Singularities of Differentiable Maps, Volume 2: Monodromy and Asymptotics of Integrals

ISBN-13: 9780817683429 / Angielski / Miękka / 2012 / 492 str.

V. I. Arnold; S. M. Gusein-Zade; A. N. Varchenko

Opis

The present volume is the second in a two-volume set entitled Singularities of Differentiable Maps. While the first volume, subtitled Classification of Critical Points and originallypublishedas Volume82 in the Monographs in Mathematics series, contained the zoology of differentiable maps, that is, it was devoted to a description of what, where, and how singularities could be encountered, this second volume concentrates on elements of theanatomy and physiology of singularities of differentiable functions. The questions considered are about the structure of singularities and how they function."

Spis treści

​Part I. The topological structure of isolated critical points of functions.- Introduction.- Elements of the theory of Picard-Lefschetz.- The topology of the non-singular level set and the variation operator of a singularity.- The bifurcation sets and the monodromy group of a singularity.- The intersection matrices of singularities of functions of two variables.- The intersection forms of boundary singularities and the topology of complete intersections.- Part II. Oscillatory integrals.- Discussion of results.- Elementary integrals and the resolution of singularities of the phase.- Asymptotics and Newton polyhedra.- The singular index, examples.- Part III. Integrals of holomorphic forms over vanishing cycles.- The simplest properties of the integrals.- Complex oscillatory integrals.- Integrals and differential equations.- The coefficients of series expansions of integrals, the weighted and Hodge filtrations and the spectrum of a critical point.- The mixed ​Hodge structure of an isolated critical point of a holomorphic function.- The period map and the intersection form.- References.- Subject Index.

Blurb

Originally published in the 1980s, Singularities of Differentiable Maps: Monodromy and Asymptotics of Integrals was the second of two volumes that together formed a translation of the authors' influential Russian monograph on singularity theory.  This uncorrected softcover reprint of the work brings its still-relevant content back into the literature, making it available—and affordable—to a global audience of researchers and practitioners.

​​​While the first volume of this title, subtitled Classification of Critical Points, Caustics and Wave Fronts, contained the zoology of differentiable maps—that is, was devoted to a description of what, where, and how singularities could be encountered—this second volume concentrates on elements of the anatomy and physiology of singularities of differentiable functions.  The questions considered here are about the structure of singularities and how they function.

In the first part the authors consider the topological structure of isolated critical points of holomorphic functions: vanishing cycles; distinguished bases; intersection matrices; monodromy groups; the variation operator; and their interconnections and method of calculation.  The second part is devoted to the study of the asymptotic behavior of integrals of the method of stationary phase, which is widely met within applications.  The third and last part deals with integrals evaluated over level manifolds in a neighborhood of the critical point of a holomorphic function.  

This monograph is suitable for mathematicians, researchers, postgraduates, and specialists in the areas of mechanics, physics, technology, and other sciences dealing with the theory of singularities of differentiable maps.