Elliptic Operators, Topology, and Asymptotic Methods

This Research Note gives an introduction to the circle of ideas surrounding the `heat equation proof' of the Atiyah-Singer index theorem. Asymptotic expansions for the solutions of partial differential equations on compact manifolds are used to obtain topological information, by means of a `supersymmetric' cancellation of eigenspaces. The analysis is worked out in the context of Dirac operators on Clifford bundles. The work includes proofs of the Hodge theorem; eigenvalue estimates; the Lefschetz theorem; the index theorem; and the Morse inequalities. Examples illustrate the general theory, and several recent results are included. This new edition has been revised to streamline some of the analysis and to give better coverage of important examples and applications. Readership: The book is aimed at researchers and graduate students with a background in differential geometry and functional analysis.