ABSTRACT
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. §1.1 and §1.2 are brief reviews of Sobolev spaces and pseudodifferential operators on Euclidean spaces. In §1.3, we transfer these notions to compact Riemannian manifolds using partitions of unity. In §1.4, we review the facts concerning Fredholm operators needed in §1.5 to prove the Hodge decomposition theorem. In §1.6, we derive the spectral theory of self-adjoint operators. In §1.7, we introduce the calculus of pseudo-differential operators depending on a complex parameter and discuss the heat equation. In §1.8, we discuss the asymptotics of the heat equation and derive a local formula for the index of an elliptic partial differential operator using heat equation methods. In §1.9, we study various variational formulas and generalize the heat equation asymptotics of §1.8. In §1.10, we discuss equivariant heat equation asymptotics and find a local formula for the Lefschetz number. In §1.11, we discuss elliptic boundary value problems for partial differential operators with partial differential boundary conditions and find a local formula for the index on a manifold with boundary. In §1.12, we discuss the zeta function and in §1.13, we discuss the eta function.
