Updated on 2022/04/12

写真a

 
BACHMANN Henrik lennart
 
Organization
Administrative Support Organizations International Education & Exchange Designated associate professor
Graduate School of Mathematics Designated associate professor
Title
Designated associate professor

Degree 1

  1. Ph.D. ( 2015.12 ) 

Research History 3

  1. Nagoya University   Institute for Advanced Research & Graduate School of Mathematics   Assistant professor (YLC)

    2017.4

  2. Max-Planck Institute for Mathematics (Bonn/Germany)   Guest ​Scientist

    2017.4 - 2018.3

  3. Nagoya University   Graduate School of Mathematics   JSPS Postdoctoral fellow

    2016.4 - 2017.3

 

Papers 10

  1. The algebra of bi-brackets and regularized multiple Eisenstein series

    Bachmann Henrik

    JOURNAL OF NUMBER THEORY   Vol. 200   page: 260-294   2019.7

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    Language:English   Publishing type:Research paper (scientific journal)  

    DOI: 10.1016/j.jnt.2018.12.006

    Web of Science

  2. The algebra of bi-brackets and regularized multiple Eisenstein series Reviewed

    H. Bachmann

    Journal of Number Theory   Vol. 200   page: 260 - 294   2019.7

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    Language:English   Publishing type:Research paper (scientific journal)  

  3. Cyclotomic analogues of finite multiple zeta values

    Bachmann Henrik, Takeyama Yoshihiro, Tasaka Koji

    COMPOSITIO MATHEMATICA   Vol. 154 ( 12 ) page: 2701-2721   2018.12

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    Language:English   Publishing type:Research paper (scientific journal)  

    DOI: 10.1112/S0010437X18007583

    Web of Science

  4. Checkerboard style Schur multiple zeta values and odd single zeta values Reviewed

    Henrik Bachmann, Yoshinori Yamasaki

    Mathematische Zeitschrift   Vol. 290 ( 3-4 ) page: 1173 - 1197   2018.12

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:Springer Science and Business Media LLC  

    DOI: 10.1007/s00209-018-2058-5

    Web of Science

    Other Link: http://link.springer.com/content/pdf/10.1007/s00209-018-2058-5.pdf

  5. Cyclotomic analogues of finite multiple zeta values Reviewed

    H. Bachmann, Y. Takeyama, K. Tasaka

    Compositio Mathematica   Vol. 154 ( 12 ) page: 2701 - 2721   2018.12

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    Language:English   Publishing type:Research paper (scientific journal)  

  6. INTERPOLATED SCHUR MULTIPLE ZETA VALUES Reviewed

    Henrik Bachmann

    Journal of the Australian Mathematical Society   Vol. 104 ( 3 ) page: 289 - 307   2018.6

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:Cambridge University Press  

    Inspired by the recent work of M. Nakasuji, O. Phuksuwan and Y. Yamasaki, we combine interpolated multiple zeta values and Schur multiple zeta values into one object, which we call interpolated Schur multiple zeta values. Our main result will be a Jacobi-Trudi formula for a certain class of these new objects. This generalizes an analogous result for Schur multiple zeta values and implies algebraic relations between interpolated multiple zeta values.

    DOI: 10.1017/S1446788717000209

    Web of Science

    Scopus

  7. THE DOUBLE SHUFFLE RELATIONS FOR MULTIPLE EISENSTEIN SERIES

    Bachmann Henrik, Tasaka Koji

    NAGOYA MATHEMATICAL JOURNAL   Vol. 230   page: 180-212   2018.6

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    Language:English   Publishing type:Research paper (scientific journal)  

    DOI: 10.1017/nmj.2017.9

    Web of Science

  8. The double shuffle relations for multiple Eisenstein series Reviewed

    H. Bachmann, K. Tasaka

    Nagoya Mathematical Journal   Vol. 230   page: 180 - 212   2018.6

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    Language:English   Publishing type:Research paper (scientific journal)  

  9. Rooted tree maps and the derivation relation for multiple zeta values Reviewed

    H. Bachmann, T. Tanaka

    International Journal of Number Theory   Vol. 14 ( 10 ) page: 2657 - 2662   2018

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:WORLD SCIENTIFIC PUBL CO PTE LTD  

    Rooted tree maps assign to an element of the Connes-Kreimer Hopf algebra of rooted trees a linear map on the noncommutative polynomial algebra in two letters. Evaluated at any admissible word, these maps induce linear relations between multiple zeta values. In this note, we show that the derivation relations for multiple zeta values are contained in this class of linear relations.

    DOI: 10.1142/S1793042118501592

    Web of Science

  10. On multiple series of Eisenstein type Reviewed

    Henrik Bachmann, Hirofumi Tsumura

    RAMANUJAN JOURNAL   Vol. 42 ( 2 ) page: 479 - 489   2017.2

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:SPRINGER  

    The aim of this paper is to study certain multiple series which can be regarded as multiple analogues of Eisenstein series. As part of a prior research, the second-named author considered double analogues of Eisenstein series and expressed them as polynomials in terms of ordinary Eisenstein series. This fact was derived from the analytic observation of infinite series involving hyperbolic functions which were based on the study of Cauchy, and also Ramanujan. In this paper, we prove an explicit relation formula among these series. This gives an alternative proof of this fact by using the technique of partial fraction decompositions of multiple series which was introduced by Gangl, Kaneko and Zagier. By the same method, we further show a certain multiple analogue of this fact and give some examples of explicit formulas. Finally we give several remarks about the relation between the results of the present and the previous works for infinite series involving hyperbolic functions.

