The purpose of this paper is to present the construction of a canonical determinant functional on elliptic pseudodifferential operators (ψdos) associated to the Guillemin-Wodzicki residue trace. The resulting residue determinant functional is multiplicative, a local invariant, and not defined by a regularization procedure. The residue determinant is consequently a quite different object to the zeta function determinant, which is non-local and non-multiplicative. Indeed, the residue determinant does not arise as the derivative of a trace on the complex power operators, and does not depend on a choice of spectral cut. The identification of a certain residue determinant with the index of an elliptic ψdo shows the residue determinant to be topologically significant. 1 1 This work arose following conversations with Steve Rosenberg concerning higher Chern-Weil invariants, my thanks to him for his support and interest. I am also indebted to Kate Okikiolu for a helpful suggestion, the essential role of [Ok1, Ok2] in the current work is evident. I am grateful to Gerd Grubb and Sylvie Paycha for numerous interesting discussions and helpful comments. 1 2 1. Definition and Properties of the Residue Determinant Let A be a ψdo of order α ∈ R acting on the space of smooth sections C ∞ (E) of a rank N vector bundle E over a compact boundaryless manifold M of dimension n. This means that in each local trivialization of E with U × R N, with U an open subset of M identified with an open set in R n, and smooth functions φ, ψ with supp(φ), supp(ψ) ⊂ U, then for x ∈ U and f ∈ C ∞ c (U, RN) (1.1) (φAψ)f(x) = 1 (2π) n

Abstract. For a holomorphic family of classical pseudodifferential operators on a closed manifold we give exact formulae for all coefficients in the Laurent expansion of its Kontsevich-Vishik canonical trace. This generalizes to all higher order terms a known result identifying the residue trace with a pole of the canonical trace.

Abstract. We prove a local index formula for cusp-pseudodifferential operators on a manifold with boundary. This is known to be equivalent to an index formula for manifolds with cylindrical ends, and hence we obtain a new proof of the classical Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary, as well as an extension of Melrose’s b-index theorem. Our approach is based on an unpublished paper by Melrose and Nistor “Homology of pseudo-differential operators I. Manifolds with boundary ” [39]. We therefore take the opportunity to review some of the results from that paper from the perspective of subsequent research on the Hochschild and cyclic homologies of algebras of pseudodifferential operators and of their applications to index theory.

Abstract. For a holomorphic family of classical pseudodifferential operators on a closed manifold we give exact formulae for all coefficients in the Laurent expansion of its Kontsevich-Vishik canonical trace. This generalizes a known result identifying the Wodzicki residue with the pole at zero to all higher order terms.