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Overview
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.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 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--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.
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.
Synopsis
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.
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 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—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.
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.
Booknews
Introduces the Picard-Lefschetz theory, which controls the ramifications and qualitative behavior of many important functions of partial differential equations and integral geometry, and reviews the fundamentals of singularity (mathematical catastrophe) theory, on which it is based. Then uses Picard-Lefschetz as a topological tool to investigate the analytic properties of three famous classes of functions: the volume functions, which appear in the Archimedes-Newton problem on integrable bodies; the Newton-Coulomb potentials; and the Green functions of hyperbolic equations as studied in the Hadamard-Pretrovskii-Atiyah-Bott-Garding lacuna theory. Assumes a familiarity with the analysis of differential forms and homology theory. Annotation c. Book News, Inc., Portland, OR (booknews.com)