Title: Polyhedral Problems in Combinatorial Convex Geometry

Abstract: Polyhedra play a special role in combinatorial convex geometry in the sense that they are both convex sets and combinatorial objects.  As such, a polyhedron can act as either the convex set of interest or the combinatorial object describing properties of another convex set.  We will examine two instances of polyhedra in combinatorial convex geometry, one exhibiting each of these two roles.  The first instance arises in the context of Ehrhart theory, and the polyhedra are the central objects of study.  We will examine the Ehrhart h*-polynomials of a family of lattice polytopes called the r-stable (n,k)-hypersimplices, providing some combinatorial interpretations of their coefficients as well as some results on unimodality of these polynomials.  The second instance arises in algebraic statistics, and the polyhedra act as a conduit through which we study a nonpolyhedral problem.  For a graph G, we study the extremal ranks of the closure of the cone of concentration matrices of G via the facet-normals of the cut polytope of G.  Along the way, we will discover that real-rooted polynomials are lurking in the background of all of these problems.