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
- (Reduced) permutation twisted cohomology for all finite groups + immersiveness (w/ J. O. Gómez) (writeup stage)
- 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)
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. The tensor-triangular geometry of some oligomorphic groups | Nate Harman | 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) |
On generative AI usage
Although I do not directly engage with generative AI in any facets of my personal life, during the 2025-2026 academic year, I experimented with using generative AI for research. Any use is clearly documented in papers that benefit from it. However, as of September 2026, I cannot in good conscience continue these experiments due to the existential threat that generative AI and AI companies pose to the practice of mathematics and society and the local and global environmental and epidemiological harm that rapid, uncontrolled data center expansion may cause.