Żoładek H. The monodromy group / Instytut matematyczny PAN. - Basel: Birkhäuser Verlag, 2006. - 580 p. - Bibliogr.: p.537-557. - (Monografie matematyczne. New series; Vol.67). - Ind.: p.559-580. - ISBN-10 3-7643-7535-3; ISBN-13 978-3-7643-7536-6  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

Preface ....................................................... vii 1 Analytic Functions and Morse Theory .......................... 1 §1 Theorem about Monodromy ................................... 1 §2 Morse Lemma ............................................... 3 §3 The Morse Theory .......................................... 7 2 Normal Forms of Functions ................................... 13 §1 Tougeron Theorem ......................................... 13 §2 Deformations ............................................. 17 §3 Proofs of Theorems 2.3 and 2.4 ........................... 23 §4 Classification of Singularities .......................... 29 3 Algebraic Topology of Manifolds ............................. 35 §1 Homology and Cohomology .................................. 35 §2 Index of Intersection .................................... 40 §3 Homotopy Theory .......................................... 55 4 Topology and Monodromy of Functions ......................... 57 §1 Topology of a Non-singular Level ......................... 57 §2 Picard-Lefschetz Formula ................................. 65 §3 Root Systems and Coxeter Groups .......................... 82 §4 Bifurcational Diagrams ................................... 88 §5 Resolution and Normalization ............................ 102 5 Integrals along Vanishing Cycles ........................... 117 §1 Analytic Properties of Integrals ........................ 117 §2 Singularities and Branching of Integrals ................ 125 §3 Picard-Fuchs Equations .................................. 128 §4 Gauss-Manin Connection .................................. 140 §5 Oscillating Integrals ................................... 150 6 Vector Fields and Abelian Integrals ........................ 159 §1 Phase Portraits of Vector Fields ........................ 159 §2 Method of Abelian Integrals ............................. 164 §3 Quadratic Centers and Bautin's Theorem .................. 189 7 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 8 Linear Differential Systems ................................ 267 §1 Introduction ............................................ 267 §2 Regular Singularities ................................... 270 §3 Irregular Singularities ................................. 279 §4 Global Theory of Linear Equations ....................... 293 §5 Riemann-Hilbert Problem ................................. 296 §6 The Bolibruch Example ................................... 307 §7 Isomonodromic Deformations .............................. 315 §8 Relation with Quantum Field Theory ...................... 324 9 Holomorphic Foliations. Local Theory ....................... 333 §1 Foliations and Complex Structures ....................... 334 §2 Resolution for Vector Fields ............................ 339 §3 One-Dimensional Analytic Diffeomorphisms ................ 346 §4 The Ecalle Approach ..................................... 360 §5 Martinet-Ramis Moduli ................................... 366 §6 Normal Forms for Resonant Saddles ....................... 378 §7 Theorems of Briuno and Yoccoz ........................... 381 10 Holomorphic Foliations. Global Aspects ..................... 393 §1 Algebraic Leaves ........................................ 393 §2 Monodromy of the Leaf at Infinity ....................... 411 §3 Groups of Analytic Diffeomorphisms ...................... 418 §4 The Ziglin Theory ....................................... 435 11 The Galois Theory .......................................... 441 §1 Picard-Vessiot Extensions ............................... 441 §2 Topological Galois Theory ............................... 471 12 Hypergeometric Functions ................................... 491 §1 The Gauss Hypergeometric Equation ....................... 491 §2 The Picard-Deligne-Mostow Theory ........................ 515 §3 Multiple Hypergeometric Integrals ....................... 527 Bibliography .................................................. 537 Index ......................................................... 559