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.