Many very important services of mathematical physics are outlined as integrals counting on parameters. The Picard-Lefschetz conception stories how analytic and qualitative homes of such integrals (regularity, algebraicity, ramification, singular issues, etc.) rely on the monodromy of corresponding integration cycles. during this publication, V. A. Vassiliev offers numerous types of the Picard-Lefschetz thought, together with the classical neighborhood monodromy conception of singularities and whole intersections, Pham's generalized Picard-Lefschetz formulation, stratified Picard-Lefschetz idea, and likewise twisted models of these kinds of theories with purposes to integrals of multivalued types. the writer additionally exhibits how those types of the Picard-Lefschetz idea are utilized in learning numerous difficulties coming up in lots of parts of arithmetic and mathematical physics. particularly, he discusses the subsequent periods of capabilities: quantity services coming up within the Archimedes-Newton challenge of integrable our bodies; Newton-Coulomb potentials; primary recommendations of hyperbolic partial differential equations; multidimensional hypergeometric services generalizing the classical Gauss hypergeometric indispensable. The booklet is aimed at a extensive viewers of graduate scholars, study mathematicians and mathematical physicists attracted to algebraic geometry, complicated research, singularity concept, asymptotic tools, capability concept, and hyperbolic operators.

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