# Graduate Students

**Title**- Ph.D. Candidate

**Division**Physical & Biological Sciences Division**Department**- Mathematics Department

**Email****Website****Office Location**- McHenry Library, n/a

**Mail Stop**McHenry Library**Faculty Areas of Expertise**Mathematics

### Research Interests

My primary field of study is the modular representation theory of finite groups. In particular, I study questions regarding invertible objects in some suitable category of representation-theoretic objects. Many of these objects are connected to a natural class of representations known as permutation representations.

For my dissertation, I completely classified the invertible objects of a certain representation category, the bounded homotopy category of p-permutation kG-modules. These objects, which I call "endotrivial complexes," are governed by a class of functions known as "Borel-Smith functions" and which show up in unexpected places. This suggests a purely geometric description of these functions which is not yet understood. Endotrivial complexes are also connected to one of the biggest unsolved conjectures in my field, Broue's abelian defect group conjecture.

### Selected Publications

*Galois descent of splendid Rickard equivalences between blocks of* p*-nilpotent groups, submitted May 2024*

*On endotrivial complexes and the generalized Dade group, submitted March 2024*

*Relatively endotrivial complexes, submitted February 2024*

*Brauer pairs for splendid Rickard complexes (with J. V. Breland), submitted December 2023*

Endotrivial complexes, Journal of Algebra, 650, 173-218 (2024)

*A proof of the optimal leapfrogging conjecture (with A. T. Benjamin), *Involve (to appear)

*Challenging Knight’s Tours (with A. T. Benjamin),* Math Horizons 25:3, 18-21 (2018)

See my website for more up-to-date information.