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Singularities of mappings. The local behaviour of smooth and complex analytic mappings. (English) Zbl 1448.58032

Grundlehren der Mathematischen Wissenschaften 357. Cham: Springer (ISBN 978-3-030-34439-9/hbk; 978-3-030-34440-5/ebook). xv, 567 p. (2020).
The book under review consists of two parts, in addition to the preface, introduction, appendices, a list of literature consisting of 10 pages, and an index. It should be noted that the bibliography includes 24 references to the original works of the authors and their collaborators (see, e.g., [D. Mond, in: Singularities and foliations. Geometry, topology and applications. BMMS 2/NBMS 3, Salvador, Brazil, 2015. Proceedings of the 3rd singularity theory meeting, ENSINO, July 8–11, 2015 and the Brazil-Mexico 2nd meeting of singularities, July 13–17, 2015. Cham: Springer. 229–258 (2018; Zbl 1405.32048); J. J. Nuño-Ballesteros et al., Collect. Math. 69, No. 1, 65–81 (2018; Zbl 1393.32014)]).
In the preface, the authors emphasize that the book “is a monograph and not a textbook – its shape reflects the subject, or rather our knowledge of it, rather than the structure of a course ...”. However, they hope that it can be used as a basis for a graduate course. In addition, the authors proclaim that the subject of their book “is not algebraic geometry, but smooth (\(\mathcal C^\infty\)) and complex analytic geometry though the gap is not all that wide”. Then they provide a brief background to the subject based on a series of classical papers by M. Morse, H. Whitney, R. Thom, J. Mather, V. I. Arnold and their followers. More precisely, namely, the first part of the book concentrates at these works; it contains a brief description of the basics of the Thom-Mather theory and its relation with many other topics and various applications. All of them are discussed in detail in seven sections, including the theory of left-right equivalence, stability, contact equivalence, versal unfoldings, finite determinacy, classification of stable germs by their local algebras, etc.
In the second part, consisting of four sections, the authors focus mainly on the complex case. This part includes recent research results; it is devoted to the study of topology of stable perturbations, stable images and discriminants, the theory of multiple points, some important properties of bifurcation sets, the theory of knots in the framework of classical studies by K. Reidemeister [Abh. Math. Semin. Univ. Hamb. 5, 24–32 (1926; JFM 52.0579.01)], and so on.
Five appendices on background material include a detailed description of necessary concepts and results from the theory of jet spaces and bundles, the theory of stratifications, commutative algebra, local analytic geometry and the theory of sheaves.
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.
At the end of some sections, the authors listed a number of open questions. 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.
The variety of topics covered makes this book also extremely valuable to researchers, lecturers, and practitioners working in the field of general theory of singularities and catastrophe theory, algebraic geometry, real and complex analysis, commutative algebra, topology, and other areas of contemporary mathematics.

MSC:

58-02 Research exposition (monographs, survey articles) pertaining to global analysis
58K15 Topological properties of mappings on manifolds
58K40 Classification; finite determinacy of map germs
58K05 Critical points of functions and mappings on manifolds
58K60 Deformation of singularities
58K70 Symmetries, equivariance on manifolds
58K25 Stability theory for manifolds
32S05 Local complex singularities
32S25 Complex surface and hypersurface singularities
32S55 Milnor fibration; relations with knot theory
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
57K10 Knot theory
14B05 Singularities in algebraic geometry
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