The Atiyah-Patodi-Singer Index Theorem

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

ISBN: 9781568810027 | 1568810024
Cover: Hardcover
Copyright: 3/31/1993

Rent

(Recommended)

$95.41

Term

Due

Price

Semester

Dec 12

$100.43

Quarter

Aug 13

$95.41

Short Term

Jul 28

$90.38

*This item is part of an exclusive publisher rental program and requires an additional convenience fee. This fee will be reflected in the shopping bag.
Buy New

Usually Ships in 3-5 Business Days

$128.57
eBook

Available Instantly

Online: 180 Days

Downloadable: 180 Days

*To support the delivery of the digital material to you, a digital delivery fee of $3.99 will be charged on each digital item.

$46.20*

Based on the lecture notes of a graduate course given at MIT, this sophisticated treatment leads to a variety of current research topics and will undoubtedly serve as a guide to further studies. Introduction and the Proof Ordinary Differential Equations Exact b-Geometry Spin Structures Small b-Calculus Full Calculus Relative Index, Cohomology and Resolvent Heat Calculus Local Index Theorem Proof Revisited and Applications.

Preface
List of Figures
Introduction and the proof	p. 1
The Atiyah-Singer index theorem	p. 2
The Atiyah-Patodi-Singer index theorem	p. 2
Boundary conditions versus b-geometry	p. 4
Preliminaries to the proof	p. 5
The proof	p. 9
Weighting	p. 10
Outline	p. 12
Ordinary differential operators	p. 15
Operators and coordinates	p. 15
Index	p. 17
General statement	p. 21
Kernels	p. 22
Exact b-geometry	p. 25
Manifolds	p. 25
The b-tangent bundle	p. 27
Exact b-metrics	p. 30
Differential operators	p. 32
Levi-Civita Connection	p. 35
Riemann curvature tensor	p. 39
Associated bundles	p. 43
Covariant differentiation	p. 44
Christoffel symbols	p. 47
Warped products	p. 50
Curvature formulae	p. 51
Orientation	p. 53
b-connections	p. 55
Characteristic classes	p. 60
Hermitian bundles	p. 63
de Rham cohomology	p. 65
b-characteristic classes	p. 66
Spin structures	p. 67
Euclidean Dirac operator	p. 67
Clifford algebra	p. 69
Periodicity	p. 71
Clifford bundle	p. 75
Clifford modules	p. 76
Clifford bundle of [superscript b]TX	p. 79
Spin group	p. 80
Spin representations	p. 84
Spin structures	p. 86
Clifford connections	p. 88
Twisted Dirac operators	p. 90
Spin structure for a b-metric	p. 90
Boundary behaviour	p. 91
Dirac operators of warped products	p. 93
Small b-calculus	p. 95
Inward-pointing spherical normal bundle	p. 95
The b-stretched product	p. 98
Submanifolds of [actual symbol not reproducible]	p. 99
Lifting vector fields	p. 101
Densities	p. 105
The space of pseudodifferential operators	p. 108
Distributions	p. 110
Kernels of b-differential operators	p. 113
The small space of b-pseudodifferential operators	p. 114
Symbol map	p. 115
Elementary mapping properties	p. 117
Asymptotic completeness	p. 121
Small parametrix	p. 122
Non-compactness	p. 124
Indicial operator	p. 125
General coefficients	p. 130
Examples	p. 132
Trace class operators	p. 134
The b-integral	p. 140
The b-trace functional	p. 142
Full calculus	p. 144
Mellin transform	p. 144
Inversion of the indicial family	p. 146
Analytic Fredholm theory	p. 150
Conjugation by powers	p. 152
Commutator identity for the b-trace	p. 153
Invertibility of the indicial operator	p. 156
Kernel of the inverse of the indicial operator	p. 160
Index formula for invariant operators	p. 161
Composition in the small calculus	p. 164
Polyhomogeneous conormal distributions	p. 166
Mellin transform and polyhomogeneity	p. 171
Boundary terms	p. 173
True parametrix	p. 174
Finitely residual terms	p. 176
Boundedness on Sobolev spaces	p. 178
Calculus with bounds	p. 180
Fredholm properties	p. 183
Extended index sets	p. 185
Formal solutions	p. 187
Finer parametrix	p. 189
Composition with boundary terms	p. 190
Residual terms	p. 196
Composition in general	p. 199
General bundles and summary	p. 201
Parametrices and null space	p. 204
Generalized inverse	p. 206
Relative index, cohomology and resolvent	p. 208
Boundary pairing	p. 208
Relative index formula	p. 213
Riemann-Roch for surfaces	p. 216
Hodge theory	p. 220
Extended L[superscript 2] null space of Dirac operators	p. 233
Resolvent family	p. 234
Analytic continuation of the resolvent	p. 238
Poles of the resolvent	p. 242
Spectral theory	p. 246
Scattering matrices	p. 248
Heat calculus	p. 252
Heat space	p. 252
Standard heat calculus	p. 261
Heat kernel	p. 263
b-heat space	p. 273
b-heat calculus	p. 276
Bundle coefficients	p. 281
Long-time behaviour, Fredholm case	p. 283
Long-time behaviour, non-Fredholm case	p. 291
Local index theorem	p. 295
Simple rescaling	p. 295
Rescaling bundles	p. 297
Rescaling and connections	p. 301
Getzler's rescaling	p. 305
Rescaled trace	p. 309
Rescaled heat calculus	p. 312
Rescaled normal operator	p. 313
Lichnerowicz' formula	p. 317
Mehler's formula	p. 322
Local index formula	p. 326
The b case	p. 328
Graded Hermitian Clifford modules	p. 329
The eta integrand	p. 332
The modified eta invariant	p. 334
Variation of eta	p. 336
Spectral flow	p. 338
The circle	p. 341
Proof revisited and applications	p. 343
The APS theorem	p. 343
Euler characteristic	p. 347
Signature formula	p. 351
Flat coefficient bundles	p. 356
Zeta function	p. 357
Analytic torsion	p. 362
The b-eta invariant	p. 362
Bibliography	p. 364
Index of Notations	p. 368
Author Index	p. 371
Index	p. 372
Table of Contents provided by Blackwell. All Rights Reserved.

Please wait while the item is added to your bag...

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

The Atiyah-Patodi-Singer Index Theorem

Summary

Table of Contents

Supplemental Materials

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

The Atiyah-Patodi-Singer Index Theorem

Summary

Table of Contents

Supplemental Materials

Digital License