Ramified Integrals, Singularities and Lacunas [Hardcover]

I. PicardLefschetzPham 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. Phams 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. Newtons 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. Newtons 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\hblck