From the basic definitions, differential topology studies the global properties of smooth manifolds, while differential geometry studies both local properties (curvature) and global properties (geodesies). This text studies how differential operators on a smooth manifold reveal deep relationships between the geometry and the topology of the manifold. This is a broad and active area of research, and has been treated in advanced research monographs such as [5], [30], [59]. This book in contrast is aimed at students knowing just the basics of smooth manifold theory, say through Stokes' theorem for differential forms. In particular, no knowledge of differential geometry is assumed.

The goal of the text is an introduction to central topics in analysis on manifolds through the study of Laplacian-type operators on manifolds. The main subjects covered are Hodge theory, heat operators for Laplacians on forms, and the Chern-Gauss-Bonnet theorem in detail. Atiyah-Singer index theory and zeta functions for Laplacians are also covered, although in less detail. The main technique used is the heat flow associated to a Laplacian. The text can be taught in a one year course, and by the conclusion the student should have an appreciation of current research interests in the field.

We now give a brief, quasi-historical overview of these topics, followed by an outline of the book's organization.

The only natural differential operator on a manifold is the exterior derivative d taking κ-forms to (κ + 1)-forms. This operator is defined purely in terms of the smooth structure.