Combinatorics related to Modular Forms and Multiple Zeta Values

In this talk we discuss combinatorial structures, encoded by Hopf algebras, which are related to modular forms and multiple zeta values. In particular we shall also consider their connections with Monstrous Moonshine. These Hopf algebras equally capture the notion of `freeness` in non-commutative probability theory. The talk is based on material from joint work with McKay.