I am primarily interested in the modular representation theory of finite groups. This means that I study vector spaces (over finite fields, such as the field $\mathbb{Z}/p\mathbb{Z}$) with some sort of additional symmetry described by a finite group. In this setting, there is an underlying “local-to-global” philosophy, which states that the representation theory of a finite group $G$ should be related to the representation theory of its “ $p$ -local” subgroups. I seek to discover and understand these connections on a structural level by working with categories and structures that appear to govern this philosophy.
Current focuses
- Local-to-global conjectures such as Broue’s abelian defect group conjecture
- Invertible objects such as endotrivial modules and complexes
- Intersections of modular representation theory and tensor-triangulated geometry
- Biset functors (i.e. global Mackey functors) and applications
- Galois descent in representation theory
- Categorification
In Preparation
- The Euler characteristic of an endotrivial complex
(with Nadia Mazza)
Publications and Preprints
- 8. On endosplit $p$-permutation resolutions and Broue’s conjecture for $p$-solvable groups
Submitted
- 7. Galois descent of splendid Rickard equivalences between blocks of $p$-nilpotent groups
Forthcoming, Proc. Amer. Math. Soc.
- 6. The classification of endotrivial complexes
Submitted
- 5. Relatively endotrivial complexes
Forthcoming, J. Pure Appl. Algebra
- 4. Brauer pairs for splendid Rickard complexes
(with Jadyn V. Breland)
Submitted - 3. Endotrivial complexes
- 2. A proof of the optimal leapfrogging conjecture
- 1. Challenging knight’s tours
(with Arthur T. Benjamin)
Math Horizons, 25(3), 18-21 (2018)
(Listed in chronological order)
Theses
- The Combinatorial Polynomial Hirsch Conjecture
Harvey Mudd College Senior Theses, 109.