Differential Equations with singularities on free divisors and related Geometry

• Project/Area Number: 17K05269
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tokyo 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: ¥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)
• Keywords: WDVV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。

Research Products

• [Journal Article] The algebraic potentials having tri-hamiltonian structures of the reflection groups of types D4, F4, H4 (2022)
  • Author(s): Jiro Sekiguchi
  • Journal Title: Formal and Analytic Solutions of Differential Equations Volume: 1 Pages: 23-85
  • 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
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Flat structure on the space of isomonodromic deformations (2020)
  • Author(s): Mitsuo Kato, Toshiyuki Mano, Jiro Sekiguchi
  • Journal Title: Symmetry, Integrability and Geometry: Methods and Applications Volume: 16 Pages: 1-36
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Solutions to the extended WDVV equations and the Painleve VI equation (2019)
  • 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
  • Peer Reviewed / Open Access
• [Journal Article] Solutions to extended WDVV equations;ST34, E8 cases (2019)
  • Author(s): J. Sekiguchi
  • Journal Title: Rev.Roumaine Math. Pures Appl. Volume: 64 Pages: 563-581
  • Peer Reviewed / Open Access
• [Journal Article] Flat structure and potential vector fields related with algebraic solutions to Painleve VI equation (2018)
  • 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
  • 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
  • Invited
• [Presentation] Differential equations associated to Frobenius manifolds with trihamiltonian structure of rank four (2022)
  • 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）
  • Invited
• [Presentation] Algebraic potentials of type En(n=6,7,8) defined by quadratic equation (2021)
  • Author(s): Jiro Sekiguchi
  • Organizer: Workshop on Frobenius Manifolds and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Algebraic potentials and reflection groups (2021)
  • Author(s): 関口次郎
  • Organizer: 日本大学特異点月曜セミナー
  • Invited
• [Presentation] Fuchsian differential equations and algebraic potentials having tri-Hamiltonian structures (2020)
  • Author(s): Jiro Sekiguchi
  • Organizer: Workshop FASnet20
  • Int'l Joint Research / Invited
• [Presentation] Examples of algebraic potentials and related topics (2019)
  • Author(s): Jiro Sekiguchi
  • Organizer: Workshop on hyperplane arrangements and reflection groups
  • Invited
• [Presentation] The construction problem of algebraic potentials and related topics (2019)
  • Author(s): Jiro Sekiguchi
  • Organizer: International symposium ''Advances and perspectives in representation theory''
  • Invited
• [Presentation] The construction problem of algebraic potentials and reflection groups (2019)
  • Author(s): 関口次郎
  • Organizer: 2019年度表現論シンポジウム
  • 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
  • Invited
• [Presentation] Uniqueness problem of potential vector fields related with reflection groups (2019)
  • Author(s): Jiro Sekiguchi
  • Organizer: Sminaire Groupes Representations et Geometrie
  • 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」
  • Invited
• [Presentation] Solutions to extended WDVV equations and Painleve VI equation (2018)
  • Author(s): Jiro Sekiguchi
  • Organizer: Complex differential and difference equations-onference
  • 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
  • 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
  • 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
  • Invited
• [Presentation] Potential vector fields related with algebraic solutions to Painleve VI equation (2017)
  • Author(s): Jiro Sekiguchi
  • Organizer: JARCS SYDNEY 2017(University of Sydney, Australia)
  • Int'l Joint Research / Invited
• [Presentation] The discriminant of Valentiner reflection group and deformations of a plane curve (2017)
  • Author(s): Jiro Sekiguchi
  • Organizer: Seminar of Algebra (University of Seville, Spain)
  • Invited
• [Presentation] Potential vector fields related with algebraic solutions to Painleve VI equation (2017)
  • Author(s): Jiro Sekiguchi
  • Organizer: FASdiff2017(University of Alcala, Spain)
  • Int'l Joint Research / Invited
• [Presentation] The discriminant of Valentiner reflection group and deformations of a plane curve (2017)
  • Author(s): 関口次郎
  • Organizer: 特異点月曜セミナー（日本大学文理学部）
• [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）