Singularities of Differentiable Maps, Volume 2

Monodromy and Asymptotics of Integrals

  • V.I. Arnold
  • S.M. Gusein-Zade
  • A.N. Varchenko

Part of the Modern Birkhäuser Classics book series (MBC)

Table of contents

  1. Front Matter
    Pages i-x
  2. The topological structure of isolated critical points of functions

    1. Front Matter
      Pages 1-8
    2. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 9-28
    3. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 29-66
    4. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 67-113
    5. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 114-138
    6. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 139-167
  3. Oscillatory integrals

    1. Front Matter
      Pages 169-169
    2. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 170-214
    3. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 215-232
    4. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 233-262
    5. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 263-267
  4. Integrals of holomorphic forms over vanishing cycles

    1. Front Matter
      Pages 269-269
    2. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 270-295
    3. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 296-315
    4. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 316-350
    5. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 394-440
    6. V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 441-463
  5. Back Matter
    Pages 465-492

About this book

Introduction

Originally published in the 1980s, Singularities of Differentiable Maps: Monodromy and Asymptotics of Integrals was the second of two volumes that together formed a translation of the authors' influential Russian monograph on singularity theory.  This uncorrected softcover reprint of the work brings its still-relevant content back into the literature, making it available—and affordable—to a global audience of researchers and practitioners.

​​​While the first volume of this title, subtitled Classification of Critical Points, Caustics and Wave Fronts, contained the zoology of differentiable maps—that is, was devoted to a description of what, where, and how singularities could be encountered—this second volume concentrates on elements of the anatomy and physiology of singularities of differentiable functions.  The questions considered here are about the structure of singularities and how they function.

In the first part the authors consider the topological structure of isolated critical points of holomorphic functions: vanishing cycles; distinguished bases; intersection matrices; monodromy groups; the variation operator; and their interconnections and method of calculation.  The second part is devoted to the study of the asymptotic behavior of integrals of the method of stationary phase, which is widely met within applications.  The third and last part deals with integrals evaluated over level manifolds in a neighborhood of the critical point of a holomorphic function.  

This monograph is suitable for mathematicians, researchers, postgraduates, and specialists in the areas of mechanics, physics, technology, and other sciences dealing with the theory of singularities of differentiable maps.

Keywords

asymptotics intersection forms mixed Hodge structures of singularities monodromy oscillatory integrals singularity theory

Authors and affiliations

  • V.I. Arnold
    • 1
  • S.M. Gusein-Zade
    • 2
  • A.N. Varchenko
    • 3
  1. 1.Russian Academy of SciencesMoscowRussia
  2. 2.Moscow State UniversityMoscowRussia
  3. 3., Department MathematicsUniversity of North CarolinaChapel HillUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-8176-8343-6
  • Copyright Information Springer Science+Business Media New York 2012
  • Publisher Name Birkhäuser, Boston
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-8176-8342-9
  • Online ISBN 978-0-8176-8343-6
  • About this book
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