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Differential Equations with singularities on free divisors and related Geometry

Research Project

Project/Area Number 17K05269
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionTokyo University of Agriculture and Technology

Principal Investigator

Sekiguchi Jiro  東京農工大学, 工学(系)研究科(研究院), 名誉教授 (30117717)

Project Period (FY) 2017-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2017: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
KeywordsWDVV方程式 / 代数的ポテンシャル / 実鏡映群 / 複素鏡映群 / 単純特異点 / potential vector field / 拡張WDVV方程式 / ポテンシャル・ベクトル場 / パンルベVI方程式 / パンルベ方程式 / フロベニウス多様体 / 平坦構造
Outline of Final Research Achievements

The principal researcher of this research studied mainly two subjects.
The first subject concerned with the construction of potential vector fields of
flat structures. The flat structure is a generalization of Frobenius manifolds established by the principal researcher and collabolators Mitsuo Kato and Toshiyuki Mano. There are two interesting examples of such structure. One is related to Painleve VI equation and the other is related to complex reflection groups. Concerning to these examples, we constructed corresponding potential vector fields. The second subject is the construction problem of algebraic potentials of Frobenius manifolds. The principal researcher constructed some examples of such potentials and in particular showed an interesting relationship between two examples related with reflection groups of types E6 and E7 and complex reflection groups No.33 and No.34. As a consequence an answer to the Arnold Problem on complex reflection groups are given to these two groups.

Academic Significance and Societal Importance of the Research Achievements

フロベニウス多様体の一般化である平坦構造の構成、その中心的対象であるポテンシャルベクトル場の構成とパンルベVI方程式の代数解との対応の明確化、また複素鏡映群の場合のポテンシャルの存在と構成について基本的な成果が得られた。平坦構造の重要性についての意義を与えることができた。代数的フロベニウス多様体のポテンシャルの例の構成はあまりなかったが、本研究ではいくつかの例を構成できた。本研究の意義は、代数的フロベニウス多様体と複素鏡映群との関係を与えたこと、1970年代に定式化された複素鏡映群についてのアーノルドの問題に対する進展を得たことである。

Report

(7 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (26 results)

All 2023 2022 2021 2020 2019 2018 2017

All Journal Article (6 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 6 results,  Open Access: 5 results) Presentation (20 results) (of which Int'l Joint Research: 5 results,  Invited: 18 results)

  • [Journal Article] The algebraic potentials having tri-hamiltonian structures of the reflection groups of types D4, F4, H42022

    • Author(s)
      Jiro Sekiguchi
    • Journal Title

      Formal and Analytic Solutions of Differential Equations

      Volume: 1 Pages: 23-85

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] The WDVV solution E8(a1)2021

    • Author(s)
      Yassir Dinar, Jiro Sekiguchi
    • Journal Title

      Journal of Geometry and Physics

      Volume: 170 Pages: 1-11

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Flat structure on the space of isomonodromic deformations2020

    • Author(s)
      Mitsuo Kato, Toshiyuki Mano, Jiro Sekiguchi
    • Journal Title

      Symmetry, Integrability and Geometry: Methods and Applications

      Volume: 16 Pages: 1-36

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Solutions to the extended WDVV equations and the Painleve VI equation2019

    • Author(s)
      M.Kato, T.Mano, J.Sekiguchi
    • Journal Title

      Complex differential and difference equations

      Volume: 1 Pages: 344-364

    • DOI

      10.1515/9783110611427-013

    • ISBN
      9783110611427
    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Solutions to extended WDVV equations;ST34, E8 cases2019

    • Author(s)
      J. Sekiguchi
    • Journal Title

      Rev.Roumaine Math. Pures Appl.

      Volume: 64 Pages: 563-581

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Flat structure and potential vector fields related with algebraic solutions to Painleve VI equation2018

    • Author(s)
      M. Kato, T. Mano, J, Sekiguchi
    • Journal Title

      Opscula Mathematica

      Volume: 38 Issue: 2 Pages: 201-252

    • DOI

      10.7494/opmath.2018.38.2.201

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] ゴセットの6次元ポリトープについて2023

    • Author(s)
      関口次郎
    • Organizer
      直観幾何学2023
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Differential equations associated to Frobenius manifolds with trihamiltonian structure of rank four2022

    • Author(s)
      関口次郎
    • Organizer
      アクセサリー・パラメータ研究会(熊本大学)
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Algebraic potentials of type En(n=6,7,8) defined by quadratic equation2021

    • Author(s)
      Jiro Sekiguchi
    • Organizer
      Workshop on Frobenius Manifolds and Related Topics
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Algebraic potentials and reflection groups2021

    • Author(s)
      関口次郎
    • Organizer
      日本大学特異点月曜セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Fuchsian differential equations and algebraic potentials having tri-Hamiltonian structures2020

    • Author(s)
      Jiro Sekiguchi
    • Organizer
      Workshop FASnet20
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Examples of algebraic potentials and related topics2019

    • Author(s)
      Jiro Sekiguchi
    • Organizer
      Workshop on hyperplane arrangements and reflection groups
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] The construction problem of algebraic potentials and related topics2019

    • Author(s)
      Jiro Sekiguchi
    • Organizer
      International symposium ''Advances and perspectives in representation theory''
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] The construction problem of algebraic potentials and reflection groups2019

    • Author(s)
      関口次郎
    • Organizer
      2019年度表現論シンポジウム
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 概Belyi写像への自由因子の理論の応用2019

    • Author(s)
      関口次郎
    • Organizer
      農工大数学セミナー2019
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Uniqueness problem of potential vector fields related with reflection groups2019

    • Author(s)
      Jiro Sekiguchi
    • Organizer
      Sminaire Groupes Representations et Geometrie
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] 複素鏡映群と平坦構造2018

    • Author(s)
      関口次郎
    • Organizer
      研究集会「微分方程式と表現論」
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Solutions to extended WDVV equations and Painleve VI equation2018

    • Author(s)
      Jiro Sekiguchi
    • Organizer
      Complex differential and difference equations-onference
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] ポテンシャル・ベクトル場について2018

    • Author(s)
      関口次郎
    • Organizer
      アクセサリー・パラメーター研究会
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 3次元の一般化されたWDVV方程式の特殊解について2018

    • Author(s)
      関口次郎
    • Organizer
      第2回古典解析・徳島研究会
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] WDVV方程式の一般化とPainleve VI方程式2017

    • Author(s)
      関口次郎
    • Organizer
      第16回岡シンポジウム
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Potential vector fields related with algebraic solutions to Painleve VI equation2017

    • Author(s)
      Jiro Sekiguchi
    • Organizer
      JARCS SYDNEY 2017(University of Sydney, Australia)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The discriminant of Valentiner reflection group and deformations of a plane curve2017

    • Author(s)
      Jiro Sekiguchi
    • Organizer
      Seminar of Algebra (University of Seville, Spain)
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Potential vector fields related with algebraic solutions to Painleve VI equation2017

    • Author(s)
      Jiro Sekiguchi
    • Organizer
      FASdiff2017(University of Alcala, Spain)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The discriminant of Valentiner reflection group and deformations of a plane curve2017

    • Author(s)
      関口次郎
    • Organizer
      特異点月曜セミナー(日本大学文理学部)
    • Related Report
      2017 Research-status Report
  • [Presentation] Painleve VI方程式の代数関数解と平坦構造2017

    • Author(s)
      関口次郎
    • Organizer
      数理解析セミナー(一橋大学)
    • Related Report
      2017 Research-status Report

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Published: 2017-04-28   Modified: 2024-01-30  

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