My current research is on the motivic homotopy theory of Stiefel varieties GL(n)/GL(n-r) with applications to algebraic vector bundles. The main project of my PhD thesis is to construct sections of a canonical map between Stiefel varieties GL(n)/GL(n-r-k) -> GL(n)/GL(n-r), or show that no such section exists, using methods from motivic homotopy theory. The existence (or nonexistence) of a section is related to splitting free summands from stably free modules.
A related project of mine concerns stable splittings of the Stiefel varieties in motivic homotopy theory, analogous to this paper of Haynes Miller.
I am also interested in classifying spaces in motivic homotopy theory and their approximations.
Motivic homotopy groups of spheres and free summands of stably free modules. With Ben Williams. Preprint (submitted). 2025.
The nonexistence of sections of Stiefel varieties and stably free modules. Preprint (submitted). 2025.
Free summands of stably free modules. With Ben Williams. In Forum of Mathematics, Sigma. A short video presentation about this paper is available here. 2025.
A geometric splitting of the motive of GLn. Preprint. 2024.
Spaces of generators for the 2x2 complex matrix algebra. With Ben Williams. In New York Journal of Mathematics. This is a concise version of my master's thesis below. 2024.
My master's thesis focused on homotopical approximations to the classifying space of PGL(n, C) (in the style of this paper) and how these approximations relate to bounding the minimal number of generators of an Azumaya algebra.
Classifying spaces for topological Azumaya algebras. MSc Thesis. 2020.