    DOI: 10.1007/s11139-015-9738-0

    Web of Science

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MISC 4

  1. Finite and symmetric Mordell-Tornheim multiple zeta values

    Henrik Bachmann, Yoshihiro Takeyama, Koji Tasaka

        2020.1

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    We introduce finite and symmetric Mordell-Tornheim type of multiple zeta
    values and give a new approach to the Kaneko-Zagier conjecture stating that the
    finite and symmetric multiple zeta values satisfy the same relations.

    arXiv

  2. Special values of finite multiple harmonic q-series at roots of unity

    H. Bachmann, Y. Takeyama, K. Tasaka

    preprint     2018

  3. Rooted t​ree maps and the Kawashima relations for multiple zeta values

    H. Bachmann, T. Tanaka

    preprint     2018

  4. Modular forms and q-analogues of modified double zeta values

    H. Bachmann

    preprint     2018

Presentations 13

  1. A simultaneous q-analogue of finite and symmetrized multiple zeta values Invited International conference

    Henrik Bachmann

    CARMA (Algebraic Combinatorics, Resurgence, Moulds and Applications), CIRM, Luminy  2017.6 

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    Language:English   Presentation type:Oral presentation (general)  

  2. Numbers, infinite sums and multiple zeta values Invited

    Henrik Bachmann

    17th IAR YLC Seminar, Institute for Advanced Research, Nagoya University  2018.12.25 

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    Language:English   Presentation type:Oral presentation (general)  

  3. Multiple harmonic q-series at roots of unity and their connection to finite & symmetrized multiple zeta values Invited

    Henrik Bachmann

    Algebra seminar, Tohoku University  2018.5 

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    Language:English   Presentation type:Oral presentation (general)  

  4. Multiple harmonic q-series at roots of unity and finite & symmetrized multiple zeta values Invited

    Henrik Bachmann

    Periods in Number Theory, Algebraic Geometry and Physics, HIM, Bonn  2018.1 

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    Language:English   Presentation type:Oral presentation (general)  

  5. Multiple harmonic q-series at primitive roots of unity and finite multiple zeta values Invited

    Henrik Bachmann

    Seminar Aachen-Bonn-Köln-Lille-Siegen on Automorphic Forms, Universität zu Köln  2017.10 

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    Language:English   Presentation type:Oral presentation (general)  

  6. Modular forms and q-analogues of modified double zeta values Invited

    Henrik Bachmann

    九大多重ゼータセミナー, Kyushu University  2018.9.14 

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    Language:English   Presentation type:Oral presentation (general)  

  7. Modular forms and q-analogues of modified double zeta values Invited

    Henrik Bachmann

    解析数論セミナー, Nagoya University  2018.10.11 

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    Language:English   Presentation type:Oral presentation (general)  

  8. Modular forms and q-analogues of modified double zeta values Invited

    Henrik Bachmann

    関西多重ゼータ研究会第42回, Kyoto Sangyo University  2018.9.29 

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    Language:English   Presentation type:Oral presentation (general)  

  9. Modular forms and multiple zeta values Invited

    Henrik Bachmann

    33rd Automorphic Forms Workshop, Duquesne University, Pittsburgh  2019.3.7 

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    Language:English   Presentation type:Oral presentation (general)  

  10. Interpolated Schur multiple zeta values Invited

    Henrik Bachmann

    解析数論セミナー, Nagoya University  2017.1 

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    Language:English   Presentation type:Oral presentation (general)  

  11. Derivatives of q-analogues of multiple zeta values Invited

    Henrik Bachmann

    Multiple zeta values research meeting,Kindai University  2017.2 

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    Language:English   Presentation type:Oral presentation (general)  

  12. Cyclotomic analogues of finite multiple zeta values Invited

    Henrik Bachmann

    The 10th Young Mathematicians Conference on Zeta Functions, Nagoya University  2017.2 

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    Language:English   Presentation type:Oral presentation (general)  

  13. Checkerboard style Schur multiple zeta values Invited

    Henrik Bachmann

    解析数論セミナー, Nagoya Unversity  2018.4 

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    Language:English   Presentation type:Oral presentation (general)  

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KAKENHI (Grants-in-Aid for Scientific Research) 2

  1. Generalizations of the double shuffle relations for multiple zeta values and the connections to modular forms

    Grant number:21K13771  2021.4 - 2023.3

    科学研究費助成事業  若手研究

    BACHMANN Henrik

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    Authorship:Principal investigator 

    Grant amount:\2210000 ( Direct Cost: \1700000 、 Indirect Cost:\510000 )

    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.

  2. q-analogues of multiple zeta values and their applications in geometry

    Grant number:19K14499  2019.4 - 2021.3

    科学研究費助成事業  若手研究

    BACHMANN Henrik

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    Authorship:Principal investigator 

    Grant amount:\2080000 ( Direct Cost: \1600000 、 Indirect Cost:\480000 )

    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.
    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.
    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.
    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).

 

Social Contribution 2

  1. Studium Generale

    Role(s):Lecturer

    Nagoya University  Numbers, infinite sums and their appearences in daily life  2018.12

  2. JSPS Science Dialogue

    Role(s):Presenter, Lecturer

    Iwate Prefectural Mizusawa High School & Japan Society for the Promotion of Science  "Hamburgers, Numbers and infinite Series"  2016.11