E-raamat: Elliptic Theory on Singular Manifolds

Using examples, the authors describe singular manifolds with differential operations defined on them, focusing on spaces that possess rich additional structures permitting definition of differential operators in a natural way. They begin with the geometry of singularities and proceed to elliptic operators on singular manifolds, then cover analytical tools such as pseudodifferential operators and localization in elliptic theory. They uncover topological problems such as index theory and elliptic edge problems, then describe applications and related topics such as Fourier integral operators on singular manifolds, relative elliptic theory, the index of geometric operators on manifolds with cylindrical ends, the homotopy classification of elliptic operators and Lefschetz formulas. They include a comprehensive bibliography. Annotation ©2005 Book News, Inc., Portland, OR (booknews.com)

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.