Overview

This work contains an introduction to the Picard-Lefschetz theory which controls the ramification and qualitative behaviour of many important functions of PDEs and integral geometry, and its foundations in singularity theory. Solutions to many problems of these theories are treated. Subjects include the proof of multidimensional analogues of Newton's theorem on the nonintegrability of ovals and extension of the proofs for the theorems of Newton, Ivory, Arnold and Givental on potentials of algebraic surfaces. Also, it is discovered for which d and n the potentials of degree d hyperbolic surfaces in Rn are algebraic outside the surfaces; the equivalence of local regularity (the so-called sharpness) of fundamental solutions of hyperbolic PDEs and the topological Petrovskii-Atiyah-Bott-Garding condition is proved. The geometrical characterization of domains of sharpness close to simple singularities of wave fronts is considered, a ""stratified"" version of the Picard-Lefschetz formula is proved and an algorithm enumerating topologically distinct Morsifications of real function singularities is given. This book should be useful to those who are interested in integral transforms, operational calculus, algebraic geometry, PDEs, manifolds and cell complexes and potential theory.

Full Product Details

Author:   V.A. Vassiliev
Publisher:   Kluwer Academic Publishers
Imprint:   Kluwer Academic Publishers
Edition:   1995 ed.
Volume:   315
Dimensions:   Width: 17.00cm , Height: 1.90cm , Length: 24.40cm
Weight:   1.370kg
ISBN:  

9780792331933

ISBN 10:   0792331931
Pages:   294
Publication Date:   30 November 1994
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

I. Picard—Lefschetz—Pham theory and singularity theory.- § 1. Gauss-Manin connection in the homological bundles. Monodromy and variation operators.- § 2. The Picard-Lefschetz formula. The Leray tube operator.- § 3. Local monodromy of isolated singularities of holomorphic functions.- § 4. Intersection form and complex conjugation in the vanishing homology of real singularities in two variables.- § 5. Classification of real and complex singularities of functions.- § 6. Lyashko-Looijenga covering and its generalizations.- § 7. Complements of discriminants of real simple singularities (after E. Looijenga).- § 8. Stratifications. Semialgebraic, semianalytic and subanalytic sets.- § 9. Pham’s formulae.- § 10. Monodromy of hyperplane sections.- § 11. Stabilization of local monodromy and variation of hyperplane sections close to strata of positive dimension (stratified Picard-Lefschetz theory).- § 12. Homology of local systems. Twisted Picard-Lefschetz formulae.- § 13. Singularities of complete intersections and their local monodromy groups.- II. Newton’s theorem on the nonintegrability of ovals.- § 1. Stating the problems and the main results.- § 2. Reduction of the integrability problem to the (generalized) PicardLefschetz theory.- § 3. The element “cap”.- § 4. Ramification of integration cycles close to nonsingular points. Generating functions and generating families of smooth hypersurfaces.- § 5. Obstructions to integrability arising from the cuspidal edges. Proof of Theorem 1.8.- § 6. Obstructions to integrability arising from the asymptotic hyperplanes. Proof of Theorem 1.9.- § 7. Several open problems.- III. Newton’s potential of algebraic layers.- § 1. Theorems of Newton and Ivory.- § 2. Potentials of hyperbolic layers are polynomialin the hyperbolicity domains (after Arnold and Givental).- § 3. Proofs of Main Theorems 1 and 2.- § 4. Description of the small monodromy group.- § 5. Proof of Main Theorem 3.- IV. Lacunas and the local Petrovski$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{I} $$ condition for hyperbolic differential operators with constant coefficients.- § 0. Hyperbolic polynomials.- § 1. Hyperbolic operators and hyperbolic polynomials. Sharpness, diffusion and lacunas.- § 2. Generating functions and generating families of wave fronts for hyperbolic operators with constant coefficients. Classification of the singular points of wave fronts.- § 3. Local lacunas close to nonsingular points of fronts and to singularities A2, A3 (after Davydova, Borovikov and Gárding).- § 4. Petrovskii and Leray cycles. The Herglotz-Petrovskii—Leray formula and the Petrovskii condition for global lacunas.- § 5. Local Petrovskii condition and local Petrovskii cycle. The local Petrovskii condition implies sharpness (after Atiyah, Bott and Gárding).- § 6. Sharpness implies the local Petrovskii condition close to discrete-type points of wave fronts of strictly hyperbolic operators.- § 7. The local Petrovskii condition may be stronger than the sharpness close to singular points not of discrete type.- § 8. Normal forms of nonsharpness close to singularities of wave fronts (after A.N. Varchenko).- § 9. Several problems.- V. Calculation of local Petrovski$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{I} $$ cycles and enumeration of local lacunas close to real function singularities.- § 1. Main theorems.- § 2. Local lacunas close to singularities from the classification tables.- § 3. Calculation of the even local Petrovskii class.- § 4. Calculation of theodd local Petrovskii class.- § 5. Stabilization of the local Petrovskii classes. Proof of Theorem 1.5.- § 6. Local lacunas close to simple singularities.- § 7. Geometrical criterion for sharpness close to simple singularities.- § 8. A program for counting topologically different morsifications of a real singularity.- § 9. More detailed description of the algorithm.- Appendix: a FORTRAN program searching for the lacunas and enumerating the morsifications of real function singularities.

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