Elliptic Operators, Topology, and Asymptotic Methods, Second Edition
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.
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Resumé of Riemannian geometry
Connections curvature and characteristic classes
Clifford algebras and Dirac operators
The Spin groups
Analytic properties of Dirac operators
asymptotic expansion Atiyah bounded operator calculation chapter characteristic class Chern character Cl(k Cl(V Clifford algebra Clifford bundle Clifford module coefficients compact manifold compute connection coordinates critical point curvature operator defined DEFINITION denote derivative differential forms differential operators dimension Dirac complex Dirac operator direct sum eigenvalues endomorphism equal exterior filtered algebra finite formal power series Fourier geodesic geometry Getzler symbol graded heat equation heat kernel Hilbert space Hilbert-Schmidt holomorphic Ind(D index theorem inner product integral isomorphic Laplacian Lefschetz number LEMMA linear matrix metric multiplication norm oriented orthonormal frame partial differential equations polynomial Pontrjagin principal bundle PROPOSITION prove rapidly decreasing Rham complex Riemannian manifold scalar curvature self-adjoint Show smooth function smooth sections smoothing kernel smoothing operator Sobolev space solution spin representation Spin(k subspace symmetric tensor theory topology trace-class unique vector bundle vector fields vector space wave equation Weitzenbock formula zero