Mathematical interests
I’m interested in questions and structures at intersections between group theory, representation theory, tensor-triangular geometry, and a smidgen of equivariant homotopy theory. My main focus involves understanding the so-called derived category of permutation modules, its relation to other categories of interest, and, in particular, its invertible objects, which I classified in my dissertation. However, I also have projects involving noncommutative tensor-triangular geometry, block theory, tensor categories, frame theory, and global Mackey functors.
Here is an informal summary of things I was thinking about in February 2026. Certain questions posed here may have or have not been answered by now.
In preparation / Ongoing projects
- Local Picard groups for permutation modules (ongoing)
- Periodicity in the stable permutation category (w/ M. Gallauer) (continuing)
- Noncommutative tensor-triangular geometry of group algebras (w/ K. Vashaw) (progressing)
- Vague projects about frames in (tensor) triangulated categories (w/ T. De Deyn, maybe G. Stevenson?) (starting)
- Permutation resolutions for oligomorphic groups
Publications and preprints
Listed in chronological order. 1 and 2 are undergraduate research.
| Title | Co-Authors | Journal | arXiv |
|---|---|---|---|
| 1. Challenging knight’s tours | Arthur T. Benjamin | Math Horizons, 25 (3), 18-21 (2018) | |
| 2. A proof of the optimal leapfrogging conjecture | Arthur T. Benjamin | Involve, 18 (1), 105–122 (2025) | 2110.08319 |
| 3. Endotrivial complexes | J. Algebra, 650, 173-218 (2024) | 2309.12138 | |
| 4. Brauer pairs for splendid Rickard equivalences | Jadyn V. Breland | J. Algebra, 691, 694-729 (2026) | 2312.10258 |
| 5. Relatively endotrivial complexes | J. Pure Appl. Algebra, 229 (2), 107867 (2025) | 2402.08042 | |
| 6. The classification of endotrivial complexes | Adv. Math. 478, 110414 (2025) | 2403.04088 | |
| 7. Galois descent of splendid Rickard equivalences between blocks of $p$-nilpotent groups | Proc. Amer. Math. Soc. 153 (5), 1893-1902 (2025) | 2405.16061 | |
| 8. On endosplit $p$-permutation resolutions and Broue’s conjecture for $p$-solvable groups | Represent. Theory 30, 151-174 (2026) | 2408.04094 | |
| 9. On functoriality and the tensor product property in noncommutative tensor-triangular geometry | Submitted | 2505.01899 | |
| 10. The Euler characteristic of an endotrivial complex | Nadia Mazza | J. Pure Appl. Algebra, 230 (10, Special Section in honor of Daniel K. Nakano’s 60th birthday), 108345 (2026) | 2508.07404 |
| 11. Permutation twisted cohomology, remixed | Submitted | 2509.00954 (updated version available upon request) | |
| 12. Re-framing the classification of ideals in noncommutative tensor-triangular geometry | Timothy De Deyn | Submitted | 2510.23767 |
| 13. A semisimple subcategory of Khovanov’s Heisenberg category | Submitted | 2512.13968 | |
| 14. The classification of integral endotrivial complexes | Juan Omar Gómez | Submitted | 2605.31128 |
| 15. The fusion-stable tom Dieck homomorphism | Submitted | 2608.12499 | |
| 16. Non-orientable representation spheres | Not submitted, under revision | 2608.18015 | |
| 17. Appendix to: Periods in equivariant and motivic contexts by Martin Gallauer | Martin Gallauer | Submitted | 2511.14325 |
| 18. Tensor-triangular geometry of Delannoy categories | Nate Harman | Draft available upon request | |
| 19. Permutation twisted cohomology, reduced and immersed | Juan Omar Gómez | Draft available upon request |
Proceedings
| Title | Proceedings |
|---|---|
| 1. Endotrivial complexes and remixed twisted cohomology | Oberwolfach Reports 23 (1), 552-556 (2026) |
| 2. Endotrivial complexes and remixed twisted cohomology | OCAMI Reports, to appear |
Theses
| Title | Advisor | Link |
|---|---|---|
| 1. The Combinatorial Polynomial Hirsch Conjecture | Mohamed Omar | Harvey Mudd College Senior Theses, 109 (2017) |
| 2. Permutation Modules and Endotrivial Complexes | Robert Boltje | Ph.D. Thesis, UC Santa Cruz (2025) |