Elliptic Operators, Topology, and Asymptotic Methods

ISBN-13: 9780582325029 / Angielski / Miękka / 1999 / 209 str.

J. Roe; John Roe

Opis

Ten years after publication of the popular first edition of this volume, the index theorem continues to stand as a central result of modern mathematics-one of the most important foci for the interaction of topology, geometry, and analysis. Retaining its concise presentation but offering streamlined analyses and expanded coverage of important examples and applications, Elliptic Operators, Topology, and Asymptotic Methods, Second Edition introduces the ideas surrounding the heat equation proof of the Atiyah-Singer index theorem.
The author builds towards proof of the Lefschetz formula and the full index theorem with four chapters of geometry, five chapters of analysis, and four chapters of topology. The topics addressed include Hodge theory, Weyl's theorem on the distribution of the eigenvalues of the Laplacian, the asymptotic expansion for the heat kernel, and the index theorem for Dirac-type operators using Getzler's direct method. As a "dessert," the final two chapters offer discussion of Witten's analytic approach to the Morse inequalities and the L2-index theorem of Atiyah for Galois coverings.
The text assumes some background in differential geometry and functional analysis. With the partial differential equation theory developed within the text and the exercises in each chapter, Elliptic Operators, Topology, and Asymptotic Methods becomes the ideal vehicle for self-study or coursework. Mathematicians, researchers, and physicists working with index theory or supersymmetry will find it a concise but wide-ranging introduction to this important and intriguing field.

Spis treści

Resumé of Riemannian GeometryConnectionsRiemannian GeometryDifferential FormsExercisesConnection, Curvature, and Characteristic ClassesPrincipal Bundles and their ConnectionsCharacteristic ClassesGeneraNotesExercisesClifford Algebras and Dirac OperatorsClifford Bundles and Dirac OperatorsClifford Bundles and CurvatureExamples of Clifford BundlesNotesExercisesThe Spin GroupsThe Clifford Algebra as a SuperalgebraGroups of Invertibles in the Clifford AlgebraRepresentation Theory of the Clifford AlgebraSpin Structures on ManifoldsSpin Bundles and Characteristic ClassesThe Complex Spin GroupNotesExercisesAnalytic Properties of Dirac OperatorsSobolev SpacesAnalysis of the Dirac OperatorThe Functional CalculusNotesExercisesHodge TheoryNotesExercisesThe Heat and Wave EquationsExistence and Uniqueness TheoremsThe Asymptotic Expansion for the Heat KernelFinite Propagation Speed for the Wave EquationNotesExercisesTraces and Eigenvalue AsymptoticsEigenvalue GrowthTrace-Class OperatorsWeyl's Asymptotic FormulaNotesExercisesSome Non-Compact ManifoldsThe Harmonic OscillatorWitten's Perturbation of the de Rham ComplexFunctional Calculus on Open ManifoldsNotesExercisesThe Lefschetz FormulaLefschetz NumbersThe Fixed-Point ContributionsNotesExercisesThe Index ProblemGradings and Clifford BundlesGraded Dirac OperatorsThe Heat Equations and the Index TheoremNotesExercisesThe Getzler Calculus and the Local Index TheoremFiltered Algebras and SymbolsGetzler SymbolsThe Getzler Symbol of the Heat KernelThe Exact SolutionThe Index TheoremNotesExercisesApplications of the Index TheoremThe Spinor Dirac OperatorThe Signature TheoremThe Hirzebruch-Riemann-Roch TheoremLocal Index TheoryNotesExercisesWitten's Approach to Morse TheoryThe Morse InequalitiesMorse FunctionsThe Contribution from the Circle PointsNotesAtiyah's -Index TheoremAn Algebra of Smoothing OperatorsRenormalized Dimensions an the Index TheoremNotesReferences

O autorze

John Roe