Singularities of Differentiable Maps, Volume 1 Classification of Critical Points, Caustics and Wave Fronts - Arnold, V.I; Gusein-Zade, S.M; Varchenko, Alexander N. (Modern Birkhäuser Classics,, 2012)
Originally published in the 1980s, Singularities of Differentiable Maps: The Classification of Critical Points, Caustics and Wave Fronts was the first of two volumes that together formed a translation of the authors' influential Russian monograph on singularity theory.  This uncorrected softcover reprint of the work brings its still-relevant content back into the literature, making it available-and affordable-to a global audience of researchers and practitioners. Singularity theory is a far-reaching extension of maxima and minima investigations of differentiable functions, with implications for many different areas of mathematics, engineering (catastrophe theory and the theory of bifurcations), and science.  The three parts of this first volume deal with the stability problem for smooth mappings, critical points of smooth functions, and caustics and wave front singularities.  Building on these concepts, the second volume (Monodromy and Asymptotics of Integrals) describes the topological and algebro-geometrical aspects of the theory, including monodromy, intersection forms, oscillatory integrals, asymptotics, and mixed Hodge structures of singularities. Singularities of Differentiable Maps: The Classification of Critical Points, Caustics and Wave Fronts accommodates the needs of non-mathematicians, presupposing a limited mathematical background and beginning at an elementary level.  With this foundation, the book's sophisticated development permits readers to explore an unparalleled breadth of applications.