Study on Saito free divisors and uniformization equations

• Project/Area Number: 23540077
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• 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) ; HARAOKA Yoshishige 熊本大学, 大学院・自然科学研究科, 教授 (30208665)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥5,200,000 (Direct Cost: ¥4,000,000、Indirect Cost: ¥1,200,000); Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2011: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
• Keywords: 自由因子 / 一意化方程式 / パンルベ方程式 / free divisor / 孤立特異点

Research Abstract

Mainly Saito free divisors in an affine 3-dimensional space are studied. It is known that the zero set of the discriminant of any irreducible complex reflection group is a Saito free divisor. In the rank 3 case, we accomplished a classification of uniformization systems of differential equations along the zero set of discriminant. The result is published in Kyushu J. Math. There are seventeen Saito free divisors in an affine 3 dimensional space constructed by 1-parameter families of curves with simple singularities. We accomplished a classification of holonomic systems of rank 2 with singularities along such Saito free divisors.

Research Products

• [Journal Article] Uniformization systems of equations with singularities along discriminant sets of complex reflection groups of rank three (2014)
  • Author(s): M. Kato and J. Sekiguchi
  • Journal Title: Kyushu J. Math Volume: vol.68 Pages: 181-221
  • NAID: 130004941524
  • Peer Reviewed
• [Journal Article] Saito free divisors in four dimensional affine space and reflection groups of rank four (2014)
  • Author(s): J. Sekiguchi
  • Journal Title: Topics on real and complex singularities Pages: 141-158
  • Peer Reviewed
• [Journal Article] Holonomic systems of differential equations of rank two with singularities along Saito free divisors of simple type (2014)
  • Author(s): J. Sekiguchi
  • Journal Title: Topics on real and complex singularities Pages: 159-188
  • Peer Reviewed
• [Journal Article] Uniformization systems of equations with singularities along the discriminant sets of complex reflection groups of rank three (2014)
  • Author(s): Mitsuo Kato, Jiro Sekiguchi
  • Journal Title: Kyushu Journal of Mathematics Volume: 68 Pages: 181-221
  • NAID: 130004941524
  • Peer Reviewed
• [Journal Article] Saito free divisors in four dimensional affine space and reflection groups of rank four (2014)
  • Author(s): Jiro Sekiguchi
  • Journal Title: Topics on real and complex singularities Volume: 1 Pages: 141-158
  • Peer Reviewed
• [Journal Article] Holonomic systems of differential equations of rank two with singularities along Saito free divisors of simple type (2014)
  • Author(s): Jiro Sekiguchi
  • Journal Title: Topics on real and complex singularities Volume: 1 Pages: 159-188
  • Peer Reviewed
• [Journal Article] The discriminant of the reflection groups of type D4 and holonomic systems with singularities along its zero locus (2013)
  • Author(s): J. Sekiguchi
  • Journal Title: Proceedings of the 19th ICFIDCAA Hiroshima 2011, Edited by K. Matsuzaki and T. Sugawa, Tohoku University Press, Sendai Pages: 223-235
  • URL: http://www.tups.jp
  • Peer Reviewed
• [Journal Article] Remarks on an algebraic solution of Painleve sixth equation obtained by Hitchin (2013)
  • Author(s): J. Sekiguchi
  • Journal Title: Josai Mathematical Monographs Volume: vol.6 Pages: 25-28
• [Journal Article] Remark on an algebraic solution of Painleve sixth equation obtained by Hitchin (2013)
  • Author(s): Jiro Sekiguchi
  • Journal Title: Josai Mathematical Monographs Volume: 6 Pages: 25-28
• [Journal Article] The discriminant of the reflection group of type D4 and holonomic systems with singularities along its zero locus (2013)
  • Author(s): Sekiguchi, Jiro
  • Journal Title: Proceedings of the 19th ICFIDCAA Hiroshima 2011 Volume: 1 Pages: 223-235
  • Peer Reviewed
• [Journal Article] Free deformations of hypersurface singularities (2011)
  • Author(s): A. G. Aleksandrov and J. Sekiguchi
  • Journal Title: J. Math. Sci Volume: vol.175 Issue: 1 Pages: 1-16
