Gilkey P.B. Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem

2nd edition. — New York: Chapman and Hall/CRC, 2018. — 531 p.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.