Welcome to Combinatorics and Optimization

Abstract: Billera, Jia, and Reiner (2009) introduced the quasisymmetric functions of matroids and showed that this defines a Hopf algebra homomorphism which is a valuative invariant, i.e., isomorphic matroids have the same quasisymmetric function and polytopal decompositions of matroid base polytopes define relations among the corresponding quasisymmetric functions. In this project we study an analogue in non-commuting variables, the word-quasisymmetric functions. To every matroid M we associate a word-quasisymmetric function ψ(M) and call this the chromatic word-quasisymmetric functions of a matroid. Matroids and word-quasisymmetric functions form Hopf algebras, and our map ψ between them is a homomorphism. We want to study the kernel, equivalently the image, of the map ψ from matroids to word-quasisymmetric functions, that is, we would like to understand which matroids are indistinguishable by the chromatic word-quasisymmetric functions. The map ψ is not an invariant, but we can show that it is valuative. Using Schubert matroids and nested matroids, special classes of matroids, we prove a lower bound of 2^d − d for the rank of the map ψ from matroids to the chromatic word-quasisymmetric functions in degree d and conjectured the upper bound of d! is tight. Recent AI supported work shows that our conjecture is true.

This is joint work with Raul Penaguiao.

There will be a pre-seminar presenting relevant background at beginning graduate level starting at 1:30pm in MC 5417.

Abstract: Vertex decomposable simplicial complexes and geometrically vertex decomposable ideals have a lot of nice structure: they are shellable and Cohen–Macaulay, and the ideal version is also radical and glicci. There are many ways to attach an ideal to a graph, such as edge ideals, toric ideals, binomial edge ideals and star configurations, and the goal is to exploit the graph structure to decide whether the ideal is (geometrically) vertex decomposable. I will introduce these notions from scratch, with no commutative algebra assumed, and survey recent joint work, including a computer search of all graphs on 11 vertices and a conjecture for binomial edge ideals.  

Abstract: I will give a turorial-style talk on Rota’s Basis Conjecture focusing on an old result, due independently to Huang and Rota and to Onn, who "all-but proved" Rota’s Basis Conjecture for real-representable matroids.