Ebook: Singularities of integrals: homology, hyperfunctions and microlocal analysis

""Contents""; ""Foreword""; ""Part I Introduction to a topological study of Landau singularities""; ""Introduction""; ""I Differentiable manifolds""; ""1 Definition of a topological manifold""; ""2 Structures on a manifold""; ""3 Submanifolds""; ""4 The tangent space of a differentiable manifold""; ""5 Differential forms on a manifold""; ""6 Partitions of unity on a C manifold""; ""7 Orientation of manifolds. Integration on manifolds""; ""8 Appendix on complex analytic sets""; ""II Homology and cohomology of manifolds""; ""1 Chains on a manifold (following de Rham). Stokes' formula"";Bringing together two fundamental texts from Frédéric Pham's research on singular integrals, the first part of this book focuses on topological and geometrical aspects while the second explains the analytic approach. Using notions developed by J. Leray in the calculus of residues in several variables and R. Thom's isotopy theorems, Frédéric Pham's foundational study of the singularities of integrals lies at the interface between analysis and algebraic geometry, culminating in the Picard-Lefschetz formulae. These mathematical structures, enriched by the work of Nilsson, are then approached using methods from the theory of differential equations and generalized from the point of view of hyperfunction theory and microlocal analysis. Providing a 'must-have' introduction to the singularities of integrals, a number of supplementary references also offer a convenient guide to the subjects covered. This book will appeal to both mathematicians and physicists with an interest in the area of singularities of integrals. Frédéric Pham, now retired, was Professor at the University of Nice. He has published several educational and research texts. His recent work concerns semi-classical analysis and resurgent functions.