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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; 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.