The main project of my PhD thesis was to construct sections of a canonical map between Stiefel varieties, or show that no 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 algebraic analogue to the vector fields on spheres problem.
Some other things I am interested in:
Motives and motivic cohomology
Motivic cell structures and stable splittings of linear algebraic groups (and their Stiefel varieties)
Classifying spaces and their homotopical approximations
Algebraic vector bundles and Jouanolou devices
The linear spectral sequence
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: