2nd Edition

Elliptic Operators, Topology, and Asymptotic Methods

By John Roe Copyright 1999
    218 Pages
    by Chapman & Hall

    218 Pages
    by Chapman & Hall

    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.

    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

    Biography

    John Roe