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
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