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