Sistema Bibliotecario de la Universidad de Quintana Roo

Part I. The topological structure of isolated critical points of functions -- Introduction -- Elements of the theory of Picard-Lefschetz -- The topology of the non-singular level set and the variation operator of a singularity -- The bifurcation sets and the monodromy group of a singularity -- The intersection matrices of singularities of functions of two variables -- The intersection forms of boundary singularities and the topology of complete intersections -- Part II. Oscillatory integrals -- Discussion of results -- Elementary integrals and the resolution of singularities of the phase -- Asymptotics and Newton polyhedra -- The singular index, examples -- Part III. Integrals of holomorphic forms over vanishing cycles -- The simplest properties of the integrals -- Complex oscillatory integrals -- Integrals and differential equations -- The coefficients of series expansions of integrals, the weighted and Hodge filtrations and the spectrum of a critical point -- The mixed Hodge structure of an isolated critical point of a holomorphic function -- The period map and the intersection form -- References -- Subject Index.

Originally published inthe 1980s, Singularities of DifferentiableMaps: Monodromy and Asymptotics of Integrals was thesecond oftwovolumes that togetherformed 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 availableand affordableto a global audience of researchers and practitioners. While the first volume of this title, subtitled Classification of Critical Points, Caustics and Wave Fronts, contained the zoology of differentiable mapsthat is, was devoted to a description of what, where, and how singularities could be encounteredthis second volume concentrates on elements of theanatomy and physiology of singularities of differentiable functions. The questions considered here are about the structure of singularities and how they function. In the first part the authors consider the topological structure of isolated critical points of holomorphic functions: vanishing cycles; distinguished bases; intersection matrices;monodromy groups;the variation operator;and their interconnections and method of calculation. The second part is devoted to the study of the asymptotic behavior of integrals of the method of stationary phase, which is widely met within applications. The third and last part deals with integrals evaluated over level manifolds in a neighborhood of the critical point of a holomorphic function. Thismonograph is suitablefor mathematicians, researchers, postgraduates, and specialists in the areas of mechanics, physics, technology, and other sciences dealing with the theory of singularities of differentiable maps.

ZDB-2-SMA