Fri, 27th Jun 2008 (Research)
My current and planned research divides into three areas: the differential topology of loop spaces, the properties of two-vector bundles, and operations in higher K-theories. This research lies in the areas of differential topology, differential geometry, and algebric topology and it has further application to mathematical physics. In addition to the techniques of the areas already mentioned, my research has a heavy reliance on functional analysis.
I have been working on the first of the three areas since my Ph.D. whilst the latter two have been developed more recently. My research into the properties of two-vector bundles is in collaboration with Nils Baas and Tore Kro of NTNU. My study of operations in higher K--theories is in collaboration with Sarah Whitehouse of Sheffield University and is currently part of an EPSRC-funded project, GR/S76823/01.
Thu, 17th Feb 2011 (Research :: Preprints)
Joint with Sarah Whitehouse
Tall-Wraith monoids were introduced in The Hunting of the Hopf Ring to describe the algebraic structure on the set of unstable operations of a suitable generalised cohomology theory. In this paper we begin the study of Tall-Wraith monoids in an algebraic and categorical setting. We show that for V a variety of algebras, applying the free V-algebra functor to a monoid in Set produces a Tall-Wraith monoid. We also study the example of the Tall-Wraith monoid defined by the self set-maps of a finite ring, an example closely related to the original motivation for Tall-Wraith monoids.
Mon, 17th Dec 2007 (Research :: Papers)
New version: fairly comprehensively rewritten.
We compare various different definitions of the category of smooth objects. The definitions compared are due to Chen, Frölicher, Sikorski, Smith, and Souriau. The purpose of the comparison is to answer two questions: which, if any, captures the essence of smoothness? and how are the various categories related to each other? The first is a subjective question and we give a subjective answer: Frölicher spaces. For the second we give a more objective answer in terms of the existence of functors between the categories with certain nice properties, such as the existence of inverses and of adjunctions. We also make some general remarks about the role that topology does or doesn't play and on how one might go about defining tangent and cotangential structures on Frölicher spaces.