# Multiple Zeta Values, Associators, and Related Topics

## Speakers:

Anton Alekseev (UNIGE)

Francis Brown (Oxford)

Annika Burmester (Bielefeld)

Jean-Luc Portner (Oxford)

Muze Ren (UNIGE)

Federico Zerbini (Madrid)

**Organizers: **Anton Alekseev and Francis Brown

## Workshop schedule:

#### Monday, March 4

Morning (Room 6-13)

11:00 - 12:00 Francis Brown

Afternoon (Room 1-15)

13:30 - 14:30 Muze Ren

15:00 - 16:00 Annika Burmester

#### Tuesday, March 5

Morning (Room 6-13)

10:00 - 11:00 Federico Zerbini

11:30 - 12:30 Jean-Luc Portner

Afternoon (Room 6-13)

14:00 - 15:00 Anton Alekseev

## Titles and abstracts:

**Anton Alekseev **

Title: Non-commutative divergence on group rings

Abstract:

Non-commutative divergence is a 1-cocycle on the Lie algebra of derivations of a free associative algebra. Similar to the ordinary divergence, it is defined modulo coboundaries. In this talk, we show that a non-commutative divergence on a group ring of a finitely generated free group is (almost) unique. For fundamental groups of oriented surfaces with boundary, we show that there is a unique non-commutative divergence cocycle associated to each framing of the surface. The talk is based on a joint work with N. Kawazumi, Y. Kuno and F. Naef.

**Francis Brown **

Title: Multiple zeta values, depth filtration, and canonical associators

Abstract:

A dream of Grothendieck was to show that elements of the Galois group of the rational numbers should have a kind of `Taylor expansion'. While this remains out of reach, there is a version for a certain motivic Galois group acting on the free Lie algebra on two generators. I will discuss how this relates to the coefficients appearing in relations between multiple zeta values, the depth filtration, and report on progress towards the problem of finding an explicit rational associator.

TBA

**Annika Burmester**

Title: Multiple Eisenstein series and an extension of the double shuffle Lie algebra

Abstract:

Multiple zeta values are an extension of Riemann zeta values possessing two distinct product expressions. Those lead to the extended double shuffle relations, which are expected to determine all algebraic relations among multiple zeta values. To study these relations, Racinet introduced the double shuffle Lie algebra dm_0 and the corresponding group scheme DM_0. By work of Furusho and Schneps, those are closely related to the Grothendieck-Teichmüller group and the Kashiwara-Vergne problem.

Multiple Eisenstein series were introduced by Gangl-Kaneko-Zagier and extend the classical Eisenstein series. To study their relations we introduce the combinatorial multiple Eisenstein series, which are essentially obtained by replacing the Fourier coefficients by a rational solution to the double shuffle equations. These objects satisfy a product formula and are invariant under some involution, and conjecturally these capture all algebraic relations. This leads us to studying a space bm_0 and a corresponding affine scheme BM_0, which we expect to admit a Lie algebra resp. group structure. The double shuffle Lie algebra dm_0 and the group scheme DM_0 embed into those structures. This is an algebraic counter part of the fact that combinatorial multiple Eisenstein series yield multiple zeta values under a limit q to 1.

**Jean-Luc Portner**

Title: graph cocycles, single-valued multiple zeta values, and associators

Abstract:

Using integrals over configuration spaces, Rossi and Willwacher constructed cocycles of the commutative graph complex. I will explain how these integrals can be evaluated in terms of single-valued multiple zeta values, demonstrating explicitly the wheel graphs. This leads to the conjecture that these integrals are equal to Brown's canonical integrals. Furthermore, I will report on explicit calculations of the images of these cocycles in the Grothendieck-Teichmueller Lie algebra grt_1 up to depth 3, extending previous results from Matteo Felder on the Alekseev-Torossian associator. As an aside, we obtain formulas for single-valued multiple zeta values in depths 2 and 3.

**Muze Ren**

Title: Generalized Pentagon Equations

Abstract:

Vladimir Drinfeld defined his KZ associator by considering the regularized holonomy of KZ equation along the real interval from 0 to 1 and proved that it satisfies the Pentagon equation.

We consider general curves possibly with self intersections and study the equations it satisfied. We will also discuss the relations between cabling operations in the generalized equations and noncommutative geometry in the sense of Kontsevich-Rosenberg, based on joint work with Anton Alekseev and Florian Naef.

**Federico Zerbini**

Title: Conical sums

Abstract: Conical sums are periods defined by certain infinite sums over lattice points contained in cones of R^n. Special cases include multiple zeta values, as well as Matsumoto-Witten zeta values associated with semisimple Lie algebras. They have also appeared in the computation of string theory amplitudes. The Q^ab-algebra generated by conical sums was proven by Terasoma to coincide with the Q^ab-algebra of cyclotomic multiple zeta values; all relations in this algebra are conjectured to follow from decompositions of cones. Open questions on Matsumoto-Witten zeta values would be answered by proving a general conjecture of Dupont about the motivic nature of conical sums. I will introduce conical sums and their algebra, and report on the current state of the art on Dupont’s conjecture.