Many vital capabilities of mathematical physics are outlined as integrals counting on parameters. The Picard-Lefschetz conception stories how analytic and qualitative houses of such integrals (regularity, algebraicity, ramification, singular issues, etc.) rely on the monodromy of corresponding integration cycles. during this ebook, V. A. Vassiliev offers a number of types of the Picard-Lefschetz conception, together with the classical neighborhood monodromy idea of singularities and entire intersections, Pham's generalized Picard-Lefschetz formulation, stratified Picard-Lefschetz conception, and in addition twisted types of a majority of these theories with purposes to integrals of multivalued types. the writer additionally indicates how those types of the Picard-Lefschetz concept are utilized in learning quite a few difficulties bobbing up in lots of components of arithmetic and mathematical physics. particularly, he discusses the next periods of features: quantity features coming up within the Archimedes-Newton challenge of integrable our bodies; Newton-Coulomb potentials; primary suggestions of hyperbolic partial differential equations; multidimensional hypergeometric features generalizing the classical Gauss hypergeometric vital. The e-book is aimed toward a wide viewers of graduate scholars, learn mathematicians and mathematical physicists attracted to algebraic geometry, advanced research, singularity conception, asymptotic tools, strength thought, and hyperbolic operators.