Liam Solus: Polynomial constraints on varieties parameterized by colored directed graphs

Abstract: A basic problem in algebraic geometry is to identify useful polynomials vanishing on an algebraic variety that is parameterized by rational map. These 'useful' polynomials could be, for example, a generating set for the ideal of the variety, or a collection of polynomials that allow us to distinguish the variety from another. When the parameterization defining the variety is combinatorially rich, identifying such sets of useful polynomials becomes a combinatorial problem. In this talk, we will examine this story for a family of algebraic varieties parameterized by colored directed graphs. The key combinatorial players will be a poset and a polytope, whose combinatorial invariants translate into closed-form expressions for polynomials vanishing on the variety of a colored directed graph.