2023/05/17 更新

写真a

バッハマン ヘンリック レナード
BACHMANN Henrik lennart
BACHMANN Henrik lennart
所属
大学院多元数理科学研究科 多元数理科学専攻 基幹数理 准教授
大学院担当
大学院多元数理科学研究科
学部担当
理学部 数理学科
職名
准教授

学位 1

  1. Ph.D. ( 2015年12月 ) 

研究キーワード 3

  1. モジュラー形式

  2. 多重ゼータ値

  3. 整数論

研究分野 1

  1. 自然科学一般 / 代数学

経歴 3

  1. 名古屋大学   Institute for Advanced Research & Graduate School of Mathematics   Assistant professor (YLC)

    2017年4月 - 現在

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

    2017年4月 - 2018年3月

  3. 名古屋大学   Graduate School of Mathematics   JSPS Postdoctoral fellow

    2016年4月 - 2017年3月

 

論文 10

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

    Bachmann Henrik

    JOURNAL OF NUMBER THEORY   200 巻   頁: 260-294   2019年7月

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    記述言語:英語   掲載種別:研究論文(学術雑誌)  

    DOI: 10.1016/j.jnt.2018.12.006

    Web of Science

  2. The algebra of bi-brackets and regularized multiple Eisenstein series 査読有り

    H. Bachmann

    Journal of Number Theory   200 巻   頁: 260 - 294   2019年7月

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    記述言語:英語   掲載種別:研究論文(学術雑誌)  

  3. Cyclotomic analogues of finite multiple zeta values

    Bachmann Henrik, Takeyama Yoshihiro, Tasaka Koji

    COMPOSITIO MATHEMATICA   154 巻 ( 12 ) 頁: 2701-2721   2018年12月

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    記述言語:英語   掲載種別:研究論文(学術雑誌)  

    DOI: 10.1112/S0010437X18007583

    Web of Science

  4. Checkerboard style Schur multiple zeta values and odd single zeta values 査読有り

    Henrik Bachmann, Yoshinori Yamasaki

    Mathematische Zeitschrift   290 巻 ( 3-4 ) 頁: 1173 - 1197   2018年12月

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    記述言語:英語   掲載種別:研究論文(学術雑誌)   出版者・発行元:Springer Science and Business Media LLC  

    DOI: 10.1007/s00209-018-2058-5

    Web of Science

    その他リンク: http://link.springer.com/content/pdf/10.1007/s00209-018-2058-5.pdf

  5. Cyclotomic analogues of finite multiple zeta values 査読有り

    H. Bachmann, Y. Takeyama, K. Tasaka

    Compositio Mathematica   154 巻 ( 12 ) 頁: 2701 - 2721   2018年12月

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    記述言語:英語   掲載種別:研究論文(学術雑誌)  

  6. INTERPOLATED SCHUR MULTIPLE ZETA VALUES 査読有り

    Henrik Bachmann

    Journal of the Australian Mathematical Society   104 巻 ( 3 ) 頁: 289 - 307   2018年6月

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    記述言語:英語   掲載種別:研究論文(学術雑誌)   出版者・発行元: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   230 巻   頁: 180-212   2018年6月

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    記述言語:英語   掲載種別:研究論文(学術雑誌)  

    DOI: 10.1017/nmj.2017.9

    Web of Science

  8. The double shuffle relations for multiple Eisenstein series 査読有り

    H. Bachmann, K. Tasaka

    Nagoya Mathematical Journal   230 巻   頁: 180 - 212   2018年6月

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    記述言語:英語   掲載種別:研究論文(学術雑誌)  

  9. Rooted tree maps and the derivation relation for multiple zeta values 査読有り

    H. Bachmann, T. Tanaka

    International Journal of Number Theory   14 巻 ( 10 ) 頁: 2657 - 2662   2018年

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    記述言語:英語   掲載種別:研究論文(学術雑誌)  

    DOI: 10.1142/S1793042118501592

    Web of Science

  10. On multiple series of Eisenstein type 査読有り

    Henrik Bachmann, Hirofumi Tsumura

    RAMANUJAN JOURNAL   42 巻 ( 2 ) 頁: 479 - 489   2017年2月

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    記述言語:英語   掲載種別:研究論文(学術雑誌)   出版者・発行元: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

