Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem (1984)

This book treats the Atiyah-Singer index theorem using the heat equation, which gives a local formula for the index of any elliptic complex. Heat equation methods are also used to discuss Lefschetz fixed point formulas, the Gauss-Bonnet theorem for a manifold with smooth boundary, and the geometrical theorem for a manifold with smooth boundary. The author uses invariance theory to identify the integrand of the index theorem for classical elliptic complexes with the invariants of the heat equation. In chapter one, we develop the analysis needed to define the index of an elliptic operator and to compute the index using heat equation methods.