Mathematical Surveys and Monographs 2002; 324 pp; hardcover Volume: 97 ISBN10: 0821829483 ISBN13: 9780821829486 List Price: US$91 Member Price: US$72.80 Order Code: SURV/97
 Many important functions of mathematical physics are defined as integrals depending on parameters. The PicardLefschetz 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 PicardLefschetz theory, including the classical local monodromy theory of singularities and complete intersections, Pham's generalized PicardLefschetz formulas, stratified PicardLefschetz 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 PicardLefschetz 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 ArchimedesNewton problem of integrable bodies;
 NewtonCoulomb potentials;
 fundamental solutions of hyperbolic partial differential equations;
 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. Readership Graduate students, research mathematicians and mathematical physicists interested in algebraic geometry, complex analysis, singularity theory, asymptotic methods, potential theory, and hyperbolic operators. Reviews "This is a book rich in ideas ..."  Mathematical Reviews Table of Contents  Introduction
 Local monodromy theory of isolated singularities of functions and complete intersections
 Stratified PicardLefschetz theory and monodromy of hyperplane sections
 Newton's theorem on the nonintegrability of ovals
 Lacunas and local Petrovskiĭcondition for hyperbolic differential operators with constant coefficients
 Calculation of local Petrovskiĭcycles 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
