Andrew Stacey :: 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.

A more detailed research plan is available in PDF or PostScript.

We consider the problem of defining the structure of a smooth manifold on the various spaces of piecewise-smooth loops in a smooth finite dimensional manifold. We succeed for a particular type of piecewise-smooth loops.

We also examine the action of the diffeomorphism group of the circle. It is not a useful action on the manifold that we define. We consider how one might fix this problem and conclude that it can only be done by completing to the space of loops of bounded variation.

In this paper we define and examine the truncated Witten genus. It is defined as the equivariant index of the Dirac operator on the manifold Map(C_p,M) with its natural C_p-action. Here, Map(C_p,M) is the space of maps from the cyclic group of order p into a closed, connected, spin manifold, M. By applying the Atiyah-Singer index theorem we give a topological formula for the truncated Witten genus which is related to the formula for the Witten genus by truncation of the infinite products. We also show that the equivariant index of the Dirac operator on the projective space ℙMap(C_p,ℂⁿ⁺¹) is closely related to the truncated Witten genus of ℂℙⁿ. The spaces ℙMap(C_p,ℂⁿ⁺¹) define a filtration of the space ℙMap(S¹,ℂⁿ⁺¹) which has been used to study equivariant objects on the smooth loop space of ℂℙⁿ.

Andrew Stacey

About Andrew Stacey Information about my research, teaching, and other interests. By: Andrew Stacey Contact details $Andrew Stacey$ Root Menu Fjord (1) Research (11) Teaching (10) Seminars (8) Professional (3) HowDidIDoThat (21) Links (15) Current Menu Preprints (8) Papers (3) $blosxom icon$	home :: 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. A more detailed research plan is available in PDF or PostScript. Wed, 5th Mar 2008 (Research :: Preprints) The Smooth Structure of the Space of Piecewise-Smooth Loops (PDF or PS) We consider the problem of defining the structure of a smooth manifold on the various spaces of piecewise-smooth loops in a smooth finite dimensional manifold. We succeed for a particular type of piecewise-smooth loops. We also examine the action of the diffeomorphism group of the circle. It is not a useful action on the manifold that we define. We consider how one might fix this problem and conclude that it can only be done by completing to the space of loops of bounded variation. See more ... [Full link] Last modified on Wed, 5th Mar 2008 Mon, 5th Apr 2004 (Research :: Papers) The Truncated Witten Genus (PDF, PS) In this paper we define and examine the truncated Witten genus. It is defined as the equivariant index of the Dirac operator on the manifold Map(C_p,M) with its natural C_p-action. Here, Map(C_p,M) is the space of maps from the cyclic group of order p into a closed, connected, spin manifold, M. By applying the Atiyah-Singer index theorem we give a topological formula for the truncated Witten genus which is related to the formula for the Witten genus by truncation of the infinite products. We also show that the equivariant index of the Dirac operator on the projective space ℙMap(C_p,ℂⁿ⁺¹) is closely related to the truncated Witten genus of ℂℙⁿ. The spaces ℙMap(C_p,ℂⁿ⁺¹) define a filtration of the space ℙMap(S¹,ℂⁿ⁺¹) which has been used to study equivariant objects on the smooth loop space of ℂℙⁿ. See more ... [Full link] Last modified on Mon, 5th Apr 2004