**Combinatorial multiple Eisenstein series & polynomial functions on partitions**

Lecture series at the Multiple Zeta Research Center, Kyushu University 6th April - 8th April 2022

**Abstracts & Schedule**

The lecture series consists of the following three parts.

In this introductory part, we will start with recalling some basic facts on multiple zeta values, their regularizations, and the double shuffle relations. After this, we define multiple Eisenstein series, defined by Gangl-Kaneko-Zagier, and explain the construction of their Fourier expansion. We will also talk about certain q-analogues of multiple zeta series, which appear as building blocks in these Fourier expansions. In the end, we interpret the Fourier expansion from a more algebraic point of view and sketch the construction of stuffle and shuffle regularized multiple Eisenstein series.

Overview

Based on the construction of Fourier expansion of multiple Eisenstein series, we will present a construction of formal q-series with rational coefficients, which we call combinatorial multiple Eisenstein series (CMES). The CMES can be seen as an interpolation between a rational solution to the double shuffle equations and (stuffle regularized) multiple zeta values. Moreover, they give a natural bridge between the theory of modular forms and multiple zeta values. The construction will use the language of (bi)moulds, which we will introduce at the beginning.

Overview (tentative)

In this talk, we generalize the idea of q-analogues of multiple zeta values and talk about functions on partitions. We introduce a special case of "polynomial functions on partitions", which give exactly q-analogues of multiple zeta values after applying the so-called q-bracket. Further, we will introduce the notion of degree and degree/weight limit of polynomial functions, which then will lead to bi-multiple zeta values. These can be seen as natural generalizations of stuffle regularized multiple zeta values and we will explain that they can also be seen as the limit as q->1 of CMES introduced in part 2.

Overview (tentative)

**(1) Multiple zeta values, Multiple Eisenstein series and q-analogues (6th - 7th April)**In this introductory part, we will start with recalling some basic facts on multiple zeta values, their regularizations, and the double shuffle relations. After this, we define multiple Eisenstein series, defined by Gangl-Kaneko-Zagier, and explain the construction of their Fourier expansion. We will also talk about certain q-analogues of multiple zeta series, which appear as building blocks in these Fourier expansions. In the end, we interpret the Fourier expansion from a more algebraic point of view and sketch the construction of stuffle and shuffle regularized multiple Eisenstein series.

Overview

- 1.1 Multiple zeta values (MZV)
- 1.2 Algebraic Setup I (Quasi-shuffle products) & Regularization
- 1.3 Multiple Eisenstein series (MES) & q-analogues of MZV (qMZV)
- 1.4 MES as maps and stuffle & shuffle regularized MES

**(2) Combinatorial multiple Eisenstein series (7th - 8th April)**Based on the construction of Fourier expansion of multiple Eisenstein series, we will present a construction of formal q-series with rational coefficients, which we call combinatorial multiple Eisenstein series (CMES). The CMES can be seen as an interpolation between a rational solution to the double shuffle equations and (stuffle regularized) multiple zeta values. Moreover, they give a natural bridge between the theory of modular forms and multiple zeta values. The construction will use the language of (bi)moulds, which we will introduce at the beginning.

Overview (tentative)

- 2.1 Motivation & Overview
- 2.2 Algebraic Setup II (Bimoulds)
- 2.3 The bimoulds b,L, and g*
- 2.4 The bimould G and combinatorial multiple Eisenstein series (CMES)
- 2.5 Proof sketch (Symmetril & Swap invariance)

**(3) Polynomial functions on partitions and bi-multiple zeta values (8th April)**In this talk, we generalize the idea of q-analogues of multiple zeta values and talk about functions on partitions. We introduce a special case of "polynomial functions on partitions", which give exactly q-analogues of multiple zeta values after applying the so-called q-bracket. Further, we will introduce the notion of degree and degree/weight limit of polynomial functions, which then will lead to bi-multiple zeta values. These can be seen as natural generalizations of stuffle regularized multiple zeta values and we will explain that they can also be seen as the limit as q->1 of CMES introduced in part 2.

Overview (tentative)

- 3.1 Introduction & Functions on partitions
- 3.2 Polynomial functions on partitions and their degree limit
- 3.3 Algebraic Setup III
- 3.4 Bi-multiple zeta values
- 3.5 Applications to MZV

**Materials**

- Lecture notes: Part 1 (MZV & MES), Part 2 (CMES), Part 3 (Fct. on partitions)
- Handout

**Main references ****(more references can be found in the handout)**

[B] H. Bachmann:

[BB] H. Bachmann, A. Burmester:

[BI] H. Bachmann, J.W. van-Ittersum:

[BKM] H. Bachmann, U. Kühn, N. Matthes:

[BT] H. Bachmann, K. Tasaka:

*q-analogues of multiple zeta values and the formal double Eisenstein space*, 2021 Waseda number theory Conference Proceedings (pdf).[BB] H. Bachmann, A. Burmester:

*Combinatorial multiple Eisenstein series*, preprint, (pdf).[BI] H. Bachmann, J.W. van-Ittersum:

*Partitions, Multiple Zeta Values, and the q-bracket*, preprint, (pdf).[BKM] H. Bachmann, U. Kühn, N. Matthes:

*Realizations of the formal double Eisenstein space*, preprint, (pdf).[BT] H. Bachmann, K. Tasaka:

*The double shuffle relations for multiple Eisenstein series*, Nagoya Math. J.**230**, 180-212. (Journal link, Arxiv link)