## Free Probability and Operators

**Date: ** November 10, 2023

**Time: ** 4:00PM - 5:00PM

**Location: ** BLOC 306

**Speaker: **Adrian Celestino, TU Graz

**Title: ***Antipode formulas, Schröder trees and cumulants in non-commutative probability*

**Abstract: **In a series of recent papers, Ebrahimi-Fard and Patras developed an algebraic approach for cumulants in non-commutative probability based on a combinatorial Hopf algebra of words on words on an alphabet. In particular, they showed that the combinatorial moment-cumulants formulas, expressed in terms of non-crossing partitions, can be retrieved from specific fixed-point equations involving linear functionals on a Hopf algebra.

In this talk, we discuss a combinatorial formula for the antipode in this Hopf algebra, which is represented in terms of Schröder trees, which have recently appeared in the context of non-commutative probability theory. Finally, we will see the implications of the antipode formula in non-commutative probability, namely, cumulant-moment formulas and free Wick polynomials in terms of Schröder trees. Based on an ongoing joint work with Yannic Vargas.