Elliptic operators, topology and asymptotic methods /
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| Format: | eBook |
| Language: | English |
| Published: |
Boca Raton, FL :
CRC Press, Taylor & Francis Group,
[2017]
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| Edition: | Second edition. |
| Series: | Chapman & Hall/CRC research notes in mathematics series ;
395. |
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| Online Access: | Connect to the full text of this electronic book |
Table of Contents:
- Cover; Half Title; Title Page; Copyright Page; Dedication; Contents; Introduction; CHAPTER 1 Resumé of Riemannian geometry; Connections; Riemannian geometry; Differential forms; Exercises; CHAPTER 2 Connections, curvature, and characteristic classes; Principal bundles and their connections; Characteristic classes; Genera; Notes; Exercises; CHAPTER 3 Clifford algebras and Dirac operators; Clifford bundles and Dirac operators; Clifford bundles and curvature; Examples of Clifford bundles; Notes; Exercises; CHAPTER 4 The Spin groups; The Clifford algebra as a superalgebra.
- Groups of invertibles in the Clifford algebraRepresentation theory of the Clifford algebra; Spin structures on manifolds; Spin bundles and characteristic classes; The complex Spin group; Notes; Exercises; CHAPTER 5 Analytic properties o f Dirac operators; Sobolev Spaces; Analysis of the Dirac operator; The functional calculus; Notes; Exercises; CHAPTER 6 Hodge theory; Notes; Exercises; CHAPTER 7 The heat and wave equations; Existence and uniqueness theorem s; The asymptotic expansion for the heat kernel; Finite propagation speed for the wave equation; Notes; Exercises.
- CHAPTER 8 Traces and eigenvalue asymptoticsEigenvalue growth; Trace-class operators; Weyl's asymptotic formula; Notes; Exercises; CHAPTER 9 Some non-compact manifolds; The harmonic oscillator; Witten's perturbation of the de Rham complex; Functional calculus on open manifolds; Notes; Exercises; CHAPTER 10 The Lefschetz formula; Lefschetz numbers; The fixed-point contributions; Notes; Exercises; CHAPTER 11 The index problem; Gradings and Clifford bundles; Graded Dirac operators; The heat equation and the index theorem; Notes; Exercises.
- Renormalized dimensions and the index theoremNotes; References; Index.