Applied Picard-Lefschetz theory / V. A. Vassiliev.

Format:

Book

Author/Creator:

Vasilʹev, V. A., 1956-

Series:

Mathematical surveys and monographs ; no. 97.

Mathematical surveys and monographs, 0076-5376 ; v. 97

Language:

English

Subjects (All):

Picard-Lefschetz theory.

Singularities (Mathematics).

Integral representations.

Physical Description:

xi, 324 pages : illustrations ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, [2002]

Contents:

1. Monodromy and its localization 1

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

3. Surface potentials 13

4. Petrovskii theory of lacunas for hyperbolic operators 18

5. Hypergeometric integrals 22

Chapter I. Local Monodromy Theory of Isolated Singularities of Functions and Complete Intersections 29

1. Gau[beta]-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 Hyperplane 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 Theorem on the Non-Integrability of Ovals 111

2. Reduction to monodromy theory 117

3. The class "cap" 119

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

6. Obstructions to integrability arising from cuspidal edges. Proof of Theorem 1.8 126

7. Ramification close to asymptotic hyperplanes. Proof of Theorem 1.9 133

8. Open problems 136

Chapter IV. Lacunas and Local Petrovskii Condition for Hyperbolic Differential Operators with Constant Coefficients 137

2. Hyperbolic polynomials 140

3. Hyperbolic operators and hyperbolic polynomials. Sharpness, diffusion, and lacunas 142

4. Generating functions and generating families of wave fronts. Classification of singular points of wave fronts 146

5. Local lacunas close to non-singular points of fronts and close to singular points of types A[subscript 2] and A[subscript 3] (after Davydova, Borovikov and Garding) 149

6. Petrovskii and Leray cycles. Herglotz-Petrovskii-Leray formula. Petrovskii condition for global lacunas 151

7. Local Petrovskii condition and local Petrovskii cycle. Local Petrovskii condition implies sharpness 155

8. Sharpness implies the local Petrovskii condition close to the finite type points of wave fronts 159

9. Local Petrovskii condition can be stronger than sharpness 162

10. Normal forms of non-sharpness at the singularities of wave fronts (after A.N. Varchenko) 162

11. Problems 164

Chapter V. Calculation of Local Petrovskii Cycles and Enumeration of Local Lacunas Close to Real Singularities 165

1. Main theorems 165

2. Local lacunas close to table singularities 174

3. Calculation of the even local Petrovskii class 182

4. Calculation of the odd local Petrovskii class 187

5. Stabilization of local Petrovskii classes 191

6. Local lacunas close to simple singularities 192

7. Geometric characterization of local lacunas at simple singularities 207

8. A program enumerating topologically distinct morsifications of real function singularities 209

Chapter VI. Homology of Local Systems, Twisted Monodromy Theory, and Regularization of Improper Integration Cycles 215

1. Local systems and their homology groups 215

2. Twisted vanishing homology of functions and complete intersections 218

3. Regularization of non-compact cycles 224

4. The "double loop" cycle 226

5. Monodromy of twisted vanishing homology for Pham singularities 234

6. Stratified Picard-Lefschetz theory with twisted coefficients 240

Chapter VII. Analytic Properties of Surface Potentials 251

2. Theorems of Newton and Ivory 254

3. Hyperbolic potentials are regular in the hyperbolicity domain (after V.I. Arnold and A.B. Givental) 256

4. Reduction to monodromy theory 260

5. Ramification of potentials and monodromy of complete intersections 265

6. Examples: curves, quadrics, and Ivory's second theorem 272

7. Description of the small monodromy group 274

8. Proof of Theorem 1.4 283

9. Proof of Theorem 1.3 284

Chapter VIII. Multidimensional Hypergeometric Functions, Their Ramification, Singularities, and Resonances 287

2. Proof of the meromorphy theorem 291

3. The hypergeometric function and its one-dimensional generalizations 295

4. Homology of complements of plane arrangements. Basic strata 297

5. The number of independent hypergeometric integrals on basic strata 305.

Notes:

Includes bibliographical references (pages 313-320) and index.

ISBN:

0821829483

OCLC:

49611064