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 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 their geometry, which 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
Current Projects
- The Euler characteristic of an endotrivial complex
(w/ Nadia Mazza) We further study the image of the group of endotrivial complexes, i.e. the Picard group of the bounded homotopy category of $p$-permutation modules, in the corresponding Grothendieck group, i.e. the trivial source ring.
- Twistier cohomology/Borel-Smith conditions
(w/ Paul Balmer, Martin Gallauer) We generalize the twisted cohomology ring constructed in Balmer, Gallauer, The geometry of permutation modules that uses the classification of endotrivial complexes. This gives a construction for which the “comparison map” is injective for any finite $p$-group. Some of these techniques hopefully generalize to abstract settings under certain compatibility conditions.
We also look for ways the Borel-Smith conditions manifest themselves in the ttg of permutation modules by computing Picard groups of localizations. - The noncommutative tensor-triangular geometry of a block algebra
We determine aspects of the noncommutative ttg of categories of (permutation) block bimodules, following Nakano, Vashaw, Yakimov. These computations could have immediate implications for the abelian defect group conjecture since a splendid Rickard equivalence induces a monoidal triangulated equivalence between the corresponding bimodule categories.
- A Rickard theorem for splendid Rickard equivalences
We (attempt to) show that an equivalence of bounded homotopy categories of $p$-permutation (bi)modules implies the existence of a splendid Rickard equivalence, a $p$-permutation analogue of Rickard’s theorem.
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
To appear, Proc. Amer. Math. Soc.
- 6. The classification of endotrivial complexes
Submitted
- 5. Relatively endotrivial complexes
- 4. Brauer pairs for splendid Rickard complexes
w/ Jadyn V. Breland
Submitted - 3. Endotrivial complexes
- 2. A proof of the optimal leapfrogging conjecture
- 1. Challenging knight’s tours
(Listed in chronological order)
Theses
- The Combinatorial Polynomial Hirsch Conjecture
Harvey Mudd College Senior Theses, 109.