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

V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko
Publisher: 
Birkhäuser
Publication Date: 
2012
Number of Pages: 
492
Format: 
Paperback
Series: 
Modern Birkhäuser Classics
Price: 
89.95
ISBN: 
9780817683429
Category: 
Monograph
We do not plan to review this book.

​​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.

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