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" Applied Picard-Lefschetz theory / "
V.A. Vassiliev.
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BL
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| Record Number
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637148
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dltt
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| Main Entry
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Vasilʹev, V. A.,1956-
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| Title & Author
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Applied Picard-Lefschetz theory /\ V.A. Vassiliev.
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| Publication Statement
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Providence, R.I. :: American Mathematical Society,, c2002.
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| Series Statement
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Mathematical surveys and monographs,; v. 97
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| Page. NO
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xi, 324 p. :: ill. ;; 26 cm.
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| ISBN
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0821829483 (acid-free paper)
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: 9780821829486 (acid-free paper)
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| Bibliographies/Indexes
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Includes bibliographical references (p. 313-320) and index.
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| Contents
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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.
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Picard-Lefschetz theory.
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Singularities (Mathematics)
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Integral representations.
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| LC Classification
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QA564.V37 2002x
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