Elliptic Theory on Singular Manifolds

Vladimir E. Nazaikinskii, Anton Yu. Savin, Bert-Wolfgang Schulze, Boris Yu. Sternin

August 12, 2005 by Chapman and Hall/CRC
Reference - 376 Pages - 29 B/W Illustrations
ISBN 9781584885207 - CAT# C5203
Series: Differential and Integral Equations and Their Applications

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  • Presents the state of the art of the analytical and topological aspects on elliptic theory on singular manifolds
  • Includes the theory of pseudodifferential operators in finite- and infinite-dimensional bundles, locality principles in index theory, the theory of index defects, and the notion of spectral flow for operator families
  • Describes the theory of Fourier integral operators on manifolds with singularities
  • Uses clear prose, many examples and numerous figures to make the presentation widely accessible
  • Summary

    The analysis and topology of elliptic operators on manifolds with singularities are much more complicated than in the smooth case and require completely new mathematical notions and theories. While there has recently been much progress in the field, many of these results have remained scattered in journals and preprints.

    Starting from an elementary level and finishing with the most recent results, this book gives a systematic exposition of both analytical and topological aspects of elliptic theory on manifolds with singularities. The presentation includes a review of the main techniques of the theory of elliptic equations, offers a comparative analysis of various approaches to differential equations on manifolds with singularities, and devotes considerable attention to applications of the theory. These include Sobolev problems, theorems of Atiyah-Bott-Lefschetz type, and proofs of index formulas for elliptic operators and problems on manifolds with singularities, including the authors' new solution to the index problem for manifolds with nonisolated singularities.

    A glossary, numerous illustrations, and many examples help readers master the subject. Clear exposition, up-to-date coverage, and accessibility-even at the advanced undergraduate level-lay the groundwork for continuing studies and further advances in the field.