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.

chapter 1|119 pages

Pseudo-differential operators

chapter 2|94 pages

Characteristic classes

chapter 3|111 pages

The index theorem

chapter 4|92 pages

Spectral geometry

chapter 5|90 pages

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