Current focuses
- (Noncommutative) tensor-triangular geometry
- Intersections of representation theory and tensor-triangulated geometry
- Endotrivial complexes and Picard groups
- Permutation modules
- Galois descent in modular representation theory
- Broue’s abelian defect group conjecture and local-global representation theory of finite groups
- Biset functors and applications
In preparation
- The Euler characteristic of an endotrivial complex (in preparation)
(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 (in preparation)
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 an embedding for any finite $p$-group. Some of these techniques hopefully generalize to abstract settings under certain compatibility conditions.
Publications and preprints
- 9. On functoriality and the tensor product property in noncommutative tensor-triangular geometry
Submitted
- 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
- 6. The classification of endotrivial complexes
- 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.