▼全件表示

MISC 4

  1. Finite and symmetric Mordell-Tornheim multiple zeta values

    Henrik Bachmann, Yoshihiro Takeyama, Koji Tasaka  

        2020年1月

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

    H. Bachmann  

    preprint   2018年

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

    H. Bachmann, Y. Takeyama, K. Tasaka  

    preprint   2018年

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

    H. Bachmann, T. Tanaka  

    preprint   2018年

講演・口頭発表等 13

  1. Numbers, infinite sums and multiple zeta values 招待有り

    ヘンリック バッハマン

    17th IAR YLC Seminar, Institute for Advanced Research, Nagoya University  2018年12月25日 

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    記述言語:英語   会議種別:口頭発表(一般)  

  2. Multiple harmonic q-series at roots of unity and their connection to finite & symmetrized multiple zeta values 招待有り

    ヘンリック バッハマン

    Algebra seminar, Tohoku University  2018年5月 

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    記述言語:英語   会議種別:口頭発表(一般)  

  3. Multiple harmonic q-series at roots of unity and finite & symmetrized multiple zeta values 招待有り

    ヘンリック バッハマン

    Periods in Number Theory, Algebraic Geometry and Physics, HIM, Bonn  2018年1月 

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    記述言語:英語   会議種別:口頭発表(一般)  

  4. Multiple harmonic q-series at primitive roots of unity and finite multiple zeta values 招待有り

    ヘンリック バッハマン

    Seminar Aachen-Bonn-Köln-Lille-Siegen on Automorphic Forms, Universität zu Köln  2017年10月 

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    記述言語:英語   会議種別:口頭発表(一般)  

  5. Modular forms and q-analogues of modified double zeta values 招待有り

    ヘンリック バッハマン

    九大多重ゼータセミナー, Kyushu University  2018年9月14日 

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    記述言語:英語   会議種別:口頭発表(一般)  

  6. Modular forms and q-analogues of modified double zeta values 招待有り

    ヘンリック バッハマン

    解析数論セミナー, Nagoya University  2018年10月11日 

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    記述言語:英語   会議種別:口頭発表(一般)  

  7. Modular forms and q-analogues of modified double zeta values 招待有り

    ヘンリック バッハマン

    関西多重ゼータ研究会第42回, Kyoto Sangyo University  2018年9月29日 

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    記述言語:英語   会議種別:口頭発表(一般)  

  8. Modular forms and multiple zeta values 招待有り

    ヘンリック バッハマン

    33rd Automorphic Forms Workshop, Duquesne University, Pittsburgh  2019年3月7日 

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    記述言語:英語   会議種別:口頭発表(一般)  

  9. Interpolated Schur multiple zeta values 招待有り

    ヘンリック バッハマン

    解析数論セミナー, Nagoya University  2017年1月 

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    記述言語:英語   会議種別:口頭発表(一般)  

  10. Derivatives of q-analogues of multiple zeta values 招待有り

    ヘンリック バッハマン

    多重ゼータ研究集会  2017年2月 

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    記述言語:英語   会議種別:口頭発表(一般)  

  11. Cyclotomic analogues of finite multiple zeta values 招待有り

    ヘンリック バッハマン

    The 10th Young Mathematicians Conference on Zeta Functions, Nagoya University  2017年2月 

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    記述言語:英語   会議種別:口頭発表(一般)  

  12. Checkerboard style Schur multiple zeta values 招待有り

    ヘンリック バッハマン

    解析数論セミナー, Nagoya Unversity  2018年4月 

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    記述言語:英語   会議種別:口頭発表(一般)  

  13. A simultaneous q-analogue of finite and symmetrized multiple zeta values 招待有り 国際会議

    ヘンリック バッハマン

    CARMA (Algebraic Combinatorics, Resurgence, Moulds and Applications), CIRM, Luminy  2017年6月 

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    記述言語:英語   会議種別:口頭発表(一般)  

▼全件表示

科研費 3

  1. Connections of (quasi)modular forms to multiple zeta values and their finite analogues

    研究課題/研究課題番号:23K03030  2023年4月 - 2026年3月

    科学研究費助成事業  基盤研究(C)

    BACHMANN Henrik

      詳細を見る

    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 )

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

    研究課題/研究課題番号:21K13771  2021年4月 - 2023年3月

    科学研究費助成事業  若手研究

    BACHMANN Henrik

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    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.

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

    研究課題/研究課題番号:19K14499  2019年4月 - 2023年3月

    科学研究費助成事業  若手研究

    BACHMANN Henrik

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    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.

 

社会貢献活動 2

  1. Studium Generale

    役割:講師

    Nagoya University  Numbers, infinite sums and their appearences in daily life  2018年12月

  2. JSPS Science Dialogue

    役割:司会, 講師

    Iwate Prefectural Mizusawa High School & Japan Society for the Promotion of Science  "Hamburgers, Numbers and infinite Series"  2016年11月