Elliptic theory and noncommutative geometry : nonlocal elliptic operators / Vladimir E. Nazaikinskii, Anton Yu. Savin, Boris Yu. Sternin.

Format:

Book

Author/Creator:

Nazaĭkinskiĭ, V. E.

Contributor:

Savin, Anton Yu.

Sternin, B. I︠U︡.

Series:

Operator theory, advances and applications ; v. 183.

Operator theory, advances and applications. Advances in partial differential equations.

Operator theory, advances and applications Advances in partial differential equations ; v. 183

Language:

English

Subjects (All):

Elliptic operators.

Noncommutative differential geometry.

Physical Description:

1 online resource (232 p.)

Edition:

1st ed. 2008.

Place of Publication:

Basel : Birkhäuser, 2008.

Language Note:

English

Summary:

This comprehensive yet concise book deals with nonlocal elliptic differential operators, whose coefficients involve shifts generated by diffeomorophisms of the manifold on which the operators are defined. The main goal of the study is to relate analytical invariants (in particular, the index) of such elliptic operators to topological invariants of the manifold itself. This problem can be solved by modern methods of noncommutative geometry. This is the first and so far the only book featuring a consistent application of methods of noncommutative geometry to the index problem in the theory of nonlocal elliptic operators. Although the book provides important results, which are in a sense definitive, on the above-mentioned topic, it contains all the necessary preliminary material, such as C*-algebras and their K-theory or cyclic homology. Thus the material is accessible for undergraduate students of mathematics (third year and beyond). It is also undoubtedly of interest for post-graduate students and scientists specializing in geometry, the theory of differential equations, functional analysis, etc. The book can serve as a good introduction to noncommutative geometry, which is one of the most powerful modern tools for studying a wide range of problems in mathematics and theoretical physics.

Contents:

Analysis of Nonlocal Elliptic Operators

Nonlocal Functions and Bundles

Nonlocal Elliptic Operators

Elliptic Operators over C*-Algebras

Homotopy Invariants of Nonlocal Elliptic Operators

Homotopy Classification

Analytic Invariants

Bott Periodicity

Direct Image and Index Formulas in K-Theory

Chern Character

Cohomological Index Formula

Cohomological Formula for the ?-Index

Index of Nonlocal Operators over C*-Algebras

Examples

Index Formula on the Noncommutative Torus

An Application of Higher Traces

Index Formula for a Finite Group ?.

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9786611491246

9781281491244

1281491241

9783764387754

3764387750

OCLC:

272313415