Overview
- 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
Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 357)
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Table of contents (11 chapters)
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Thom-Mather Theory: Right-Left Equivalence, Stability, Versal Unfoldings, Finite Determinacy
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Images and Discriminants: The Topology of Stable Perturbations
Keywords
About this book
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.
Reviews
“Exercises at the end of each section are intended to help the interested reader to study the material in depth. … In general, this book is a nice supplement to the classical and modern monographs devoted to various sides of singularity theory. It comprises a lot of material scattered throughout numerous papers and presents it in a systematic and rigorous way. No doubt, it will become a common reference for various issues covered in this book.” (Eugenii Shustin, Mathematical Reviews, January, 2023)
“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)
Authors and Affiliations
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
Book Title: Singularities of Mappings
Book Subtitle: The Local Behaviour of Smooth and Complex Analytic Mappings
Authors: David Mond, Juan J. Nuño-Ballesteros
Series Title: Grundlehren der mathematischen Wissenschaften
DOI: https://doi.org/10.1007/978-3-030-34440-5
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2020
Hardcover ISBN: 978-3-030-34439-9Published: 24 January 2020
Softcover ISBN: 978-3-030-34442-9Published: 24 January 2021
eBook ISBN: 978-3-030-34440-5Published: 23 January 2020
Series ISSN: 0072-7830
Series E-ISSN: 2196-9701
Edition Number: 1
Number of Pages: XV, 567
Topics: Algebraic Geometry, Several Complex Variables and Analytic Spaces, Manifolds and Cell Complexes (incl. Diff.Topology)