On the Algebraic Structures of Multiple Zeta Values: The Zagier-Hoffman Conjectures and Beyond
ABSTRACT
The study of special values of the Riemann zeta function traces back to Euler's zeta values and a natural multivariate generalization is given by multiple zeta values (MZVs). Despite their elementary definition, the $\mathbb{Q}$-linear relations among MZVs encode deep structural properties.
In the first part of this talk, we explore the algebraic landscape of classical MZVs. We will discuss the Zagier-Hoffman conjectures—which predict the dimensions and explicit bases of the spaces spanned by MZVs of a fixed weight—and the underlying Hopf algebra structures arising from their "stuffle" and "shuffle" products.
We then transition to the arithmetic of positive characteristic to examine the function field analogs of MZVs introduced by Thakur. We will explore how these Hopf algebra frameworks manifest in this setting, highlighting Todd's contributions regarding the dimensions and relations of these Thakur MZVs.
