Abstract:
The purpose of this paper is to present the construction of a canonical determinant functional on elliptic pseudodifferential operators associated to the Guillemin-Wodzicki residue trace. The resulting 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 (quasi-) determinant, which is non-local and non-multiplicative.

Abstract:
We study the zeta determinant of global boundary problems of APS-type through a general theory for relative spectral invariants. In particular, we compute the zeta determinant for Dirac-Laplacian boundary problems in terms of a scattering Fredholm determinant over the boundary.

Abstract:
For a smooth family F of admissible elliptic pseudodifferential operators with differential form coefficients associated to a geometric fibration of manifolds M--> B we show that there is a natural zeta-form z(F,s) and zeta-determinant- form det(F) in the de-Rham algebra of smooth differential forms, generalizing the classical single operator spectral zeta function and determinant. In the case where F is the curvature of a superconnection the zeta form is exact, extending to families the Atiyah-Bott-Seeley zeta function formula for the pointwise index, and equivalent to the transgression formula for the graded Chern character. The zeta-determinant form leads to the definition of the graded zeta-Chern class form. For a family of compatible Dirac operators D with index bundle Ind(D) we prove a transgression formula leading to a local density representing the Chern class c(Ind(D))in terms of the A-hat genus and twisted Chern character. Globally the meromorphically continued zeta form and zeta determinant form exist only at the level of K-theory as characteristic class maps K(B)->H*(B).

Abstract:
We show that there is a canonical construction of a zeta (Bismut-Quillen) connection on the determinant line bundle of a family of APS elliptic boundary problems and that it has curvature equal to the 2-form part of a relative eta form.

Abstract:
Logarithmic representations of the bordism category are considered as a framework for capturing a class of additive invariants defined by generalised Reidemeister torsions.

Abstract:
We compute the relative zeta-function metric on the determinant line bundle for a family of elliptic boundary value problems of Dirac-type. To do this we prove a general formula relating the zeta-determinant to a Fredholm determinant over the boundary for a class of higher-order elliptic boundary value problems.

Abstract:
We explore the convergence of Kergin interpolation polynomials of holomorphic functions in Banach spaces, which need not be of bounded type. We also investigate a case where the Kergin series diverges.

Abstract:
It is shown that the determinant line bundle associated to a family of Dirac operators over a closed partitioned manifold has a canonical Hermitian metric with compatible connection whose curvature satisfies an additivity formula with contributions from the families of Dirac operators over the two halves. This curvature form is the natural differential representative which satisifies the same splitting principle as the Chern class of the determinant line bundle.

Abstract:
Tendo Achilles ruptures are generally traumatic in origin and while bilateral tendo Achilles ruptures are a rare occurrence, most of them are associated with risk factors or pre-existing disease and generally involve trauma or sporting activities. Most of the cases of bilateral rupture are generally treated operatively. A spontaneous onset case of bilateral tendo achilles rupture is reported in a healthy man and its conservative (non operative) management discussed with a review of the literature.