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