The Index Theorem and The Heat Equation Method. (World Scientific, 2001)
This volume provides a self-contained represenation of the local version of the Atiyah-Singer index theorem. It contains proofs of the Hodge theorem, the local index theorems for the Dirac operator and some first-order geometric elliptic operators by using the heat-equation method. The proofs are up to the standard of pure mathematics. In addition, a Chern root algorithm is introduced for proving the local index theorems.