**Lecture series: Formalization of multiple Eisenstein series**

Kyushu University, 6/7th June 2023

Lecturer: Henrik Bachmann (Nagoya University), Annika Burmester (Bielefeld University)

In this lecture series, we will introduce a formalization of multiple Eisenstein series. We will start with a historical overview of multiple Eisenstein series and related conjectures, which will give a motivation for the definition of their formal analogues. We will see that in the lowest depth, the formal objects satisfy the same relations as the classical Eisenstein series, which leads to a definition of formal modular forms. The second part consists of the presentation of another formal setup for multiple Eisenstein series, which simplifies the description of the relations between them. Then, inspired by Racinet's approach to formal multiple zeta values, we will rephrase the relations between formal multiple Eisenstein series in terms of non-commutative power series. This allows us to show that our formal setup is also a generalization of formal multiple zeta values. Finally, we introduce some derivations on the algebra of formal multiple Eisenstein series. In particular, we obtain an sl2-action, which generalizes the one on (formal) quasimodular forms.

**Lecture notes:**

Part 1 (Bachmann): (Formal) Multiple Eisenstein series, History / Motivation, Definition

Part 2 (Burmester): The balanced setup

Part 3 (Burmester): An analogue of Racinet's approach to formal multiple zeta values

Part 4 (Bachmann): Derivatives and sl2-action on G^f