Periods of the motivic fundamental groupoid of $\boldsymbol{\mathbb{P}^{1} \diagdown \lbrace 0, \mu_{N}, \infty \rbrace}$

TL;DR

This thesis explores the Hopf algebra structure of motivic cyclotomic multiple zeta values associated with the fundamental groupoid of a punctured projective line, aiming to develop a Galois theory for periods.

Contribution

It provides a detailed analysis of the motivic Hopf algebra of cyclotomic multiple zeta values and its Galois action, extending the understanding of periods and their symmetries.

Findings

Explicit combinatorial formula for the coaction of the Hopf algebra

Generation of families and identities for cyclotomic multiple zeta values

Insights into the motivic Galois group's action on periods

Abstract

In this thesis, following F. Brown's point of view, we look at the Hopf algebra structure of motivic cyclotomic multiple zeta values, which are motivic periods of the fundamental groupoid of $\mathbb{P}^{1} \diagdown \lbrace 0, \mu_{N}, \infty \rbrace $. By application of a surjective \textit{period} map (which, under Grothendieck's period conjecture, is an isomorphism), we deduce results (such as generating families, identities, etc.) on cyclotomic multiple zeta values, which are complex numbers. The coaction of this Hopf algebra (explicitly given by a combinatorial formula from A. Goncharov and F. Brown's works) is the dual of the action of a so-called \textit{motivic} Galois group on these specific motivic periods. This entire study was actually motivated by the hope of a Galois theory for periods, which should extend the usual Galois theory for algebraic numbers.