2020 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 – 2022-03-31
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| Keywords | multiple zeta values / q-analogues of MZV / modular forms / Eisenstein series |
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| Outline of Annual Research Achievements |
The research project this year consisted of give projects: (1) Formal multiple zeta values, (2) Functions on partitions and q-analogues of multiple zeta values, (3) Formal double Eisenstein space, (4) Combinatorial multiple Eisenstein-series and (5) Sum formulas for Schur multiple zeta values. In (1), which is a joint work with N/ Matthes and J.W. van Ittersum we introduce the notion of formal multiple Eisenstein series which gives a connection of multiple zeta values and modular forms on a purely formal level. In (2), in joint work with J.W. van Ittersum, we introduced the notion of partitions analogue of multiple zeta values. 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. In (5) I investigate sum formulas for Schur multiple zeta values together with four Japanese collaborators. These generalize the usual sum formulas for multiple zeta values.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
All projects are done with foreign researchers or domestic researchers outside of Nagoya. Due to the special circumstances in these times it was not possible to meet in person to work together on the projects. All meetings are done via online conferences, which works well but which can not replace a meeting in real life.
The preprint for (1) is almost done. 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. 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. The main results for (5) are also finished and we are currently trying to consider a few more special cases. It is expected that preprints for all projects will appear this summer.
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| Strategy for Future Research Activity |
The focus now is to finish writing up all the preprints for (1)-(5). For all projects the main results are proven, but for some we are currently trying to extend the results and give special cases. 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 summer by writing up the results we obtained so far. For the project (4) we are still trying to prove the construction of combinatorial multiple Eisenstein series in arbitrary depth.
Besides the above projects I am also currently preparing a project on finite alternating double zeta values in which I want to prove a connection to period polynomials of modular forms. For this I obtained numerical and partial results in the last weeks and I would like to investigate it more in the coming months.
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| Causes of Carryover |
If possible the remaining amount will be used to meet/invite some of the collaborators of the joint research projects to finish writing the projects up. In addition it will be used for a mathematica and zoom license.
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Research Products
(5 results)
