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Tables of Contents for Elliptic Operators, Topology and Asymptotic Methods
Chapter/Section Title
Page #
Page Count
Resume of Riemannian geometry
9
14
Connections
9
3
Riemannian geometry
12
5
Differential forms
17
4
Exercises
21
2
Connections, curvature, and characteristic classes
23
18
Principal bundles and their connections
23
6
Characteristic classes
29
5
Genera
34
3
Notes
37
1
Exercises
37
4
Clifford algebras and Dirac operators
41
14
Clifford bundles and Dirac operators
41
5
Clifford bundles and curvature
46
3
Examples of Clifford bundles
49
3
Notes
52
1
Exercises
53
2
The Spin groups
55
16
The Clifford algebra as a superalgebra
55
2
Groups of invertibles in the Clifford algebra
57
2
Representation theory of the Clifford algebra
59
3
Spin structures on manifolds
62
2
Spin bundles and characteristic classes
64
3
The complex Spin group
67
1
Notes
68
1
Exercises
68
3
Analytic properties of Dirac operators
71
16
Sobolev Spaces
71
4
Analysis of the Dirac operator
75
7
The functional calculus
82
2
Notes
84
1
Exercises
84
3
Hodge theory
87
8
Notes
91
1
Exercises
92
3
The heat and wave equations
95
14
Existence and uniqueness theorems
95
The asymptotic expansion for the heat kernel
88
16
Finite propagation speed for the wave equation
104
3
Notes
107
1
Exercises
108
1
Traces and eigenvalue asymptotics
109
10
Eigenvalue growth
109
1
Trace-class operators
110
4
Weyl's asymptotic formula
114
3
Notes
117
1
Exercises
117
2
Some non-compact manifolds
119
14
The harmonic oscillator
119
6
Witten's perturbation of the de Rham Complex
125
2
Functional Calculus on open manifolds
127
3
Notes
130
1
Exercises
130
3
The Lefschetz formula
133
8
Lefschetz numbers
133
3
The fixed-point contributions
136
4
Notes
140
1
Exercises
140
1
The index problem
141
10
Gradings and Clifford bundles
141
2
Graded Dirac operators
143
4
The heat equation and the index theorem
147
1
Notes
148
1
Exercises
149
2
The Getzler calculus and the local index theorem
151
18
Filtered algebras and symbols
151
3
Getzler symbols
154
3
The Getzler symbol of the heat kernel
157
5
The exact solution
162
2
The index theorem
164
1
Notes
165
1
Exercises
166
3
Applications of the index theorem
169
14
The spinor Dirac operator
169
3
The signature theorem
172
3
The Hirzebruch-Riemann-Roch theorem
175
2
Local index theory
177
2
Notes
179
1
Exercises
180
3
Witten's approach to Morse theory
183
10
The Morse inequalities
183
2
Morse functions
185
4
The contribution from the critical points
189
3
Notes
192
1
Atiyah's Λ-index theorem
193
10
An algebra of smoothing operators
193
5
Renormalized dimensions and the index theorem
198
3
Notes
201
2
References
203