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Applications of Algebraic Topology

Graphs and Networks. The Picard-Lefschetz Theory and Feynman Integrals

  • Book
  • © 1975

Overview

Authors:
  1. S. Lefschetz

    1. Princeton University, USA

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Part of the book series: Applied Mathematical Sciences (AMS, volume 16)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-viii
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  2. Application of Classical Topology to Graphs and Networks

    1. Front Matter

      Pages 1-4
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    2. A Résumé of Linear Algebra

      • S. Lefschetz
      Pages 5-12

    3. Duality in Vector Spaces

      • S. Lefschetz
      Pages 13-21

    4. Topological Preliminaries

      • S. Lefschetz
      Pages 22-33

    5. Graphs. Geometric Structure

      • S. Lefschetz
      Pages 34-42

    6. Graph Algebra

      • S. Lefschetz
      Pages 43-50

    7. Electrical Networks

      • S. Lefschetz
      Pages 51-59

    8. Complexes

      • S. Lefschetz
      Pages 61-69

    9. Surfaces

      • S. Lefschetz
      Pages 71-87

    10. Planar Graphs

      • S. Lefschetz
      Pages 89-106

  3. The Picard-Lefschetz Theory and Feynman Integrals

    1. Front Matter

      Pages 109-117
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    2. Topological and Algebraic Considerations

      • S. Lefschetz
      Pages 119-134

    3. The Picard-Lefschetz Theory

      • S. Lefschetz
      Pages 135-148

    4. Extension to Higher Varieties

      • S. Lefschetz
      Pages 149-153

    5. Feynman Integrals

      • S. Lefschetz
      Pages 154-176

    6. Feynman Integrals. B.

      • S. Lefschetz
      Pages 177-180

  4. Back Matter

    Pages 181-191
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Keywords

About this book

This monograph is based, in part, upon lectures given in the Princeton School of Engineering and Applied Science. It presupposes mainly an elementary knowledge of linear algebra and of topology. In topology the limit is dimension two mainly in the latter chapters and questions of topological invariance are carefully avoided. From the technical viewpoint graphs is our only requirement. However, later, questions notably related to Kuratowski's classical theorem have demanded an easily provided treatment of 2-complexes and surfaces. January 1972 Solomon Lefschetz 4 INTRODUCTION The study of electrical networks rests upon preliminary theory of graphs. In the literature this theory has always been dealt with by special ad hoc methods. My purpose here is to show that actually this theory is nothing else than the first chapter of classical algebraic topology and may be very advantageously treated as such by the well known methods of that science. Part I of this volume covers the following ground: The first two chapters present, mainly in outline, the needed basic elements of linear algebra. In this part duality is dealt with somewhat more extensively. In Chapter III the merest elements of general topology are discussed. Graph theory proper is covered in Chapters IV and v, first structurally and then as algebra. Chapter VI discusses the applications to networks. In Chapters VII and VIII the elements of the theory of 2-dimensional complexes and surfaces are presented.

Authors and Affiliations

  • Princeton University, USA

    S. Lefschetz

Bibliographic Information

  • Book Title: Applications of Algebraic Topology

  • Book Subtitle: Graphs and Networks. The Picard-Lefschetz Theory and Feynman Integrals

  • Authors: S. Lefschetz

  • Series Title: Applied Mathematical Sciences

  • DOI: https://doi.org/10.1007/978-1-4684-9367-2

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1975

  • Softcover ISBN: 978-0-387-90137-4Published: 13 May 1975

  • eBook ISBN: 978-1-4684-9367-2Published: 06 December 2012

  • Series ISSN: 0066-5452

  • Series E-ISSN: 2196-968X

  • Edition Number: 1

  • Number of Pages: VIII, 191

  • Topics: Algebraic Topology, Algebra, Graph Theory

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