Preface 1. Partial differential operators 1.1 Characteristic classes 1.1.1 Cohomology groups 1.1.2 Vector bundles 1.1.3 Chern-Weil theory 1.1.4 The Chern classes 1.1.5 The Todd, A, and Hirzebruch L polynomials 1.1.6 The Stiefel-Whitney classes 1.1.7 K theory and the Chern character 1.2 Clifford algebras and spinors 1.2.1 Clifford algebras 1.2.2 Principal G bundles 1.2.3 Spin structures on vector bundles 1.2.4 The spin bundles 1.2.5 The spin connection 1.2.6 The Dirac operator and the Lichnerowicz formula 1.3 Spectral theory of self-adjoint elliptic operators 1.3.1 Basic notational conventions 1.3.2 The Fourier transform and Sobolev spaces 1.3.3 Partial differential operators 1.3.4 Pseudo-differential operators on Euclidean space 1.3.5 Pseudo-differential operators on compact manifolds 1.3.6 Elliptic operators 1.3.7 The parametrix 1.3.8 Spectral theory 1.3.9 The Hodge decomposition theorem 1.3.10 The Hodge-de Rham theorem 1.4 The heat equation 1.4.1 Smooth kernels 1.4.2 Asymptotics of the heat equation 1.4.3 A local formula for the index of an elliptic operator 1.5 The Atiyah-Singer Index Theorem 1.5.1 The de Rham complex 1.5.2 The Hirzebrueh Signature Theorem 1.5.3 The spin complex 1.5.4 The spin-e complex 1.5.5 The Dolbeault complex 1.5.6 Tile geometrical index theorem 1.5.7 Product formulas 1.5.8 The general Atiyah-Singer index theorem 1.5.9 The Bott bundle 1.6 The eta invariant and zeta functions 1.6.1 Local invariants 1.6.2 The Mellin transform 1.6.3 The poles of the eta function 1.6.4 The normalized eta residue 1.7 Regularity of the eta function at the origin 1.7.1 A result in invariance theory 1.7.2 Analytical facts concerning the eta residue