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2011 | Buch

Differentiable manifolds Kapitel lesen Erstes Kapitel lesen

Singularities of integrals

Homology, hyperfunctions and microlocal analysis

verfasst von: Frédéric Pham

Verlag: Springer London

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Introduction to a topological study of Landau singularities

I. Differentiable manifolds
Abstract
Definition of a topological manifold, structures on a manifold, morphisms of manifolds, examples of differentiable manifolds. Submanifolds, embeddings, immersions. The tangent space of a differentiable manifold, the tangent map. Differential forms on a manifold, transformations of differential forms, complex-valued differential forms. Partitions of unity on a manifold, construction of a Riemannian metric, construction of a tubular neighbourhood of a closed submanifold. Orientation of manifolds. Every complex analytic manifold has a canonical orientation. The integral of a differential form on an oriented manifold. Integration on a manifold with boundary and Stokes’ formula. Appendix on complex analytic sets.
Frédéric Pham
II. Homology and cohomology of manifolds
Abstract
Chains on a manifold (following De Rham). Stokes’ formula, the boundary, chain transformations. Homology, torsion, retractions, homotopy, deformation-retractions, relative homology. Cohomology, cochains, homotopy, relative cohomology, de Rham duality. Families of supports. Poincaré’s isomorphism and duality, intersection index, Leray coboundary. Currents, the support of a current, the boundary and differential of a current, homology of currents, homologies between currents and differential forms, homologies between currents and chains. A useful example: the current defined by a closed oriented submanifold. Intersection indices.
Frédéric Pham
III. Leray’s theory of residues
Abstract
Division and derivatives of differential forms. The residue theorem in the case of a simple pole, Leray coboundary. The residue theorem in the case of a multiple pole. Relation with the notion of the derivative of a form. Composed residues, skew symmetry of the composed coboundary, calculation of composed residues. Generalization to relative homology.
Frédéric Pham
IV. Thom’s isotopy theorem
Abstract
Ambient Isotopy. Fiber bundles, fiber bundle map, the inverse image of a bundle, sections of a bundle, the trivial bundle, locally trivial fiber bundles, isotopy class of a path. Stratified sets, regular incidence, examples. Thom’s isotopy theorem, critical sets, apparent contours. Landau varieties, singularities of Landau varieties, incidence relations for Landau varieties.
Frédéric Pham
V. Ramification around Landau varieties
Abstract
The fundamental group of a topological space. Simple pinching, description of the vanishing chains, characterization of the vanishing classes, localization, Picard’s formula, Lefschetz’ formula, ramification type, invariant classes, proof of the Picard-Lefschetz formula. Study of certain singular points of Landau varieties, application of the Picard-Lefschetz formula.
Frédéric Pham
VI. Analyticity of an integral depending on a parameter
Abstract
Holomorphy of an integral depending on a parameter. The singular part of an integral which depends on a parameter, a practical calculation of the coefficient.
Frédéric Pham
VII. Ramification of an integral whose integrand is itself ramified
Abstract
Ramification of an integral whose integrand is itself ramified. Generalities on covering spaces, regular covering spaces, construction of all possible covering spaces of a space. Generalized Picard-Lefschetz formulae, application to the ramification of an integral, generalization to “relative” homology. Appendix on relative homology and families of supports.
Frédéric Pham

Introduction to the study of singular integrals and hyperfunctions

Frontmatter
VIII. Functions of a complex variable in the Nilsson class
Abstract
Functions of a complex variable in the Nilsson class. Differential equations with regular singular points.
Frédéric Pham
IX. Functions in the Nilsson class on a complex analytic manifold
Abstract
Functions in the Nilsson class on a complex analytic manifold. A local study of functions in the Nilsson class.
Frédéric Pham
X. Analyticity of integrals depending on parameters
Abstract
Analyticity of integrals depending on parameters. Single-valued integrals. Multivalued integrals. An example.
Frédéric Pham
XI. Sketch of a proof of Nilsson’s theorem
Abstract
Sketch of a proof of Nilsson’s theorem.
Frédéric Pham
XII. Examples: how to analyze integrals with singular integrands
Abstract
Examples: how to analyze integrals with singular integrands.
Frédéric Pham
XIII. Hyperfunctions in one variable, hyperfunctions in the Nilsson class
Abstract
Hyperfunctions in one variable. Differentiation of a hyperfunction. The local nature of the notion of a hyperfunction. The integral of a hyperfunction. Hyperfunctions whose support is reduced to a point. Hyperfunctions in the Nilsson class.
Frédéric Pham
XIV. Introduction to Sato’s microlocal analysis
Abstract
Introduction to Sato’s microlocal analysis. Functions analytic at a point and in a direction. Functions analytic in a field of directions on a real vector space. Boundary values of a function which is analytic in a field of directions. Boundary values in the sense of distributions. Boundary values in the sense of hyperfunctions. The microsingular support of a hyperfunction. The microsingular support of an integral. Oscillatory integrals. The behaviour of light waves in the neighbourhood of caustics.
Frédéric Pham
Backmatter
Metadaten
Titel
Singularities of integrals
verfasst von
Frédéric Pham
Copyright-Jahr
2011
Verlag
Springer London
Electronic ISBN
978-0-85729-603-0
Print ISBN
978-0-85729-602-3
DOI
https://doi.org/10.1007/978-0-85729-603-0

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