Abstract: The chromatic symmetric function X(G) of a graph G, introduced by Stanley in 1995, is an algebraic invariant that generalizes the chromatic polynomial of G. I'll explain all this, plus what a symmetric function is.
Abstract: This will be a continuation of last week's talk. I'll explain how to use X(G) to recover all sorts of other information about G. In particular, I'll discuss recent work of Matthew Morin, Jennifer Wagner and myself on the problem of whether a tree is determined up to isomorphism by its chromatic symmetric function.
Abstract: A reflexive polytope is a lattice polytope P such that the dual of P is also a lattice polytope. There are several other (equivalent) definitions of reflexive polytopes arising from Ehrhart theory and toric geometry. We will first survey some general properties of these objects and then discuss some recent results regarding their Ehrhart h* vectors. More details can be found here.
Abstract: We define the vector space of immanants and study the transition matrix between two bases of this vector space, one defined by Rhoades and Skandera via Kazhdan-Lusztig polynomials and one defined by Desarmenien, Kung, and Rota. We prove that this matrix can be taken to be unitriangular. Then, we generalize to the full polynomial ring C[x11, . . . , xnn] and the dual canonical basis. The proofs of both of these results rely heavily on a partial order on Sn defined using the dominance order. We define a ’P-filtration’ of the vector space of immanants and determine where irreducible character immanants fit in this structure. This is joint work with Mark Skandera at Lehigh University.
Abstract: There are several different ways of extending the definition of a tree from graphs to simplicial complexes. My favorite, due to Kalai, is both topological (the definition relies on simplicial homology) and enumerative (you can "count" them using a souped-up version of the Matrix-Tree Theorem). I'll explain the definition, give examples, and perhaps discuss ongoing work of Duval, Klivans and myself on counting simplicial spanning trees of shifted complexes. The talk will also serve as background for Molly Maxwell's talk next week.
Abstract: Simplicial trees are subcomplexes of simplicial complexes that provide a higher dimensional analogue of spanning trees in graphs. Kalai (1983) proved a version of Cayley's Theorem and the Cayley-Prufer Theorem for these higher dimensional trees.
A tree is said to be self-dual if it equals its Alexander dual (or blocker). When k is odd, an enumerator for the k-dimensional trees is the square of an enumerator for the self-dual trees. This result generalizes a graph theory result of Tutte to higher dimensions and partially solves a problem posed by Kalai.
We discuss a matroid framework that explains the common enumerative thread in these two results.
Abstract: In this talk, we give a new generalization of parking functions called G-multiparking functions. We show how this generalization is in bijection to rooted spanning forests, and how a generalized graph-searching algorithm can generate a family of such bijections. This algorithm also yields a categorization of the edges of the underlying graph. Time permitting, we will demonstrate how this categorization provides a new presentation of the Tutte polynomial. This is joint work with Catherine Yan.
Last updated Wed 8/23/06 10:00 PM