Colin Defant (MIT)

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

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 called the pure shard monoid, which we believe deserves further attention. 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; in this case, our three maps are related by Snap=Crackle Pop. This talk is based on joint work with Nathan Williams.

Location: 201, avenue du Président-Kennedy, PK-4323, UQAM, Montréal