担当区分：研究代表者 

配分額：3640000円 （ 直接経費：2800000円 、 間接経費：840000円 ）

担当区分：研究代表者 

配分額：2210000円 （ 直接経費：1700000円 、 間接経費：510000円 ）

In the first part of the research the algebraic setup of above objects will be defined. The goal will be to give an explicit connection to previous works on modular forms and multiple zeta values. In the second part the connection to other areas related to multiple zeta values will be made explicit.
In a joint work with A. Burmester we finished a preprint on "Combinatorial multiple Eisenstein series". In this work we introduce a generalization of the extended double shuffle relations and give a solution to these in terms of formal power series with rational coefficients. The construction is inspired by the classical calculation of the Fourier expansion of multiple Eisenstein series, but needs some extra ingredients. As an application one obtains purely combinatorial proofs of relations among modular forms. In another project, joint with U. Kuehn and N. Matthes, we give another definition of combinatorial multiple Eisenstein series in depth two. This is done by generalizing a construction of Gangl-Kaneko-Zagier for rational solutions to the double shuffle relations in depth two. In our project we show that power series satisfying the so-called Fay idenity can be used to obtain solutions to the generalized double shuffle relations in depth two. Applying this construction to the Kronecker function then yields a definition of double Eisenstein series.
The current research plan is going as expected. The work on combinatorial multiple Eisenstein series gave a lot of new open questions for further projects.
Currently in a joint project with J.-W. van Ittersum and N. Matthes we are investigating formal multiple Eisenstein series. These can be seen as a formal analogue of the combinatorial multiple Eisenstein series. In this work we give the algebraic describtion of generalized double shuffle relations and we show how these are related to the classical extended double shuffle relations of multiple zeta values. Based on computer based experiments we also have a conjectured sl_2 action on our space, which seems to be a natural extension of the usual sl_2 action on quasi-modular forms. The proof of this conjectured action is still work in progress but also seems to be in reach.

担当区分：研究代表者 

配分額：2080000円 （ 直接経費：1600000円 、 間接経費：480000円 ）

This research project deals with the intersection of multiple zeta values (numbers), their q-analogues (q-series), modular forms (functions) and their connections to objects in enumerative and algebraic geometry.
One goal is to clarify the connection of q-analogues of multiple zeta values to counting square tiled surfaces. In particular, the question when a linear combination of q-analogues of multiple zeta values is modular will be adressed.
In a joint work with Jan-Willem van Ittersum I finished a project on functions on partitions and their connection to q-analogues of multiple zeta values. In this project we introduce the space of polynomial functions on partitions, which is a subspace of all functions on partitions. This space can be equipped with three different products, which can be seen as natural generalizations of the harmonic and shuffle products of multiple zeta values. We show that, after applying the so-called q-bracket, that polynomial functions on partitions give rise to q-analogues of multiple zeta values. Further we show that the limits of q->1 give (generalization) of multiple zeta values. As an application we show, that other well-known families of functions on partitions, such as shifted-symmetric functions, are contained in our space. This gives relations among multiple zeta values and provides a possible bridge between enumerative geometry and the theorey of multiple zeta values.
Eventhough the current situation made it impossible to meet my collaborator overseas, we were able to smootly finishing our research project due to various online meeting. Further I also presented these results at various seminar around Japan.
It is planned to continue several small side projects related to the above mentioned project on functions on partitions. For this it is planned to visit my collaborators in Germany to discuss possible future directions. One possible future direction of the current project is to clarify the exact relationship of functions on partitions appearing in enumerative geometry and our newly introduced space of polynomial functions.