The Localization Problem in Index Theory of Elliptic Operators

• Main Authors: Nazaikinskii, Vladimir (Author), Schulze, Bert-Wolfgang (Author), Sternin, Boris (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Basel : Springer Basel : Imprint: Birkhäuser, 2014.
• Series: Pseudo-Differential Operators, Theory and Applications ; 10
• Subjects: Mathematics.; K-theory.; Functional analysis.; Global analysis (Mathematics).; Manifolds (Mathematics).; Partial differential equations.; Global Analysis and Analysis on Manifolds.; K-Theory.; Functional Analysis.; Partial Differential Equations.

This book deals with the localization approach to the index problem for elliptic operators. Localization ideas have been widely used for solving various specific index problems for a long time, but the fact that there is actually a fundamental localization principle underlying all these solutions ha...

Table of Contents:

• Preface
• Introduction
• 0.1 Basics of Elliptic Theory
• 0.2 Surgery and the Superposition Principle
• 0.3 Examples and Applications
• 0.4 Bibliographical Remarks
• Part I: Superposition Principle
• 1 Superposition Principle for the Relative Index
• 1.1 Collar Spaces
• 1.2 Proper Operators and Fredholm Operators
• 1.3 Superposition Principle
• 2 Superposition Principle for K-Homology
• 2.1 Preliminaries
• 2.2 Fredholm Modules and K-Homology
• 2.3 Superposition Principle
• 2.4 Fredholm Modules and Elliptic Operators
• 3 Superposition Principle for KK-Theory
• 3.1 Preliminaries
• 3.2 Hilbert Modules, Kasparov Modules, and KK
• 3.3 Superposition Principle
• Part II: Examples
• 4 Elliptic Operators on Noncompact Manifolds
• 4.1 Gromov–Lawson Theorem
• 4.2 Bunke Theorem
• 4.3 Roe’s Relative Index Construction
• 5 Applications to Boundary Value Problems
• 5.1 Preliminaries
• 5.2 Agranovich–Dynin Theorem
• 5.3 Agranovich Theorem
• 5.4 Bojarski Theorem and Its Generalizations
• 5.5 Boundary Value Problems with Symmetric Conormal Symbol
• 6 Spectral Flow for Families of Dirac Type Operators
• 6.1 Statement of the Problem
• 6.2 Simple Example
• 6.3 Formula for the Spectral Flow
• 6.4 Computation of the Spectral Flow for a Graphene Sheet
• Bibliography.