My current research is on the motivic homotopy theory of Stiefel varieties. Haynes Miller proved that there is a filtration on the Stiefel manifolds over R, C, and the quaternions which splits in the stable homotopy category. It is conjectured that there is an analogous splitting for the Stiefel varieties GL(n)/GL(n-r) over a field in the motivic stable homotopy category of Morel and Voevodsky. I am also interested in constructing sections of certain maps between Stiefel varieties, and how such sections are related to splitting trivial summands from stably trivial algebraic vector bundles.
My master's thesis was on homotopical approximations to the classifying space of PGL(n,C) (in the style of Burt Totaro's work) and how these approximations relate to bounding the number of generators of an Azumaya algebra.
Free summands of stably free modules. With Ben Williams. To appear in Forum of Math: Sigma. 2025.
A geometric splitting of the motive of GLn. Preprint (submitted). 2024.
Spaces of generators for the 2x2 complex matrix algebra. With Ben Williams. In New York Journal of Mathematics. 2024.
Classifying spaces for topological Azumaya algebras. MSc Thesis. 2020.