Elliptic Operators, Topology, and Asymptotic Methods

ISBN: 9780582325029 出版年：2013 页码：218 Roe, John CRC Press

Resume 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