Speaker:  Colin Defant (MIT)

Title:  Pop, Crackle, Snap (and Pow): Some Facets of Sharts

Abstract:

Reading cut the hyperplanes in a real central arrangement H into pieces called shards, which reflect order-theoretic properties of the arrangement. We show that shards have a natural interpretation as certain generators of the fundamental group of the complement of the complexification of H. Taking only positive expressions in these generators yields a new poset that we call the pure shard monoid. When H is simplicial, its poset of regions is a lattice, so it comes equipped with a pop-stack sorting operator Pop. In this case, we use Pop to define an embedding Crackle of Reading's shard intersection order into the pure shard monoid. When H is the reflection arrangement of a finite Coxeter group, we also define a poset embedding Snap of the shard intersection order into the positive braid monoid. This talk is based on joint work with Nathan Williams.