Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem

Content: Cover
 Half Title
 Title Page
 Copyright Page
 Preface
 Table of Contents
 1: Pseudo-differential operators
 1.0 Introduction
 1.1 Fourier transform and Sobolev spaces
 1.1.1 Convolution product
 1.1.2 Fourier transform
 1.1.3 Sobolev spaces
 1.1.4 Duality and interpolation
 1.2 Pseudo-Differential Operators on Rm
 1.2.1 Continuity properties
 1.2.2 Equivalence of symbols
 1.2.3 A wider class of symbols
 1.2.4 Adjoints and compositions
 1.2.5 Operators defined by kernels
 1.2.6 Pseudo-locality
 1.2.7 Completeness
 1.3 Pseudo-differential operators on manifolds
 1.3.1 Ellipticity 1.3.2 Change of coordinates1.3.3 Operators on manifolds
 1.3.4 Sobolev spaces on manifolds
 1.3.5 Extension to vector bundles
 1.4 Index of Fredholm Operators
 1.4.1 Compact operators
 1.4.2 Fredholm operators
 1.4.3 Compositions and adjoints
 1.4.4 Index of Fredholm operators
 1.4.5 Properties of the index
 1.4.6 Elliptic pseudo-differential operators
 1.5 Elliptic complexes
 1.5.1 Hodge decomposition theorem
 1.5.2 de Rham complex
 1.6 Spectral theory
 1.6.1 Self-adjoint compact operators
 1.6.2 Self-adjoint elliptic operators
 1.6.3 Bounding the spectrum from below
 1.6.4 Heat equation 1.6.5 Trace and kernel1.6.6 Heat equation and index theory
 1.7 The heat equation
 1.7.1 Dependence on a complex parameter
 1.7.2 Spectral theory
 1.7.3 Heat equation
 1.8 Local index formula
 1.8.1 Asymptotic expansions
 1.8.2 Index theory
 1.9 Variational formulas
 1.9.1 Generalized heat equation asymptotics
 1.9.2 Properties of the trace
 1.9.3 Conformal geometry
 1.10 Lefschetz fixed point theorems
 1.10.1 Generalized Lefschetz number
 1.10.2 Equivariant asymptotics
 1.10.3 Isolated fixed points
 1.10.4 Heat asymptotics and Lefschetz number
 1.11 Elliptic boundary value problems 1.11.1 Notational conventions1.11.2 Operators of Dirac and Laplace type
 1.11.3 Spectral theory
 1.11.4 Heat equation
 1.11.5 Index theory of operators of Dirac type
 1.11.6 Non-local boundary conditions
 1.12 The Zeta function
 1.12.1 Notational conventions
 1.12.2 Zeta function and heat equation
 1.12.3 Zeta function of powers
 1.12.4 Positive semi-definite operators
 1.12.5 Eigenvalue growth estimates
 1.13 The Eta function
 1.13.1 Eta invariant and spectral asymmetry
 2: Characteristic classes
 2.0 Introduction
 2.1 Characteristic classes of complex bundles
 2.1.1 Notational conventions 2.1.2 Chern-Weil homomorphism2.1.3 Functorial constructions
 2.1.4 Chern classes
 2.1.5 Chern character
 2.2 Characteristic classes of real bundles
 2.2.1 Generating functions
 2.2.2 Euler class
 2.2.3 Directional covariant derivative
 2.3 Complex projective space
 2.3.1 Holomorphic manifolds
 2.3.2 Fiber metrics and connections
 2.3.3 Complex projective space
 2.3.4 Characteristic classes of complex projective space
 2.3.5 Dual basis to the characteristic forms
 2.3.6 Todd class and Hirzebruch L polynomial
 2.4 Invariance theory
 2.4.1 Notational conventions
 2.4.2 Dimensional analysis