Elliptic Theory on Singular Manifolds

Elliptic Theory on Singular Manifolds

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Description

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.
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Product details

  • Hardback | 376 pages
  • 172 x 244 x 24mm | 759.99g
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 29 Illustrations, black and white
  • 1584885203
  • 9781584885207

Table of contents

I Singular Manifolds and Differential Operators

GEOMETRY OF SINGULARITIES
Preliminaries
Manifolds with conical singularities
Manifolds with edges
ELLIPTIC OPERATORS ON SINGULAR MANIFOLDS
Operators on manifolds with conical singularities
Operators on manifolds with edges
Examples of elliptic edge operators

II Analytical Tools

PSEUDODIFFERENTIAL OPERATORS
Preliminary remarks
Classical theory
Operators in sections of Hilbert bundles
Operators on singular manifolds
Ellipticity and finiteness theorems
Index theorems on smooth closed manifolds
LOCALIZATION (SURGERY) IN ELLIPTIC THEORY
The index locality principle
Localization in index theory on smooth manifolds
Surgery for the index of elliptic operators on singular manifolds
Relative index formulas on manifolds with isolated singularities

III Topological Problems

INDEX THEORY
Statement of the problem
Invariants of interior symbol and symmetries
Invariants of the edge symbol
Index theorems
Index on manifolds with isolated singularities
Supplement. Classification of elliptic symbols with symmetry and K-theory
Supplement. Proof of Proposition 5.16
ELLIPTIC EDGE PROBLEMS
Morphisms
The obstruction to ellipticity
A formula for the obstruction in topological terms
Examples. Obstructions for geometric operators

IV Applications and Related Topics

FOURIER INTEGRAL OPERATORS ON SINGULAR MANIFOLDS
Homogeneous canonical (contact) transformations
Definition of Fourier integral operators
Properties of Fourier integral operators
The index of elliptic Fourier integral operators
Application to quantized contact transformations
Example
RELATIVE ELLIPTIC THEORY
Analytic aspects of relative elliptic theory
Topological aspects of relative elliptic theory
INDEX OF GEOMETRIC OPERATORS ON MANIFOLDS WITH CYLINDRICAL ENDS
Operators on manifolds with cylindrical ends
Index formulas
HOMOTOPY CLASSIFICATION OF ELLIPTIC OPERATORS
The homotopy classification problem
Classification on smooth manifolds
Atiyah-de Rham duality
Abstract elliptic operators and analytic K-homology
Classification on singular manifolds
Some applications
LEFSCHETZ FORMULAS
Main result
Proof of the theorem
Contributions of conical points as sums of residues
Supplement. The Lefschetz number
Supplement. The Sternin-Shatalov method

APPENDICES
Spectral Flow
Eta Invariants
Index of Parameter-Dependent Elliptic Families

Bibliographic Remarks
Bibliography
Index
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