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© 2020

Singularities of Mappings

The Local Behaviour of Smooth and Complex Analytic Mappings

  • Provides a self-contained introduction to the theory of singularities of mappings and ends at the frontier of current research

  • Includes hundreds of exercises, with many designed to help master techniques of calculation

  • A self-contained reference with 5 appendices on background material

Book

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 357)

Table of contents

  1. Front Matter
    Pages i-xv
  2. David Mond, Juan J. Nuño-Ballesteros
    Pages 1-10
  3. Thom-Mather Theory: Right-Left Equivalence, Stability, Versal Unfoldings, Finite Determinacy

    1. Front Matter
      Pages 11-11
    2. David Mond, Juan J. Nuño-Ballesteros
      Pages 13-44
    3. David Mond, Juan J. Nuño-Ballesteros
      Pages 45-96
    4. David Mond, Juan J. Nuño-Ballesteros
      Pages 97-139
    5. David Mond, Juan J. Nuño-Ballesteros
      Pages 141-179
    6. David Mond, Juan J. Nuño-Ballesteros
      Pages 181-216
    7. David Mond, Juan J. Nuño-Ballesteros
      Pages 217-250
  4. Images and Discriminants: The Topology of Stable Perturbations

    1. Front Matter
      Pages 251-251
    2. David Mond, Juan J. Nuño-Ballesteros
      Pages 253-301
    3. David Mond, Juan J. Nuño-Ballesteros
      Pages 303-368
    4. David Mond, Juan J. Nuño-Ballesteros
      Pages 369-412
    5. David Mond, Juan J. Nuño-Ballesteros
      Pages 413-468
  5. Back Matter
    Pages 469-567

About this book

Introduction

The first monograph on singularities of mappings for many years, this book provides an introduction to the subject and an account of recent developments concerning the local structure of complex analytic mappings.

Part I of the book develops the now classical real C and complex analytic theories jointly. Standard topics such as stability, deformation theory and finite determinacy, are covered in this part. In Part II of the book, the authors focus on the complex case. The treatment is centred around the idea of the "nearby stable object" associated to an unstable map-germ, which includes in particular the images and discriminants of stable perturbations of unstable singularities. This part includes recent research results, bringing the reader up to date on the topic.

By focusing on singularities of mappings, rather than spaces, this book provides a necessary addition to the literature. Many examples and exercises, as well as appendices on background material, make it an invaluable guide for graduate students and a key reference for researchers. A number of graduate level courses on singularities of mappings could be based on the material it contains.

Keywords

singularity theory Multiplicity Milnor-Tjurina relation vanishing homology left-right equivalence Finite determinancy Image and discriminant Multiple point spaces

Authors and affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Departament de MatemàtiquesUniversitat de ValènciaBurjassotSpain

About the authors

David Mond received his PhD at the University of Liverpool and has held positions at the National University of Colombia, Bogotá, and at the University of Warwick, where he is a full professor. His main field of research is the theory of singularities of mappings, especially the geometry and topology of images and discriminants, and their relation to the deformation theory of unstable germs map-germs.

Juan J. Nuño-Ballesteros is Professor at the University of Valencia. His research is in the field of singularities of real and complex mappings and his main contributions to the subject include the topological classification of real analytic map-germs and also the Whitney equisingularity of complex analytic families of map-germs. He has also other important contributions in generic geometry (applications of Singularity Theory to the differential geometry of submanifolds in Euclidean spaces) and about global aspects of singularities of smooth mappings.

Bibliographic information

Reviews

“The book is written in a clear pedagogical style; it contains many examples, exercises, comments, remarks, nice pictures, very useful instructive and systematic references, computational algorithms with implementation in the computer algebra software systems Macaulay 2 and Mathematica, etc. … Without a doubt, the book is understandable, interesting and useful for graduate students and can serve as a good starting point for those who are interested in various aspects of both pure and applied mathematics.” (Aleksandr G. Aleksandrov, zbMATH 1448.58032, 2020)