2nd Edition

Elliptic Operators, Topology, and Asymptotic Methods





ISBN 9780582325029
Published January 6, 1999 by Chapman and Hall/CRC
209 Pages

USD $125.00

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Book Description

Ten years after publication of the popular first edition of this volume, the index theorem continues to stand as a central result of modern mathematics-one of the most important foci for the interaction of topology, geometry, and analysis. Retaining its concise presentation but offering streamlined analyses and expanded coverage of important examples and applications, Elliptic Operators, Topology, and Asymptotic Methods, Second Edition introduces the ideas surrounding the heat equation proof of the Atiyah-Singer index theorem.
The author builds towards proof of the Lefschetz formula and the full index theorem with four chapters of geometry, five chapters of analysis, and four chapters of topology. The topics addressed include Hodge theory, Weyl's theorem on the distribution of the eigenvalues of the Laplacian, the asymptotic expansion for the heat kernel, and the index theorem for Dirac-type operators using Getzler's direct method. As a "dessert," the final two chapters offer discussion of Witten's analytic approach to the Morse inequalities and the L2-index theorem of Atiyah for Galois coverings.
The text assumes some background in differential geometry and functional analysis. With the partial differential equation theory developed within the text and the exercises in each chapter, Elliptic Operators, Topology, and Asymptotic Methods becomes the ideal vehicle for self-study or coursework. Mathematicians, researchers, and physicists working with index theory or supersymmetry will find it a concise but wide-ranging introduction to this important and intriguing field.

Table of Contents

Resumé of Riemannian Geometry
Connections
Riemannian Geometry
Differential Forms
Exercises
Connection, Curvature, and Characteristic Classes
Principal Bundles and their Connections
Characteristic Classes
Genera
Notes
Exercises
Clifford Algebras and Dirac Operators
Clifford Bundles and Dirac Operators
Clifford Bundles and Curvature
Examples of Clifford Bundles
Notes
Exercises
The Spin Groups
The Clifford Algebra as a Superalgebra
Groups of Invertibles in the Clifford Algebra
Representation Theory of the Clifford Algebra
Spin Structures on Manifolds
Spin Bundles and Characteristic Classes
The Complex Spin Group
Notes
Exercises
Analytic Properties of Dirac Operators
Sobolev Spaces
Analysis of the Dirac Operator
The Functional Calculus
Notes
Exercises
Hodge Theory
Notes
Exercises
The Heat and Wave Equations
Existence and Uniqueness Theorems
The Asymptotic Expansion for the Heat Kernel
Finite Propagation Speed for the Wave Equation
Notes
Exercises
Traces and Eigenvalue Asymptotics
Eigenvalue Growth
Trace-Class Operators
Weyl's Asymptotic Formula
Notes
Exercises
Some Non-Compact Manifolds
The Harmonic Oscillator
Witten's Perturbation of the de Rham Complex
Functional Calculus on Open Manifolds
Notes
Exercises
The Lefschetz Formula
Lefschetz Numbers
The Fixed-Point Contributions
Notes
Exercises
The Index Problem
Gradings and Clifford Bundles
Graded Dirac Operators
The Heat Equations and the Index Theorem
Notes
Exercises
The Getzler Calculus and the Local Index Theorem
Filtered Algebras and Symbols
Getzler Symbols
The Getzler Symbol of the Heat Kernel
The Exact Solution
The Index Theorem
Notes
Exercises
Applications of the Index Theorem
The Spinor Dirac Operator
The Signature Theorem
The Hirzebruch-Riemann-Roch Theorem
Local Index Theory
Notes
Exercises
Witten's Approach to Morse Theory
The Morse Inequalities
Morse Functions
The Contribution from the Circle Points
Notes
Atiyah's -Index Theorem
An Algebra of Smoothing Operators
Renormalized Dimensions an the Index Theorem
Notes
References

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