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Inhaltsangabe
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$$overset{lower0.5emhbox{$smash{scriptscriptstylesmile}$}}{I}$$ 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$$overset{lower0.5emhbox{$smash{scriptscriptstylesmile}$}}{I}$$ 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.
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$$overset{lower0.5emhbox{$smash{scriptscriptstylesmile}$}}{I}$$ 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$$overset{lower0.5emhbox{$smash{scriptscriptstylesmile}$}}{I}$$ 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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