Algebra and Combinatorics Seminar

Date: October 28, 2016

Time: 3:00PM - 3:50PM

Location: BLOC 628

Speaker: Jacob White, UTRGV

Title: An introduction to Coloring Problems

Abstract: Many combinatorial objects, such as graphs, posets, and matroids, can be studied through numerical polynomial invariants, such as the chromatic polynomial and the order polynomial. Moreover, these invariants often have similar properties, which can be properly understood via combinatorial Hopf algebras. In studying Hopf monoids in Joyal's category of species, I have discovered an interesting combinatorial Hopf monoid, involving a new combinatorial object, called a coloring problem. We introduce the notion of an abstract coloring problem, which generalizes various aspects of graphs, posets, and matroids. We will discuss chromatic polynomials, which count the number of solutions of a coloring problem. We show that chromatic polynomials satisfy many nice identities, including having positive h-vectors. We mention relationships between this work and poset topology. If there is time remaining, we may discuss how coloring problems form a terminal combinatorial Hopf monoid.