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Singularities of Differentiable Maps

Volume II Monodromy and Asymptotic Integrals

  • Book
  • © 1988

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Editors:
  1. V. I. Arnold
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  2. S. M. Gusein-Zade
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  3. A. N. Varchenko
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Part of the book series: Monographs in Mathematics (MMA, volume 83)

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Table of contents (16 chapters)

  1. Front Matter

    Pages I-VIII
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  2. The topological structure of isolated critical points of functions

    1. Introduction

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 1-8

    2. Elements of the theory of Picard-Lefschetz

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 9-28

    3. The topology of the non-singular level set and the variation operator of a singularity

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 29-66

    4. The bifurcation sets and the monodromy group of a singularity

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 67-113

    5. The intersection matrices of singularities of functions of two variables

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 114-138

    6. The intersection forms of boundary singularities and the topology of complete intersections

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 139-167

  3. Oscillatory integrals

    1. Front Matter

      Pages 169-267
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    2. Discussion of results

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 170-214

    3. Elementary integrals and the resolution of singularities of the phase

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 215-232

    4. Asymptotics and Newton polyhedra

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 233-262

    5. The singular index, examples

      • 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 269-463
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    2. The simplest properties of the integrals

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 270-295

    3. Complex oscillatory integrals

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 296-315

    4. Integrals and differential equations

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 316-350

    5. The coefficients of series expansions of integrals, the weight and Hodge filtrations and the spectrum of a critical point

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 351-393

    6. The mixed Hodge structure of an isolated critical point of a holomorphic function

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 394-440

    7. The period map and the intersection form

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 441-463

  5. Back Matter

    Pages 465-488
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Keywords

About this book

The present. volume is the second volume of the book "Singularities of Differentiable Maps" by V.1. Arnold, A. N. Varchenko and S. M. Gusein-Zade. The first volume, subtitled "Classification of critical points, caustics and wave fronts", was published by Moscow, "Nauka", in 1982. It will be referred to in this text simply as "Volume 1". Whilst the first volume contained the zoology of differentiable maps, that is it was devoted to a description of what, where and how singularities could be encountered, this volume contains the elements of the anatomy and physiology of singularities of differentiable functions. This means that the questions considered in it are about the structure of singularities and how they function. Another distinctive feature of the present volume is that we take a hard look at questions for which it is important to work in the complex domain, where the first volume was devoted to themes for which, on the whole, it was not important which field (real or complex) we were considering. Such topics as, for example, decomposition of singularities, the connection between singularities and Lie algebras and the asymptotic behaviour of different integrals depending on parameters become clearer in the complex domain. The book consists of three parts. In the first part we consider the topological structure of isolated critical points of holomorphic functions. We describe the fundamental topological characteristics of such critical points: vanishing cycles, distinguished bases, intersection matrices, monodromy groups, the variation operator and their interconnections and method of calculation.

Bibliographic Information

  • Book Title: Singularities of Differentiable Maps

  • Book Subtitle: Volume II Monodromy and Asymptotic Integrals

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

  • Series Title: Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4612-3940-6

  • Publisher: Birkhäuser Boston, MA

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1988

  • Softcover ISBN: 978-1-4612-8408-6Published: 01 August 2012

  • eBook ISBN: 978-1-4612-3940-6Published: 06 December 2012

  • Series ISSN: 1017-0480

  • Series E-ISSN: 2296-4886

  • Edition Number: 1

  • Number of Pages: VIII, 492

  • Number of Illustrations: 5 b/w illustrations

  • Topics: Differential Geometry, Manifolds and Cell Complexes (incl. Diff.Topology)

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