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

Format:

Book

Author/Creator:

Vasilʹev, V. A., 1956- author.

Series:

Mathematical surveys and monographs ; no. 97.

Mathematical surveys and monographs, 0076-5376 ; volume 97

Language:

English

Subjects (All):

Picard-Lefschetz theory.

Singularities (Mathematics).

Integral representations.

Physical Description:

1 online resource (338 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2002]

Language Note:

English

Summary:

Many important functions of mathematical physics are defined as integrals depending on parameters. The Picard-Lefschetz theory studies how analytic and qualitative properties of such integrals (regularity, algebraicity, ramification, singular points, etc.) depend on the monodromy of corresponding integration cycles. In this book, V. A. Vassiliev presents several versions of the Picard-Lefschetz theory, including the classical local monodromy theory of singularities and completeintersections, Pham's generalized Picard-Lefschetz formulas, stratified Picard-Lefschetz theory, and also twisted versions of all these theories with applications to integrals of multivalued forms. The author also shows how these versions of the Picard-Lefschetz theory are used in studying a variety ofproblems arising in many areas of mathematics and mathematical physics. In particular, he discusses the following classes of functions: volume functions arising in the Archimedes-Newton problem of integrable bodies; Newton-Coulomb potentials; fundamental solutions of hyperbolic partial differential equations; multidimensional hypergeometric functions generalizing the classical Gauss hypergeometric integral. The book is geared toward a broad audience of graduate students, research mathematiciansand mathematical physicists interested in algebraic geometry, complex analysis, singularity theory, asymptotic methods, potential theory, and hyperbolic operators.

Contents:

""4. Dynkin diagrams of real singularities of functions of two variables (after S. M. Gusein-Zade and N. A'Campo)""""5. Classification of singularities of smooth functions""; ""6. Lyashko�Looijenga covering""; ""7. Complements of discriminants of real simple singularities (after E. Looijenga)""; ""8. Pham singularities""; ""9. Singularities and local monodromy of complete intersections""; ""CHAPTER II. STRATIFIED PICARD�LEFSCHETZ THEORY AND MONODROMY OF HYPERPLANE SECTIONS""; ""1. Stratifications of semianalytic and subanalytic sets""; ""2. Monodromy of hyperplane sections""

""3. Simplest facts on intersection homology theory""""4. Stratified Picard�Lefschetz theory""; ""CHAPTER III. NEWTON'S THEOREM ON THE NON-INTEGRABILITY OF OVALS""; ""1. Introduction""; ""2. Reduction to monodromy theory""; ""3. The class ""cap""""; ""4. Ramification of integration chains at non-singular points""; ""5. Examples""; ""6. Obstructions to integrability arising from cuspidal edges. Proof of Theorem 1.8""; ""7. Ramification close to asymptotic hyperplanes. Proof of Theorem 1.9""; ""8. Open problems""

""CHAPTER IV. LACUNAS AND LOCAL PETROVSKII CONDITIONFOR HYPERBOLIC DIFFERENTIAL OPERATORS WITH CONSTANT COEFFICIENTS""""1. Introduction""; ""2. Hyperbolic polynomials""; ""3. Hyperbolic operators and hyperbolic polynomials. Sharpness, diffusion, and lacunas""; ""4. Generating functions and generating families of wave fronts. Classification of singular points of wave fronts""; ""5. Local lacunas close to non-singular points of fronts and close tosingular points of types A[sub(2)] and A[sub(3)] (after Davydova, Borovikov and G°arding)""

""6. Petrovskil and Leray cycles. Herglotz�Petrovskh�Leray formula. Petrovskil condition for global lacunas""""7. Local Petrovskil condition and local Petrovskil cycle. Local Petrovskil condition implies sharpness""; ""8. Sharpness implies the local Petrovskil condition close to the finite type points of wave fronts""; ""9. Local Petrovskil condition can be stronger than sharpness""; ""10. Normal forms of non-sharpness at the singularities of wave fronts (after A.N. Varchenko)""; ""11. Problems""

""CHAPTER V. CALCULATION OF LOCAL PETROVSKII CYCLES AND ENUMERATION OF LOCAL LACUNAS CLOSE TO REAL SINGULARITIES""

Notes:

Description based upon print version of record.

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

Description based on print version record.

ISBN:

1-4704-1324-8