Ramified Integrals, Singularities and Lacunas

Many special functions occuring in physics and partial differential equations can be represented by integral transformatIons: the fundamental solutions of many PDE's, Newton-Coulomb potentials, hypergeometric functions, Feynman integrals, initial data of (inverse) tomography problems, etc. The general picture of such transfor mations is as follows. There is an analytic fibre bundle E --+ T, a differential form w on E, whose restrictions on the fibres are closed, and a family of cycles in these fibres, parametrized by the points of T and depending continuously on these points. Then the integral of the form w along these cycles is a function on the base. The analytic properties of such functions depend on the monodromy action, i.e., on the natural action of the fundamental group of the base in the homology of the fibre: this action on the integration cycles defines the ramification of the analytic continuation of our function. The study of this action (which is a purely topological problem) can answer questions about the analytic behaviour of the integral function, for instance, is this function single-valued or at least algebraic, what are the singular points of this function, and what is its asymptotics close to these points. In this book, we study such analytic properties of three famous classes of func tions: the volume functions, which appear in the Archimedes-Newton problem on in tegrable bodies; the Newton-Coulomb potentials, and the Green functions of hyperbolic equations (studied, in particular, in the Hada mard-Petrovskii-Atiyah-Bott-Garding lacuna theory).

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This volume 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. br/ 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; extension of the proofs for the theorems of Newton, Ivory, Arnold and Givental on potentials of algebraic surfaces. Also, it is discovered for which emd/em and emn/em the potentials of degree emd/em hyperbolic surfaces in Rsupn/sup 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--Gårding condition is proved, and 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. br/ This book will be valuable to those who are interested in integral transforms, operational calculus, algebraic geometry, PDEs, manifolds and cell complexes and potential theory. br/

Contenu
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. 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-PetrovskiiLeray 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.