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. My personal feeling is that 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, culture, and society, and the local and global environmental and epidemiological harm that rapid, uncontrolled data center expansion may cause. However, I cannot pretend these tools do not exist.
Recent news about Navier–Stokes and Tristan Buckmaster’s statement on the results have made two things clear, however. First, OpenAI and Anthropic appear to be participating in mathematics as if it were a contest, constantly competing to see who can out-prove each other. These competitions are not collegial, nor do they resemble any semblance of open science. This approach, I imagine, is likely for publicity’s sake, to raise the public’s perception of how capable these tools are. I believe this strategy will only cause harm to the greater mathematics community, as Terrance Tao put it, open problems are now being mined as a non-renewable resource, leaving a scarcity of resources for mathematicians, especially early-career researchers. These AI endeavors also potentially shut down new avenues of research, as often, new breakthroughs are discovered during the research phase in exploration of ideas, even if they do not contribute to the final product of the question at hand. Moreover, we seem to be entering a situation similar to pharmaceutical or computer science research, where industry and academia compete, but where industry has more funding and proprietary tools to conduct research.
Secondly, Pandora’s box has opened, and once it has opened, it cannot be shut. As much as I would like to tell everybody to stop using generative AI, prohibition movements never work. There are obvious benefits to using these tools, both for performing research at the boundaries of mathematics and for expanding one’s toolbox as a researcher. Therefore, I can understand any mathematician’s desire to use these tools despite the ethical concerns. Moreover, hypothetically, if all mathematicians collectively decided to stop using generative AI, this would only lead to undisclosed use. After talking with numerous colleagues about potentially boycotting these tools and companies, I realize that at present, that is unfeasible, at least until the technology becomes good enough for me to self-host my own generative AI.
Rather, I am choosing to adopt a harm-reduction approach. I may use generative AI, but actively keep its use to a bare minimum, and encourage others to do the same. I enjoy the process of doing mathematics, both when things are going well and when I am stuck bashing my head against a logical wall. The practice of mathematics is as much an art as it is a science to me, and I would like for it to stay that way as long as possible. However, effective uses of these tools, such as learning a new topic when I don’t have a colleague to teach me, or checking a paper for errors, do not obstruct and may actively aid in my vision of doing mathematics.