The Monodromy Group (eBook)

Henryk Zoladek (Autor)

eBook Download: PDF

2006 | 2006
XI, 583 Seiten
Springer Basel (Verlag)
978-3-7643-7536-2 (ISBN)

Infos

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 16^th 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.

Inhalt

Contents 5
Preface 7
1 Analytic Functions and Morse Theory 12
§1 Theorem about Monodromy 12
§2 Morse Lemma 14
§3 The Morse Theory 18
2 Normal Forms of Functions 23
§1 Tougeron Theorem 23
§2 Deformations 27
§3 Proofs of Theorems 2.3 and 2.4 33
§4 Classi.cation of Singularities 39
3 Algebraic Topology of Manifolds 45
§1 Homology and Cohomology 45
§2 Index of Intersection 50
§3 Homotopy Theory 65
4 Topology and Monodromy of Functions 67
§1 Topology of a Non-singular Level 67
§2 Picard-Lefschetz Formula 75
§3 Root Systems and Coxeter Groups 92
§4 Bifurcational Diagrams 98
§5 Resolution and Normalization 112
5 Integrals along Vanishing Cycles 127
§1 Analytic Properties of Integrals 127
§2 Singularities and Branching of Integrals 135
§3 Picard–Fuchs Equations 138
§4 Gauss–Manin Connection 150
§5 Oscillating Integrals 160
6 Vector Fields and Abelian Integrals 169
§1 Phase Portraits of Vector Fields 169
§2 Method of Abelian Integrals 174
§3 Quadratic Centers and Bautin’s Theorem 199
7 Hodge Structures and Period Map 205
§1 Hodge Structure on Algebraic Manifolds 206
§2 Hypercohomologies and Spectral Sequences 213
§3 Mixed Hodge Structures 220
§4 Mixed Hodge Structures and Monodromy 234
§5 Period Mapping in Algebraic Geometry 262
8 Linear Di.erential Systems 277
§1 Introduction 277
§2 Regular Singularities 280
§3 Irregular Singularities 289
§4 Global Theory of Linear Equations 303
§5 Riemann–Hilbert Problem 306
§6 The Bolibruch Example 317
§7 Isomonodromic Deformations 325
§8 Relation with Quantum Field Theory 334
9 Holomorphic Foliations. Local Theory 342
§1 Foliations and Complex Structures 343
§2 Resolution for Vector Fields 348
§3 One-Dimensional Analytic Di.eomorphisms 355
§4 The Ecalle Approach 369
§5 Martinet–Ramis Moduli 375
§6 Normal Forms for Resonant Saddles 387
§7 Theorems of Briuno and Yoccoz 390
10 Holomorphic Foliations. Global Aspects 401
§1 Algebraic Leaves 401
§2 Monodromy of the Leaf at In.nity 419
§3 Groups of Analytic Di.eomorphisms 426
§4 The Ziglin Theory 443
11 The Galois Theory 449
§1 Picard–Vessiot Extensions 449
§2 Topological Galois Theory 479
12 Hypergeometric Functions 499
§1 The Gauss Hypergeometric Equation 499
§2 The Picard–Deligne–Mostow Theory 523
§3 Multiple Hypergeometric Integrals 535
Bibliography 545
Index 566

Produkt-Details

• Erscheint lt. Verlag: 10.8.2006
• Reihe/Serie: Monografie Matematyczne
• Reihe/Serie: Monografie Matematyczne
• Zusatzinfo: XI, 583 p.
• Verlagsort: Basel
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: Algebra • Algebraic Geometry • algebraic topology • Analytic Functions • Dynamical Systems • Foliations • Galois Theory • Hodge structures • Monodromy • Monodromy group • Morse Theory • Ordinary differential equations • singularity theory
• ISBN-10: 3-7643-7536-1 / 3764375361
• ISBN-13: 978-3-7643-7536-2 / 9783764375362