My current research focus lies within modular representation theory, in particular the block theory of finite group algebras and the theory of biset functors developed by Serge Bouc. My dissertation problem involves determining when a $p$-permutation equivalence, an isomorphism of Grothendieck groups induced by virtual $p$-permutation bimodules, lifts to a splendid equivalence, an equivalence of derived categories first studied in detail by Jeremy Rickard. These equivalences are further connected to isotypies, a “bouquet” of perfect isometries first studied by Michel Broué. Broué’s celebrated Abelian Defect Group Conjecture predicts the existence of these equivalences, and though it has been verified in many cases, the most general case is still unknown.
However, my undergraduate interests were primarily combinatorics-focused. In particular, my research projects involved studying combinatorial puzzles, proving either the existence of solutions or proving bounds on optimal play. Though this is far removed from my current focuses, my hope in the future is to combine the two, studying more combinatorial aspects of representation theory. I have a growing interest in the theory of Soergel bimodules, a powerful tool in geometric representation theory with far-reaching connections.
- A. T. Benjamin, S. K. Miller (2018): Challenging Knight’s Tours, Math Horizons 25:3, 18-21.
S. K. Miller, Endotrivial Complexes. In progress (2023)
A. T. Benjamin, S. K. Miller, Optimal Leapfrogging. Submitted for publication (2023)
arXiv. Most of this work was completed 2017-2018 when I was a software engineer! It’s possible to do research even when not in an academic setting. 6 years later, the work will finally see the light of day.
- S. K. Miller (2017): The Combinatorial Polynomial Hirsch Conjecture, Harvey Mudd College Senior Theses, 109.