Contents

PREFACE ix

INTRODUCTION 1

1. Monodromy and its localization 1

2. Newton's problem on the integrability of ovals 6

3. Surface potentials 13

4. Petrovskh theory of lacunas for hyperbolic operators 18

5. Hypergeometric integrals 22

Chapter I. LOCAL MONODROMY THEORY OF ISOLATED

SINGULARITIES OF FUNCTIONS AND

COMPLET E INTERSECTIONS 29

1. GauB-Manin connection in homological bundles. Monodromy and

variation operators 29

2. Picard-Lefschetz formula 32

3. Monodromy theory of isolated function singularities 37

4. Dynkin diagrams of real singularities of functions of two variables

(after S.M. Gusein-Zade and N. A'Campo) 51

5. Classification of singularities of smooth functions 56

6. Lyashko-Looijenga covering 62

7. Complements of discriminants of real simple singularities (after

E. Looijenga) 65

8. Pham singularities 67

9. Singularities and local monodromy of complete intersections 71

Chapter II. STRATIFIED PICARD-LEFSCHETZ THEORY AND

MONODROMY OF HYPERPLAN E SECTIONS 75

1. Stratifications of semianalytic and subanalytic sets 76

2. Monodromy of hyperplane sections 79

3. Simplest facts on intersection homology theory 89

4. Stratified Picard-Lefschetz theory 91

Chapter III. NEWTON'S THEORE M ON THE NON-INTEGRABILITY

OF OVALS 111

1. Introduction 111

2. Reduction to monodromy theory 117

3. The class "cap" 119

4. Ramification of integration chains at non-singular points 121

5. Examples 124