The main project of my PhD thesis was to construct sections of a canonical map between Stiefel varieties, 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. This project may be seen as an analogue to the vector fields on spheres problem, solved by Adams.
Two related projects are calculating the stable homotopy groups of motivic spheres and establishing a motivic stable splitting of the Stiefel varieties, analogous to this paper by Haynes Miller.
Here are some other things I like to think about:
Classifying spaces in motivic homotopy theory and their approximations
Classifying algebraic vector bundles and the Jouanolou device
The linear spectral sequence and Suslin's factorial conjecture
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. To appear in Mathematische Zeitschrift. 2026.
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:
My PhD thesis: