I may as well offer some updates since the last posts. Unfortunately, my dissertation topic has shifted a bit away working directly with the abelian defect group conjecture, and instead pivoted towards working with a class of invertible chain complexes which I’ve coined “Endotrivial complexes,” so the older blog posts are no longer as relevant for my dissertation (though they still play a role, and I expect I can apply my findings to the original question). Over the course of three papers, I’ve:

- Introduced the notion of an endotrivial copmlex (and hopefully given some reasons as to why they are interesting), and cooked up some numerical invariants which arise from them (h-marks).
- Generalized them to a relative projectivity setting, similar to what Caroline Lassueur did in her Ph.D. thesis for endotrivial modules. I thank Caroline and Nadia Mazza for their suggestion to consider this back at the ANTIC conference in Manchester.
- Completely classified endotrivial complexes, assigned biset functor structure to their corresponding groups, and used these results to answer a few other questions. I thank Ergun Yalcin, some of his work helped inspire these results. Some of the techniques used take inspiration from his work with G-Moore spaces.

There are still questions to be asked here, however, and I believe that these complexes are closely tied to the abelian defect group conjecture. Every endotrivial complex induces a splendid Rickard autoequivalence, and moreover, this induces a group action on the set of splendid Rickard equivalences between two block algebras. Correspondingly, the orthogonal unit group of the trivial source ring induces a group action on the set of $p$-permutation equivalences between two block algebras. Endotrivial complexes induce orthogonal units in the same way splendid Rickard equivalences induce $p$-permutation equivalences, and I believe that understanding the image of this Lefschetz map will allow us to understand better the differences between splendid Rickard equivalences and $p$-permutation equivalences.

In general, the Lefschetz invariant map is not surjective (this is proven in Endotrivial complexes), but surprisingly, it is surjective for $p$-groups (this is proven in On endotrivial complexes and the generalized Dade group). Perhaps the biggest question I still have is determining this image in general. There appears to be descent as well - after modding out by top-level characters, it appears that endotrivial complexes always descend to the finite field of $p$ elements. This ties into current work of mine, working on Galois descent of splendid Rickard equivalences motivated by my advisor’s work related to Galois descent of $p$-permutation equivalences. This work is motivated by a refined version of the defect group conjecture proposed by Kessar and Linckelmann, which predicts the conjecture holds over any finite field. Also relevant to the conjecture, I, with Jadyn Breland, extended our advisor’s work on Brauer paris for $p$-permutation equivalences to the splendid Rickard equivalence setting.

In non-mathematical news, I was awarded the UC Santa Cruz Dissertation Year Fellowship, completely buying out my teaching duties for the 2024-2025 academic year. This gives me much more freedom to travel to give talks and collaborate, which I am immensely excited for. Additionally, I will be in Europe for two months, June and July 2024, to attend numerous conferences and some research visits (see those tabs for details)!

Thanks for reading! sam