** Speaker:** Ben Braun, University of Kentucky

** Title:** Geometric properties of r-stable hypersimplices

** Abstract: ** Hypersimplices are well-studied objects in combinatorics, optimization, and
representation theory. For each hypersimplex, I'll define a new family of subpolytopes, called r-stable
hypersimplices. I'll discuss how a standard unimodular triangulation of the hypersimplex restricts to a
unimodular triangulation of each r-stable hypersimplex. For the case of the second hypersimplex defined by
the two-element subsets of an n-set with n odd, I will describe a shelling of this triangulation that
sequentially shells each r-stable sub-hypersimplex. In this case, there are also interesting connections
between the Ehrhart h*-vector of the second hypersimplex and the r-stable sub-hypersimplices that I will
talk about if time permits. This is joint work with Liam Solus.