In non-commutative probability there are different notions of independence (free, Boolean and monotone), each with a notion of cumulants (analogue of classic cumulants) that linearize the addition of independent random variables. Formulas relating moments and cumulants can be expressed as a sum indexed by set partitions. Our goal is to construct a Hopf algebra T on non-crossing partitions NC that allows us to systematically study the transitions between distinct brands of cumulants in non-commutative probability. The Hopf algebra T is such that its character group can be identified with a group of 'semi-multiplicative' functions on the incidence algebra of NC, used to encode the formulas. While a basic tool of the Hopf Algebra, such as the antipode of T, helps in inverting such formulas. We will explain how T relates to other (more famous) Hopf algebras and explain some extensions we have worked on. This is a joint work with Celestino, Ebrahimi-Fard, Nica and Witzman.

An A-path is a non-trivial path that intersects a vertex set A exactly at its endpoints. Beginning with a classical result of Gallai from 1961, several families of A-paths have been shown to satisfy an approximate packing-covering duality known as the Erdos-Posa property. However, there is very little known about the structures of graphs where this property fails for A-paths, which is in contrast to many similar situations where one can salvage a half-integral version of the Erdos-Posa property. In this talk, we prove a structure theorem that characterizes the obstructions to the Erdos-Posa property of A-paths in group-labelled graphs. This gives a general half-integral Erdos-Posa result as well as a characterization of the full Erdos-Posa property for A-paths in group-labelled graphs. Joint work with O-joung Kwon.

Motivated by a conjecture of Maglione—Voll concerning Igusa zeta functions, we introduce and study the Poincaré-extended ab-index. This polynomial generalizes both the ab-index and the Poincaré polynomial. For posets admitting R-labelings, we prove that the coefficients are nonnegative and give a combinatorial description of the coefficients. This proves Maglione—Voll’s conjecture as well as a conjecture of the Kühne—Maglione. We also recover, generalize, and unify results from Billera—Ehrenborg—Readdy, Ehrenborg, and Saliola—Thomas. This is joint work with Joshua Maglione and Christian Stump.

The relative new and evolving Topological Data Analysis = TDA is using algebraic topology to analyze the geometric structure of data, ranging from tissue in the brain to pieces of text. One end product for TDA is a module over a ring. How this module breaks into indecomposable ones describes/characterizes the stability of the topological structures associated to the underlying data. In this talk I discuss one approach to this via the structure theorem for finitely generated modules over a principal ideal domain.

We study an extension of the k-Disjoint Paths Problem where, in addition to finding k disjoint paths joining k given pairs of vertices in a graph, we ask that those paths satisfy certain constraints expressable by abelian groups. We give an O(n^8) time algorithm to solve this problem under the assumption that the constraint can be expressed as avoiding a bounded number of group elements; moreover, our O(n^8) algorithm allows any bounded number of such constraints to be combined. Group-expressable constraints include, but not limited to: (1) paths of length r modulo m for any fixed r and m, (2) paths passing through any bounded number of prescribed sets of edges and/or vertices, and (3) paths that are long detours (paths of length at least r more than the distance between their ends for fixed r). The k=1 case with the modularity constraint solves problems of Arkin, Papadimitriou and Yannakakis from 1991. Our work also implies a polynomial time algorithm for testing the existence of a subgraph isomorphic to a subdivision of a fixed graph, where each path of the subdivision between branch vertices satisfies any combination of a bounded number of group-expressable constraints. In addition, our work implies similar results addressing edge-disjointness. It is joint work with Youngho Yoo.