Catalogue


The monodromy group /
Henryk *Zoł*adek.
imprint
Basel, Switzerland ; Boston : Birkhäuser, c2006.
description
xi, 580 p. : ill. ; 24 cm.
ISBN
3764375353 (alk. paper)
format(s)
Book
Holdings
More Details
imprint
Basel, Switzerland ; Boston : Birkhäuser, c2006.
isbn
3764375353 (alk. paper)
standard identifier
9783764375355
catalogue key
5889379
 
Includes bibliographical references (p. [537]-557) and index.
A Look Inside
Summaries
Main Description
In singularity theory and algebraic geometry the monodromy group is embodied in the Picard-Lefschetz formula and the Picard-Fuchs equations. It has applications in the weakened 16th Hilbert problem and in mixed Hodge structures.
Main Description
In singularity theory and algebraic geometry, the monodromy group is embodied in the Picard-Lefschetz formula and the Picard-Fuchs equations. It has applications in the weakened 16th Hilbert problem and in mixed Hodge structures. There is a deep connection of monodromy theory with Galois theory of differential equations and algebraic functions. In covering these and other topics, this book underlines the unifying role of the monogropy group.
Long Description
In singularity theory and algebraic geometry the monodromy group is embodied in the Picard-Lefschetz formula and the Picard-Fuchs equations. It has applications in the weakened 16th Hilbert problem and in mixed Hodge structures. In the theory of systems of linear differential equations one has the Riemann-Hilbert problem, the Stokes phenomena and the hypergeometric functions with their multidimensional generalizations. In the theory of homomorphic foliations there appear the Ecalle-Voronin-Martinet-Ramis moduli. On the other hand, there is a deep connection of monodromy theory with Galois theory of differential equations and algebraic functions. All this is presented in this book, underlining the unifying role of the monodromy group. The material is addressed to a wide audience, ranging from specialists in the theory of ordinary differential equations to algebraic geometers. The book contains a lot of results which are usually spread in many sources. Readers can quickly get introduced to modern and vital mathematical theories, such as singularity theory, analytic theory of ordinary differential equations, holomorphic foliations, Galois theory, and parts of algebraic geometry, without searching in vast literature.
Back Cover Copy
In singularity theory and algebraic geometry the monodromy group is embodied in the Picard-Lefschetz formula and the Picard-Fuchs equations. It has applications in the weakened 16th Hilbert problem and in mixed Hodge structures. In the theory of systems of linear differential equations one has the Riemann-Hilbert problem, the Stokes phenomena and the hypergeometric functions with their multidimensional generalizations. In the theory of homomorphic foliations there appear the Ecalle-Voronin-Martinet-Ramis moduli. On the other hand, there is a deep connection of monodromy theory with Galois theory of differential equations and algebraic functions. All this is presented in this book, underlining the unifying role of the monodromy group.The material is addressed to a wide audience, ranging from specialists in the theory of ordinary differential equations to algebraic geometers. The book contains a lot of results which are usually spread in many sources. Readers can quickly get introduced to modern and vital mathematical theories, such as singularity theory, analytic theory of ordinary differential equations, holomorphic foliations, Galois theory, and parts of algebraic geometry, without searching in vast literature.
Table of Contents
Prefacep. vii
Analytic Functions and Morse Theoryp. 1
Theorem about Monodromyp. 1
Morse Lemmap. 3
The Morse Theoryp. 7
Normal Forms of Functionsp. 13
Tougeron Theoremp. 13
Deformationsp. 17
Proofs of Theorems 2.3 and 2.4p. 23
Classification of Singularitiesp. 29
Algebraic Topology of Manifoldsp. 35
Homology and Cohomologyp. 35
Index of Intersectionp. 40
Homotopy Theoryp. 55
Topology and Monodromy of Functionsp. 57
Topology of a Non-singular Levelp. 57
Picard-Lefschetz Formulap. 65
Root Systems and Coxeter Groupsp. 82
Bifurcational Diagramsp. 88
Resolution and Normalizationp. 102
Integrals along Vanishing Cyclesp. 117
Analytic Properties of Integralsp. 117
Singularities and Branching of Integralsp. 125
Picard-Fuchs Equationsp. 128
Gauss-Manin Connectionp. 140
Oscillating Integralsp. 150
Vector Fields and Abelian Integralsp. 159
Phase Portraits of Vector Fieldsp. 159
Method of Abelian Integralsp. 164
Quadratic Centers and Bautin's Theoremp. 189
Hodge Structures and Period Mapp. 195
Hodge Structure on Algebraic Manifoldsp. 196
Hypercohomologies and Spectral Sequencesp. 203
Mixed Hodge Structuresp. 210
Mixed Hodge Structures and Monodromyp. 224
Period Mapping in Algebraic Geometryp. 252
Linear Differential Systemsp. 267
Introductionp. 267
Regular Singularitiesp. 270
Irregular Singularitiesp. 279
Global Theory of Linear Equationsp. 293
Riemann-Hilbert Problemp. 296
The Bolibruch Examplep. 307
Isomonodromic Deformationsp. 315
Relation with Quantum Field Theoryp. 324
Holomorphic Foliations. Local Theoryp. 333
Foliations and Complex Structuresp. 334
Resolution for Vector Fieldsp. 339
One-Dimensional Analytic Diffeomorphismsp. 346
The Ecalle Approachp. 360
Martinet-Ramis Modulip. 366
Normal Forms for Resonant Saddlesp. 378
Theorems of Briuno and Yoccozp. 381
Holomorphic Foliations. Global Aspectsp. 393
Algebraic Leavesp. 393
Monodromy of the Leaf at Infinityp. 411
Groups of Analytic Diffeomorphismsp. 418
The Ziglin Theoryp. 435
The Galois Theoryp. 441
Picard-Vessiot Extensionsp. 441
Topological Galois Theoryp. 471
Hypergeometric Functionsp. 491
The Gauss Hypergeometric Equationp. 491
The Picard-Deligne-Mostow Theoryp. 515
Multiple Hypergeometric Integralsp. 527
Bibliographyp. 537
Indexp. 559
Table of Contents provided by Ingram. All Rights Reserved.

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