Arborified Zeta Values and its regularized double shuffle relations

30.10.2026, 11:00  –  Campus Golm, Building 09, Room 2.09.2.22
Freitag Seminar

Pierre Catoire (Mulhouse)

In order to study the zeta function, L. Euler introduced in 1775 Multiple Zeta Values (MZVs)
as a generalization defined for words on non-zero integers. Since then, those functions appeared
in many different fields in mathematics : from arithmetic geometry, number theory to quantum
field theory. Hence, their study can be made using different approaches like complex analysis,
motivic integration or other algebraic technics. One of the main goal is to find the dimension of
the rational vector space generated by MZVs of the same weight. Such relations maybe obtained
thanks to regularized double shuffle relations as conjectured by Hoffman in the 90’s.
In this talk, we focus on the study of another generalization of MZVs introduced by J. Ecalle
in 1981 and highlighted by D. Manchon in 2016 : Arborified Zeta Values (AVZs for short).
First, after a historical introduction to MZVs, we detail the importance of the integral and
series representations of MZVs (which are morphisms respectively for the shuffle and quasi-shuffle
of words) and their relation through the Kontsevich map. It enables us to state what is a double
shuffle regularisation relation.
Then, we introduce AZVs as a map associating any appropriate decorated tree to a real
number expressed by an integral/series representation. Using algebraic technics, we will define the
notion of shuffles, quasi-shuffles and Kontsevich map for trees. Finally, from all those definitions
we state a regularized double shuffle relations conjecture for AZVs and mention it implies the
usual one for MZVs.

This talk is based on joint work with  P. Clavier and K-Y. Fan