Mathematical Sciences Department Discrete Math Seminar - Bill Martin, WPI "Association Schemes, Planar Scaffolds, and Duality"

Friday, April 5th, 2024

12:00 PM - 1:00 PM

Olin Hall 109

Title: Association schemes, planar scaffolds, and duality

Abstract: There are various concepts of duality that arise in the theory of association schemes. For example, the formal duality of Bose-Mesner algebras becomes concrete in the presence of a regular action under an abelian group. In this sense, the octahedron and the complete bipartite graph $K_{3,3}$ are duals to one another. Linear programming duality also dovetails nicely with association scheme duality when working with bounds on the size of various codes and designs. But I was surprised to see evidence of a connection between duality of Bose-Mesner algebras and topological duality of circular planar graphs. I will discuss the duality exhibited in the example below, a flawed conjecture of mine, and an elegant proof of a special case due to Xiaoye Liang (former visiting PhD student), Hajime Tanaka (former postdoc at WPI), and their co-authors.