Study on free divisors and differential equations

• Project/Area Number: 26400111
• 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), 教授 (30117717)
• Co-Investigator(Renkei-kenkyūsha): KATO Mitsuo 琉球大学, 教育学部, 教授 (50045043) ; MANO Toshiyuki 琉球大学, 理学部, 准教授 (60378594)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2014: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
• Keywords: 自由因子 / 平坦構造 / 複素鏡映群 / パンルベ方程式 / パンルベVI方程式 / 代数関数解 / ポテンシャル・ベクトル場 / 大久保型微分方程式

Outline of Final Research Achievements

We intruduce the notion of flat sturucture which is a generalization of Frobenius manifold defined by B. Dubrovin.Potentials play a basic role in the definition of Frobenius manifold. Potential vector fields play a similar role in the definition of the flat structures. We obtain the following three results as its application. The first one is to introduce the flat coordinate on the quotient space of the affine space under the action of the well-generated　irreducible complex reflection group.The second one is to establish a correspondence between solutions of generalized WDVV equtins of three variables to those of Painleve VI equation. The third one is to obtain the description of potential vector fields for algebraic solutions to Painleve VI equation.

Research Products

• [Journal Article] The discriminant of Valentiner reflection group and deformations of a plane curve (2017)
  • Author(s): Jiro Sekiguchi
  • Journal Title: Saitama Mathematical Journal Volume: 31
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Flat structures and algebraic solutions to Painleve VI equations (2017)
  • Author(s): Mitsuo Kato, Toshiyuki Mano and Jiro Sekiguchi
  • Journal Title: Analytic, Algebraic and Geometric Aspects of Differential Equations, Trends in Mathematics Volume: 1 Pages: 383-398
  • DOI: 10.1007/978-3-319-52842-7_11
  • ISBN: 9783319528410, 9783319528427
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Flat structures without potentials (2015)
  • Author(s): Mitsuo Kato, Toｓｈyuki Mano, Jiro Sekiguchi
  • Journal Title: Revue Roumaine Math. Pures Appl. Volume: 60 Pages: 481-505
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Presentation] Uniformization model of type A_3 and almost Belyi functions (2017)
  • Author(s): 関口次郎
  • Organizer: 農工大数学セミナー
  • Place of Presentation: 東京農工大学(東京都小金井市）
  • Year and Date: 2017-03-21
  • Invited
• [Presentation] A_3型一意化モデルについて (2017)
  • Author(s): 関口次郎
  • Organizer: 近畿大学数学談話会
  • Place of Presentation: 近畿大学(大阪府、東大阪市）
  • Year and Date: 2017-03-10
  • Invited
• [Presentation] 平坦構造と鏡映群 (2017)
  • Author(s): 関口次郎
  • Organizer: 東京電機大学数学講演会
  • Place of Presentation: 東京電機大学（東京都足立区）
  • Year and Date: 2017-02-20
  • Invited
• [Presentation] Flat structure and potential vector fields related with algebraic solutions to Painlve VI equation (2016)
  • Author(s): Jiro Sekiguchi
  • Organizer: Integrable Systems 2016
  • Place of Presentation: University of Sydney(Australia, Sydney)
  • Year and Date: 2016-12-01
  • Int'l Joint Research / Invited
• [Presentation] パンルベVI方程式の代数関数解と対応するベクトル・ポテンシャルについて (2016)
  • Author(s): 関口次郎
  • Organizer: Workshop on Accessory Parameters
  • Place of Presentation: 東京大学玉原国際セミナーハウス（群馬県沼田市）
  • Year and Date: 2016-08-23
  • Invited
• [Presentation] 鏡映群の平坦構造と関連する話題 (2016)
  • Author(s): 関口次郎
  • Organizer: 2016年度RIMS研究集会「表現論と非可換解析をめぐる諸問題」
  • Place of Presentation: 京都大学数理解析研究所（京都府、京都市）
  • Year and Date: 2016-06-28
  • Invited
• [Presentation] Examples of algebraic solutions to Painleve VI equation treated by Dubrovin and Mazzocco (2016)
  • Author(s): 関口次郎
  • Organizer: アクセサリー・パラメーター研究会
  • Place of Presentation: 熊本大学理学部(熊本県熊本市）
  • Year and Date: 2016-03-22
• [Presentation] 自由因子と微分方程式 (2016)
  • Author(s): 関口次郎
  • Organizer: ２０１６年度日本数学会年会
  • Place of Presentation: 筑波大学(茨城県つくば市）
  • Year and Date: 2016-03-16
  • Invited
• [Presentation] Flat structures and algebraic solutions to Painleve VI equation (2016)
  • Author(s): Jiro Sekiguchi
  • Organizer: Seminar at Taida Institute for Mathematical Sciences
  • Place of Presentation: 国立台湾大学(台湾)
  • Year and Date: 2016-02-22
• [Presentation] Deformation of a plane curve related with the Valentiner group (2016)
  • Author(s): Jiro Sekiguchi
  • Organizer: 中正大学数学系セミナー
  • Place of Presentation: 国立中正大学(台湾)
  • Year and Date: 2016-02-18
• [Presentation] A generalization of WDVV equation and its applications (2015)
  • Author(s): Jiro Sekiguchi
  • Organizer: Mirror Symmetry and Algebraic Geometry 2015
  • Place of Presentation: 京都大学数学教室(京都府京都市）
  • Year and Date: 2015-12-07
  • Invited
• [Presentation] Deformations of a plane curve related with the Valentiner group (2015)
  • Author(s): Jiro Sekiguchi
  • Organizer: The sixth Japanese-Australian Workshop on Real and Complex Singularities
  • Place of Presentation: 鹿児島大学理学部(鹿児島県鹿児島市）
  • Year and Date: 2015-11-23
  • Int'l Joint Research / Invited
• [Presentation] 平坦構造入門 (2015)
  • Author(s): 関口次郎
  • Organizer: 大域解析セミナー
  • Place of Presentation: 熊本大学理学部（熊本県熊本市）
  • Year and Date: 2015-10-27
• [Presentation] A generalisation of WDVV equation and flat structures (2015)
  • Author(s): Jiro Sekiguchi
  • Organizer: Analytic, Algebraic and Geometric Aspects of Differential Equations
  • Place of Presentation: Bedlewo (Poland)
  • Year and Date: 2015-09-07
  • Int'l Joint Research / Invited
• [Presentation] Flat structure on the space of isomonodromic deformations (2015)
  • Author(s): 眞野智行、加藤満生、関口次郎
  • Organizer: 2015日本数学会年会
  • Place of Presentation: 明治大学駿河台キャンパス
  • Year and Date: 2015-03-21 – 2015-03-24
• [Presentation] Flat structure on the space of isomonodromic deformations (2015)
  • Author(s): 加藤満生、眞野智行、関口次郎
  • Organizer: The 10th Kagoshima Algebra-Analysis-Geometry Seminar
  • Place of Presentation: 鹿児島大学理学部
  • Year and Date: 2015-02-16 – 2015-02-19
  • Invited
• [Presentation] Flat structure on the space of isomonodromic deformations (2015)
  • Author(s): 加藤満生、眞野智行、関口次郎
  • Organizer: 超幾何方程式研究会2015
  • Place of Presentation: 神戸大学瀧川記念学術交流会館
  • Year and Date: 2015-01-05 – 2015-01-07
  • Invited