Multizeta values: Lie algebras and periods on $\mathfrak{M}_{0,n}$

TL;DR

This thesis explores algebraic and geometric relations of multizeta values, providing new dimension results, explicit cohomology bases, and a novel presentation of the Picard group of moduli spaces.

Contribution

It offers new dimension calculations for the double shuffle Lie algebra and explicit descriptions of cohomology bases for moduli spaces related to multizeta values.

Findings

Dimension of depth-graded pieces in depths 1 and 2 determined.

Explicit basis for top-dimensional de Rham cohomology constructed.

New presentation of the Picard group of ar{\u2113}M_{0,n} obtained.

Abstract

This thesis is a study of algebraic and geometric relations between multizeta values. In chapter 2, we prove a result which gives the dimension of the associated depth-graded pieces of the double shuffle Lie algebra in depths 1 and 2. In chapters 3 and 4, we study geometric relations between multizeta values coming from their expression as periods on $\mathfrak{M}_{0,n}$. The key ingredient in this study is the top dimensional de Rham cohomology of special partially compactified moduli spaces associated to multizeta values. In chapter 3, we give an explicit expression for a basis, represented by polygons, of this cohomology. In chapter 4, we generalize this method to explicitly describe the bases of the cohomology of other partially compactified moduli spaces. This thesis concludes with a result which gives a new presentation of $Pic(\overline{\mathfrak{M}}_{0,n})$.