Many important functions of mathematical physics are defined as integrals depending on parameters. The Picard-Lefschetz theory studies how analytic and qualitative properties of such integrals (regularity, algebraicity, ramification, singular points, etc.) depend on the monodromy of corresponding integration cycles. In this book, V. A. Vassiliev presents several versions of the Picard-Lefschetz theory, including the classical local monodromy theory of singularities and complete intersections, Pham's generalized Picard-Lefschetz formulas, stratified Picard-Lefschetz theory, and also twisted versions of all these theories with applications to integrals of multivalued forms. The author also shows how these versions of the Picard-Lefschetz theory are used in studying a variety of problems arising in many areas of mathematics and mathematical physics.In particular, he discusses the following classes of functions: volume functions arising in the Archimedes-Newton problem of integrable bodies; Newton-Coulomb potentials; fundamental solutions of hyperbolic partial differential equations; and, multidimensional hypergeometric functions generalizing the classical Gauss hypergeometric integral. The book is geared toward a broad audience of graduate students, research mathematicians and mathematical physicists interested in algebraic geometry, complex analysis, singularity theory, asymptotic methods, potential theory, and hyperbolic operators.
Introduction Local monodromy theory of isolated singularities of functions and complete intersections Stratified Picard-Lefschetz theory and monodromy of hyperplane sections Newton's theorem on the non-integrability of ovals Lacunas and local Petrovskiicondition for hyperbolic differential operators with constant coefficients Calculation of local Petrovskiicycles and enumeration of local lacunas close to real singularities Homology of local systems, twisted monodromy theory, and regularization of improper integration cycles Analytic properties of surface potentials Multidimensional hypergeometric functions, their ramification, singularities, and resonances Bibliography Index.