2021 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 – 2023-03-31
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| Keywords | multiple zeta values / q-analogues of MZV / modular forms |
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| Outline of Annual Research Achievements |
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.
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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
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.
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| Strategy for Future Research Activity |
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.
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| Causes of Carryover |
The remaining amount will be used for traveling inside Japan and attending conferences/seminars on which the research results will be presented.
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