A tale of two worlds: parking functions & reduction algebras (Dwight Anderson Williams II, Pomona)

“A Tale of Two Cities” is a novel told in three books/parts. Here we describe three projects related both to published work and ongoing pieces:

PROJECT 1: In the world of combinatorics, parking functions are combinatorial objects arising from the situation of parking cars under a parking strategy. In this part of the talk, we will refresh the notion of classical parking functions given by the classical parking rules/strategy. We will then state an interesting correspondence between certain classical parking functions and so-called ideal states of the famous Tower of Hanoi game. This work is to appear in The American Mathematical Monthly with the following co-authors: Y. Aguillon, D. Alvarenga, P.E. Harris, S. Kotapati, J.C. Martinez Mori, C. Monroe, Z. Saylor, and C. Tieu.

PROJECT 2: In the world of algebra, we shed light on representation theory of Lie superalgebras by constructing reduction algebras. These algebras provide structures to study in their own right, and we give an example in presenting the diagonal reduction algebra of $osp(1|2)$, first described in a joint paper with Jonas T. Hartwig.

PROJECT 3: Continuing down an algebraic pathway, we summarize the general framework given by Zhelobenko to apply representation theory of reduction algebras as a method to solve equations. Fixing equations important to the study of physics has led to recent work with Jonas T. Hartwig and Erin Dolecheck, as well, Irmak Bukey.