Alternative

      • B. Booss, D. D. Bleecker
      Pages 26-34

    5. The Main Theorems

      • B. Booss, D. D. Bleecker
      Pages 34-47

    6. Families of Invertible Operators. Kuiper’s Theorem

      • B. Booss, D. D. Bleecker
      Pages 47-60

    7. Families of Fredholm Operators. Index Bundles

      • B. Booss, D. D. Bleecker
      Pages 60-78

    8. Fourier Series and Integrals (Fundamental Principles)

      • B. Booss, D. D. Bleecker
      Pages 79-85

    9. Wiener-Hopf Operators

      • B. Booss, D. D. Bleecker
      Pages 85-102

  3. Analysis on Manifolds

    1. Partial Differential Equations

      • B. Booss, D. D. Bleecker
      Pages 103-125

    2. Differential Operators Over Manifolds

      • B. Booss, D. D. Bleecker
      Pages 126-143

    3. Pseudo-Differential Operators

      • B. Booss, D. D. Bleecker
      Pages 144-172

    4. Sobolev Spaces (Crash Course)

      • B. Booss, D. D. Bleecker
      Pages 172-181

    5. Elliptic Operators over Closed Manifolds

      • B. Booss, D. D. Bleecker
      Pages 182-188

    6. Elliptic Boundary-Value Systems I (Differential Operators)

      • B. Booss, D. D. Bleecker
      Pages 189-198

    7. Elliptic Differential Operators of First Order with Boundary Conditions

      • B. Booss, D. D. Bleecker
      Pages 199-208

    8. Elliptic Boundary-Value Systems II (Survey)

      • B. Booss, D. D. Bleecker
      Pages 208-217

  4. The Atiyah-Singer Index Formula

    1. Introduction to Algebraic Topology (K-Theory)

      • B. Booss, D. D. Bleecker
      Pages 218-246

    2. The Index Formula in the Euclidean Case

      • B. Booss, D. D. Bleecker
      Pages 246-256
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Keywords

About this book

The Motivation. With intensified use of mathematical ideas, the methods and techniques of the various sciences and those for the solution of practical problems demand of the mathematician not only greater readi­ ness for extra-mathematical applications but also more comprehensive orientations within mathematics. In applications, it is frequently less important to draw the most far-reaching conclusions from a single mathe­ matical idea than to cover a subject or problem area tentatively by a proper "variety" of mathematical theories. To do this the mathematician must be familiar with the shared as weIl as specific features of differ­ ent mathematical approaches, and must have experience with their inter­ connections. The Atiyah-Singer Index Formula, "one of the deepest and hardest results in mathematics", "probably has wider ramifications in topology and analysis than any other single result" (F. Hirzebruch) and offers perhaps a particularly fitting example for such an introduction to "Mathematics": In spi te of i ts difficulty and immensely rich interrela­ tions, the realm of the Index Formula can be delimited, and thus its ideas and methods can be made accessible to students in their middle * semesters. In fact, the Atiyah-Singer Index Formula has become progressively "easier" and "more transparent" over the years. The discovery of deeper and more comprehensive applications (see Chapter 111. 4) brought with it, not only a vigorous exploration of its methods particularly in the many­ facetted and always new presentations of the material by M. F.

Authors and Affiliations

  • IMFUFA, Roskilde Universitetscenter, Roskilde, Denmark

    B. Booss

  • Department of Mathematics, University of Hawaii, Honolulu, USA

    D. D. Bleecker

Bibliographic Information

  • Book Title: Topology and Analysis

  • Book Subtitle: The Atiyah-Singer Index Formula and Gauge-Theoretic Physics

  • Authors: B. Booss, D. D. Bleecker

  • Series Title: Universitext

  • DOI: https://doi.org/10.1007/978-1-4684-0627-6

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag New York Inc. 1985

  • Softcover ISBN: 978-0-387-96112-5Published: 21 December 1984

  • eBook ISBN: 978-1-4684-0627-6Published: 06 December 2012

  • Series ISSN: 0172-5939

  • Series E-ISSN: 2191-6675

  • Edition Number: 1

  • Number of Pages: XVI, 451

  • Topics: Real Functions, Topology

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