Singularities of differentiable maps, Volume 1. Classification of critical points, caustics and wave fronts. Transl. from the Russian by Ian Porteous, edited by V. I. Arnol'd
This volume is a reprint of the 1985 edition of volume I (see [Zbl 0554.58001]) of one of the most famous monographs in Singularity Theory consisting of two volumes. The topics developed in this book have had a tremendous impact in many fields of mathematics and theoretical physics concerned with singularities and bifurcations, like Algebraic and Analytic Geometry, Simplectic and Contact Topology, Tropical Geometry, Geometrical Optics, Robotics etc. This first volume contains the following. Part I, the basic theory of deformation and stability of maps, based on John Mather's work. Boardman numbers and classes. Part II, devoted to the classification of critical points of smooth functions. Basic results on quasihomogeneous and semi-quasihomogeneous singularities: Poincaré polynomial, Newton filtration, classification. List of singularities (simple, unimodal and bimodal) and their adjacencies, with indication of proof. Extension of the classification to boundary singularities and to symmetric singularities. Part III, the study of singularities of caustics and wave-fronts. Basic results in the theory of Lagrangian and Legendrian singularities. Classification of singularities of generic caustics in spaces of dimension \(\leq 10\), and of generic singularities of wave-fronts in spaces of dimension \(\leq 11\). Classification of bifurcation of wave-fronts in generic 1-parameter families of spaces of dimension \(\leq 5\), and the same for caustics in generic 1-parameter families of spaces of dimension \(\leq 3\). Many examples and lists of classified singularities are given. This book has been used and cited in a huge number of articles ever since its first printing. New research streams still find some of their inspiration and basic references in this outstanding monograph.
scientific article; zbMATH DE number 6034389