2019 Fiscal Year Research-status Report
q-analogues of multiple zeta values and their applications in geometry
| Project/Area Number |
19K14499
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| Research Institution | Nagoya University |
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Principal Investigator |
BACHMANN Henrik 名古屋大学, 多元数理科学研究科, 特任助教 (20813372)
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| Project Period (FY) |
2019-04-01 – 2021-03-31
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| Keywords | multiple zeta values / q-analogues of MZV / modular forms / functions on partitions / mult. Eisenstein series |
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| Outline of Annual Research Achievements |
The research project this year consisted of four projects: (1) Finite Mordell-Tornheim values, (2) Functions on partitions and q-analogues of multiple zeta values, (3) Generalized double shuffle relations and (4) Combinatorial multiple Eisenstein-series. For (1), which is a joint work with Y. Takeyama and K. Tasaka, we introduced finite Mordell-Tornheim values and gave a variant of the Kaneko-Zagier conjecture for these values. In (2), in joint work with J.W. van Ittersum, we introduced the notion of partitions analogue of multiple zeta values. The goal of this project is to connect functions on partitions, which appear in various counting problems in enumerative geometry, to the theory of q-analogues of multiple zeta values. To any function on partitions one can assign a q-analogue, and we show that a large class of these can be seen as q-analogues of multiple zeta values. Moreover, we describe an stuffle and shuffle product analogue on the space of functions on partitions. In (3), in joint work with U. Kuehn and N. Matthes, we introduce a general notion of the double shuffle relations of multiple zeta values, which can be seen for the correct family of relations when dealing with functions instead of numbers. In depth two we give explicit constructions of solutions for these equations. Closely related to this project is (4), joint with A. Burmester, in which we give explicit solutions in depth 2 and 3 for these generalized double shuffle equations given by so-called combinatorial multiple Eisenstein series.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Project (1) is finished and resulted in one preprint, which is submitted. Projects (2) is still work in progress, but it already contains a lot new results. The current goal is to make some formulas more explicit and, if possible, prove one open big conjecture. This conjecture states that almost all functions on partitions give rise to q-analogues of multiple zeta values and not just the "obvious" ones. This would imply a really nice application, for example, to calculate Masur-Veech volumes in terms of multiple zeta values. Project (3) is also still work in progress but also already contains a lot of new results. It currently just remains to write everything up, before we can submit it. Similar to (4), where the results in depth 2 and depth 3 are done. But if possible we would like to extend our construction of combinatorial multiple Eisenstein series to higher depths.
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| Strategy for Future Research Activity |
The focus now is to finish projects (2), (3), and (4). In (2) the focus is now to prove the open conjecture, for which we already obtained several numerical evidence. With this conjecture, we would try then to give applications in calculations coming from the algebraic & enumerative geometry side of the story. For the project (3) the plan is to finish this project this coming August by writing up the results we obtained so far. For the project (4) we recently made a new discovery concerning the construction of combinatorial multiple Eisenstein series in arbitrary depth. This we will try to make more precise in the upcoming months. Besides these projects, there are also plans of two new projects on t-adic finite multiple zeta values and the derivative of multiple Eisenstein series and their Fourier coefficients. The first one is currently in planning with Y. Takeyama and K. Tasaka. In the second one, I want to study the relations among multiple Eisenstein series and their implication for relations among multiple zeta values. I want also to put into connection with the projects (3) and (4).
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| Causes of Carryover |
To finish the started project I plan to visit my collaborators in Europe at the end of this summer. In addition, I plan to invite A. Burmester at the end of this year to finish our projects. Besides this, I plan to attend domestic conferences in Japan to present the research results obtained during this year. If the current situation allows it, it is also planned to organize a small conference/seminar in Nagoya targeted at people related to the described research projects.
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Research Products
(5 results)
