The Monodromy Group

Przednia okładka
Springer Science & Business Media, 10 sie 2006 - 583
0 Recenzje

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.

 

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Spis treści

Linear Differential Systems
267
2 Regular Singularities
270
3 Irregular Singularities
279
4 Global Theory of Linear Equations
293
5 RiemannHilbert Problem
296
6 The Bolibruch Example
307
7 Isomonodromic Deformations
315
8 Relation with Quantum Field Theory
324

2 Index of Intersection
40
3 Homotopy Theory
55
Topology and Monodromy of Functions
57
2 PicardLefschetz Formula
65
3 Root Systems and Coxeter Groups
82
4 Bifurcational Diagrams
88
5 Resolution and Normalization
102
Integrals along Vanishing Cycles
117
2 Singularities and Branching of Integrals
125
3 PicardFuchs Equations
128
4 GaussManin Connection
140
5 Oscillating Integrals
150
Vector Fields and Abelian Integrals
159
2 Method of Abelian Integrals
164
3 Quadratic Centers and Bautins Theorem
189
Hodge Structures and Period Map
195
1 Hodge Structure on Algebraic Manifolds
196
2 Hypercohomologies and Spectral Sequences
203
3 Mixed Hodge Structures
210
4 Mixed Hodge Structures and Monodromy
224
5 Period Mapping in Algebraic Geometry
252
Holomorphic Foliations Local Theory
332
1 Foliations and Complex Structures
334
2 Resolution for Vector Fields
339
3 OneDimensional Analytic Diffeomorphisms
346
4 The Ecalle Approach
360
5 MartinetRamis Moduli
366
6 Normal Forms for Resonant Saddles
378
7 Theorems of Briuno and Yoccoz
381
Holomorphic Foliations Global Aspects
393
2 Monodromy of the Leaf at Infinity
411
3 Groups of Analytic Diffeomorphisms
418
4 The Ziglin Theory
435
The Galois Theory
441
2 Topological Galois Theory
471
Hypergeometric Functions
491
2 The PicardDeligneMostow Theory
515
3 Multiple Hypergeometric Integrals
527
Bibliography
537
Index
558
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