About
Andrew Stacey
Information about my research, teaching, and other interests.
By: Andrew Stacey
Contact details
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.
A more detailed research plan is available in PDF or PostScript.
Mon, 26 Nov 2007 (Research :: Preprints)
The Hunting of the Hopf Ring (PDF or PS)
Joint with Sarah Whitehouse
We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E*. Our description is as a graded and completed version of a Tall-Wraith monoid. The E*-cohomology of a space X is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another.Constructing Smooth Loop Spaces (PDF, PS)
We consider the general problem of constructing the structure of a smooth manifold on a given space of loops in a smooth finite dimensional manifold. By generalising the standard construction for smooth loops, we derive a list of conditions for the model space which, if satisfied, mean that a smooth structure exists.
We also show how various desired properties can be derived from the model space; for example, topological properties such as paracompactness. We pay particular attention to the fact that the loop spaces that can be defined in this way are all homotopy equivalent; and also to the action of the circle by rigid rotations.
Stable and Unstable Operations in mod p Cohomology Theories (PDF, PS)
Joint with Sarah Whitehouse
We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. In the main example, where the target theory is one of the Morava K-theories, this provides a simple and explicit description of a splitting arising from the Bousfield-Kuhn functor.
Operations in the First Morava K--Theory (PDF, PS)
We describe the families of degree zero operations in the first Morava K-theory, and consequently in mod p K-theory, in terms of Adams' operations.
Determinants of Matrices from Pascal's Triangle (PDF, PS)
When devising problems for a linear algebra course it is desirable to have an extensive source of integral matrices of determinant one. In this short paper we give a straightforward method of generating some such matrices from binomial coefficients.
The Geometry of the Loop Space and a Construction of a Dirac Operator (PDF, PS)
We construct a natural inner product on the cotangent spaces of the free loop space of a Riemannian manifold. As an application of, and motivation for, this construction we show how to construct a Dirac operator on the loop space of a string manifold.