Generalization of flat structure and isomonodromic deformation

• Project/Area Number: 17K05335
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: University of the Ryukyus
• Principal Investigator: Mano Toshiyuki 琉球大学, 理学部, 教授 (60378594)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 線形微分方程式 / 平坦構造 / モノドロミー / 微分方程式 / モノドロミ / モノドロミ保存変形 / 複素鏡映群 / パンルヴェ方程式

Outline of Final Research Achievements

We studied a kind of geometric structure called 'flat structure' based on its relationship with differential equations. We obtained the following results:
(1) We gave a correspondence between solutions to the Painleve equations (which are nonlinear differential equations) and potential vector fields of flat structures in terms of algebraic and analytic descriptions. (2) We gave a proof of freeness of hyperplane multi-arrangements defined by reflections of a complex reflection group. This proof is an application of our construction of a flat structure associated with a complex reflection group. (3) We published a book which mainly explained our research findings. Particularly, this book contains a new framework on the theory of flat structures.

Academic Significance and Societal Importance of the Research Achievements

本研究の最大の特色は、「平坦構造」と「線形微分方程式」という一見異なる分野の対象について、本質的に同じ対象であるという観点から研究を行うことにある。その結果として、パンルヴェ方程式と呼ばれる線形微分方程式の解と平坦構造のポテンシャルベクトル場と呼ばれる対象とのいくつかの面で新しい対応が明らかになった。また、複素鏡映群に対する平坦構造の構成は「超平面配置」という分野の未解決問題の解決に応用された。
さらに、本研究の成果をまとめたものを主な内容とする専門書を出版した。この本では、平坦構造について従来とは異なる新しい理論構成が与えられている。今後この新しい理論構成を基礎とした研究の進展も期待される。

Research Products

• [Int'l Joint Research] ルール大学ボーフム/ハノーファー大学(ドイツ)
• [Int'l Joint Research] ルール大学ボーフム/ハノーファー大学(ドイツ)
• [Int'l Joint Research] ルール大学ボーフム/ベルリン自由大学/ハノーファー大学(ドイツ)
• [Journal Article] Flat Structure on the Space of Isomonodromic Deformations (2020)
  • Author(s): Kato Mitsuo, Mano Toshiyuki, Sekiguchi Jiro
  • Journal Title: Symmetry, Integrability and Geometry: Methods and Applications Volume: 16
  • DOI: 10.3842/sigma.2020.110
  • Peer Reviewed / Open Access
• [Journal Article] Freeness of multi-reflection arrangements via primitive vector fields (2019)
  • Author(s): Torsten Hoge, Toshiyuki Mano, Gerhard Roehrle, Christian Stump
  • Journal Title: Advances in Mathematics Volume: 350 Pages: 63-96
  • DOI: 10.1016/j.aim.2019.04.044
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Potential vector fields and isomonodromic tau functions in terms of flat coordinates (2019)
  • Author(s): Mano Toshiyuki
  • Journal Title: "Complex Differential and Difference Equations" in the series De Gruyter Proceedings in Mathematics Volume: - Pages: 327-342
  • DOI: 10.1515/9783110611427-012
  • ISBN: 9783110611427
  • Peer Reviewed
• [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] Regular Flat Structure and Generalized Okubo System (2019)
  • Author(s): Hiroshi Kawakami, Toshiyuki Mano
  • Journal Title: Communications in Mathematical Physics Volume: 印刷中 Issue: 2 Pages: 403-431
  • DOI: 10.1007/s00220-019-03330-w
  • Peer Reviewed
• [Journal Article] Determinant Structure for τ-Function of Holonomic Deformation of Linear Differential Equations (2018)
  • Author(s): Ishikawa Masao、Mano Toshiyuki、Tsuda Teruhisa
  • Journal Title: Communications in Mathematical Physics Volume: 363 Issue: 3 Pages: 1081-1101
  • DOI: 10.1007/s00220-018-3256-z
  • Peer Reviewed
• [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] Flat structures on solutions to the sixth Painleve equation (2023)
  • Author(s): Toshiyuki Mano
  • Organizer: Web-Seminar on Painleve Equations and related topics
  • Int'l Joint Research
• [Presentation] Period of primitive forms, the space of Okubo-Saito potentials and the sixth Painleve equation (2022)
  • Author(s): Toshiyuki Mano
  • Organizer: Painleve Equations: From Classical to Modern Analysis
  • Int'l Joint Research / Invited
• [Presentation] 複素鏡映群の平坦不変式と多重鏡映面配置の自由性 (2020)
  • Author(s): 眞野智行
  • Organizer: 日本数学会九州支部例会・特別講演
  • Invited
• [Presentation] Flat structure on the orbit space of a complex refrection group (2019)
  • Author(s): Toshiyuki Mano
  • Organizer: Silver Workshop: Complex Geometry and Non-Commutative Geometry
  • Int'l Joint Research / Invited
• [Presentation] Analytic representation of potential vector fields and isomonodromic tau functions (2018)
  • Author(s): Toshiyuki Mano
  • Organizer: Complex differential and difference equations
  • Int'l Joint Research / Invited
• [Presentation] Flat structure on the orbit space of a complex reflection group (2018)
  • Author(s): Toshiyuki Mano
  • Organizer: Matroids, Reflection Groups, and Free Hyperplane Arrangements
  • Int'l Joint Research / Invited
• [Presentation] パンルヴェ方程式と平坦座標 (2018)
  • Author(s): 眞野智行
  • Organizer: 超幾何ワークショップ
• [Presentation] パンルヴェ方程式と平坦座標 (2018)
  • Author(s): 眞野智行
  • Organizer: 第2回古典解析・徳島研究会 ～パンルヴェ首相百年記念～
• [Presentation] Potential vector fields associated with solutions to Painleve equations (2017)
  • Author(s): Toshiyuki Mano
  • Organizer: Conformal field theory, isomonodromy tau-functions and Painleve equations
  • Int'l Joint Research / Invited
• [Presentation] 複素鏡映群に関する平坦構造と多重鏡映面配置の自由性 (2017)
  • Author(s): 眞野智行
  • Organizer: 研究集会「不変式・超平面配置と平坦構造」
  • Invited
• [Presentation] 平坦構造の一般化と大久保型方程式のモノドロミ保存変形 (2017)
  • Author(s): 眞野智行
  • Organizer: 第69回 Encounter with Mathematics
  • Invited
• [Presentation] Flat structures and Painleve equations (2017)
  • Author(s): Toshiyuki Mano
  • Organizer: The XXVth International Conference on Integrable Systems and Quantum symmetries
  • Int'l Joint Research
• [Book] 平坦構造と複素鏡映群・パンルヴェ方程式 (2022)
  • Author(s): 眞野 智行
  • Total Pages: 416
  • Publisher: 森北出版
  • ISBN: 9784627083813