Colloquium

Date: May 1, 2023

Time: 4:00PM - 5:00PM

Location: Bloc 117

Speaker: Galen Dorpalen-Barry

Description: Title: Cones of Hyperplane Arrangements

Abstract: Hyperplane arrangements dissect R^n into connected components called regions. A well-known theorem of Zaslavsky counts regions as a sum of nonnegative integers called Whitney numbers of the first kind. A generalization of this theorem counts regions within any cone defined as the intersection of a collection of halfspaces from the arrangement, leading to a notion of Whitney numbers for each cone. This talk concerns Whitney numbers for arrangements coming from reflection groups (the braid arrangement and Shi arrangements). In order to describe these Whitney numbers, we define the Varchenko-Gel’fand ring of a cone of an arbitrary arrangement and use techniques inspired by Gröbner bases to obtain a general presentation for this ring.