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Theory of Singularities and its Applications
Edited by: V. I. Arnol'd
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Advances in Soviet Mathematics
1990; 333 pp; hardcover
Volume: 1
ISBN-10: 0-8218-4100-9
ISBN-13: 978-0-8218-4100-6
List Price: US$159
Member Price: US$127.20
Order Code: ADVSOV/1
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The theory of singularities lies at the crossroads between those branches of mathematics which are the most abstract and those which are the most applied. Algebraic and differential geometry and topology, commutative algebra and group theory are as intimately connected to singularity theory as are dynamical systems theory, control theory, differential equations, quantum mechanical and quasi-classical asymptotics, optics, and functional analysis.

This collection of papers incorporates recent results of participants in the editor's ongoing seminar in singularity theory, held in the Mechanics and Mathematics Department of Moscow University for over twenty years. With its broad range of subject matter, this volume will appeal to a wide range of readers in various areas of the mathematical sciences. Among the topics covered are: construction of new knot invariants, stable cohomology of complementary spaces to diffusion diagrams, topological properties of spaces of Legendre maps, application of Weierstrass bifurcation points in projective curve flattenings, classification of singularities of projective surfaces with boundary, nonsmoothness of visible contours of smooth convex hypersurfaces, flag manifolds, hyperbolic partial differential systems, and control theory.

Table of Contents

  • V. I. Arnold -- Ten problems
  • V. A. Vassiliev -- Topology of complements to discriminants and loop spaces
  • V. A. Vassiliev -- Cohomology of knot spaces
  • A. B. Givental -- Nonlinear generalization of the Maslov index
  • B. A. Khesin -- Singularities of light hypersurfaces and structure of hyperbolicity sets for systems of partial differential equations
  • I. A. Bogaevsky -- Degree of smoothness for visible contours of convex hypersurfaces
  • Yu. M. Baryshnikov -- Real vanishing inflections and boundary singularities
  • Yu. M. Baryshnikov -- Indices for extremal embeddings of 1-complexes
  • M. E. Kazarian -- Bifurcation of flattenings and Schubert cells
  • V. V. Goryunov -- Projections of generic surfaces with boundaries
  • V. M. Zakalyukin -- Generating ideals of Lagrangian varieties
  • A. G. Aleksandrov -- Nonisolated hypersurface singularities
  • V. N. Karpushkin -- Structure of uniform estimates in partial phase deformation
  • V. P. Kostov -- On the stratification and singularities of the Stokes hypersurface of one- and two-parameter families of polynomials
  • B. Z. Shapiro and A. D. Vainshtein -- Euler characteristics for links of Schubert cells in the space of complete flags
  • V. I. Bakhtin -- Weierstrass preparation theorem for finitely smooth modules
  • A. N. Shoshitaishvili -- Singularities for projections of integral manifolds with applications to control and observation problems
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