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Singularities of Differentiable Maps

Volume I: The Classification of Critical Points Caustics and Wave Fronts

  • Book
  • © 1985

Overview

Authors:
  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 82)

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Table of contents (22 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. Basic Concepts

    1. Front Matter

      Pages 1-1
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    2. The simplest examples

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 3-26

    3. The classes ΣI

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 27-59

    4. The quadratic differential of a map

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 60-71

    5. The local algebra of a map and the Weierstrass preparation theorem

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 72-83

    6. The local multiplicity of a holomorphic map

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 84-114

    7. Stability and infinitesimal stability

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 115-132

    8. The proof of the stability theorem

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 133-144

    9. Versal deformations

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 145-156

    10. The classification of stable germs by genotype

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 157-172

    11. Review of further results

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 173-182

  3. Critical Points of Smooth Functions

    1. Front Matter

      Pages 183-186
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    2. A start to the classification of critical points

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 187-191

    3. Quasihomogeneous and semiquasihomogeneous singularities

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 192-216

    4. The classification of quasihomogeneous functions

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 217-230

    5. Spectral sequences for the reduction to normal forms

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 231-241

    6. Lists of singularities

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 242-257

    7. The determinator of singularities

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 258-271

    8. Real, symmetric and boundary singularities

      • V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
      Pages 272-284
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Keywords

About this book

... there is nothing so enthralling, so grandiose, nothing that stuns or captivates the human soul quite so much as a first course in a science. After the first five or six lectures one already holds the brightest hopes, already sees oneself as a seeker after truth. I too have wholeheartedly pursued science passionately, as one would a beloved woman. I was a slave, and sought no other sun in my life. Day and night I crammed myself, bending my back, ruining myself over my books; I wept when I beheld others exploiting science fot personal gain. But I was not long enthralled. The truth is every science has a beginning, but never an end - they go on for ever like periodic fractions. Zoology, for example, has discovered thirty-five thousand forms of life ... A. P. Chekhov. "On the road" In this book a start is made to the "zoology" of the singularities of differentiable maps. This theory is a young branch of analysis which currently occupies a central place in mathematics; it is the crossroads of paths leading from very abstract corners of mathematics (such as algebraic and differential geometry and topology, Lie groups and algebras, complex manifolds, commutative algebra and the like) to the most applied areas (such as differential equations and dynamical systems, optimal control, the theory of bifurcations and catastrophes, short-wave and saddle-point asymptotics and geometrical and wave optics).

Bibliographic Information

  • Book Title: Singularities of Differentiable Maps

  • Book Subtitle: Volume I: The Classification of Critical Points Caustics and Wave Fronts

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

  • Series Title: Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4612-5154-5

  • Publisher: Birkhäuser Boston, MA

  • eBook Packages: Springer Book Archive

  • Copyright Information: Birkhäuser Boston, Inc. 1985

  • Softcover ISBN: 978-1-4612-9589-1Published: 01 October 2011

  • eBook ISBN: 978-1-4612-5154-5Published: 06 December 2012

  • Series ISSN: 1017-0480

  • Series E-ISSN: 2296-4886

  • Edition Number: 1

  • Number of Pages: 396

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

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