  • DOI: 10.1007/s10958-011-0328-4
  • Peer Reviewed
• [Journal Article] Systems of uniformization equations and hyperelliptic integrals (2011)
  • Author(s): J. Sekiguchi
  • Journal Title: J. Math. Sci Volume: vol.175 Issue: 1 Pages: 57-79
  • DOI: 10.1007/s10958-011-0333-7
  • Peer Reviewed
• [Journal Article] Systems of uniformization equations and hyperelliptic integrals (2011)
  • Author(s): Sekiguchi, Jiro
  • Journal Title: Journal of Mathematical Sciences Volume: 175 Pages: 57-79
  • Peer Reviewed
• [Journal Article] Free deformations of hypersurface singularities (2011)
  • Author(s): Aleksandrov, A. G. and Sekiguchi, Jiro
  • Journal Title: Journal of Mathematical Sciences Volume: 175 Pages: 1-16
  • Peer Reviewed
• [Presentation] Free divisors and almost Belyi maps (joint work with R. Vidunas) (2014)
  • Author(s): 関口次郎
  • Organizer: Accessory Parameter 研究会
  • Place of Presentation: 熊本大学
  • Year and Date: 2014-03-08
• [Presentation] Free divisors and almost Belyi maps (joint work with R. Vidunas) (2014)
  • Author(s): 関口次郎
  • Organizer: Accessory Parameter研究会
  • Place of Presentation: 熊本大学理学部
  • Invited
• [Presentation] 自由因子の構成と微分方程式 (2013)
  • Author(s): 関口次郎
  • Organizer: 琉球大学理学部数理科学科談話会
  • Place of Presentation: 琉球大学
  • Year and Date: 2013-12-26
• [Presentation] 自由因子と微分方程式 (2013)
  • Author(s): 関口次郎
  • Organizer: 近畿大学数学教室講演会
  • Place of Presentation: 近畿大学
  • Year and Date: 2013-12-10
• [Presentation] A free divisor which gives an algebraic solution of Painleve VI equation constructed by Hitchin (2013)
  • Author(s): J. Sekiguchi
  • Organizer: The mini-symposium (Japanese-Turkish Joint Geometry Meeting)
  • Place of Presentation: Istanbul, Turkey
  • Year and Date: 2013-11-25
• [Presentation] Free divisors nad differential equations (2013)
  • Author(s): J. Sekiguchi
  • Organizer: Japanese-Turkish Joint Geometry Meeting
  • Place of Presentation: Istanbul, Turkey
  • Year and Date: 2013-11-24
• [Presentation] Free divisors and algebraic solutions of Painleve sixth equation (2013)
  • Author(s): J. Sekiguchi
  • Organizer: JARCS Sydney 2013
  • Place of Presentation: Sydney, Australia
  • Year and Date: 2013-09-11
• [Presentation] Free divisors, holonomic systems and algebraic solutions of Painleve VI (2013)
  • Author(s): J. Sekiguchi
  • Organizer: Seminar at Department of Mathematics
  • Place of Presentation: University of Seville, Seville, Spain
  • Year and Date: 2013-03-11
• [Presentation] Free divisors and differential equations (2013)
  • Author(s): Jiro Sekiguchi
  • Organizer: Japanese-Turkish Joint Geometry Meeting
  • Place of Presentation: Istanbul, Turkey
  • Invited
• [Presentation] A free divisor which gives an algebraic solution of Painleve VI equation constructed by Hitchin (2013)
  • Author(s): Jiro Sekiguchi
  • Organizer: The mini-symposium (Japanes Turkish Joint Geometry Meeting)
  • Place of Presentation: Istanbul, Turkey
  • Invited
• [Presentation] 自由因子の構成と微分方程式 (2013)
  • Author(s): 関口次郎
  • Organizer: 琉球大学理学部数理科学科談話会
  • Place of Presentation: 琉球大学理学部
  • Invited
• [Presentation] Free divisors, holonomic systems and algebraic solutions of Painleve sixth solution (2012)
  • Author(s): J. Sekiguchi
  • Organizer: Workshop on Free Divisors and Differential Equations
  • Place of Presentation: 東京農工大学
  • Year and Date: 2012-11-04
• [Presentation] Free divisors, holonomic systems and algebraic Painleve sixth solutions (2012)
  • Author(s): 関口次郎
  • Organizer: 古典解析セミナー
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2012-07-18
• [Presentation] Free divisors and Painleve equation after M. Kato (2012)
  • Author(s): Sekiguchi, Jiro
  • Organizer: Galatasaray University Mathematics Seminar
  • Place of Presentation: Galatasaray University, Turkey
• [Presentation] Reflection group of type D4 and holonomic systems with singularities along its discriminant locus (2011)
  • Author(s): J. Sekiguchi
  • Organizer: The 19th International Conference on Finite and Infinite Dimensional Complex Analysis and Applications
  • Place of Presentation: Aster Plaza, Hiroshima
  • Year and Date: 2011-12-14
• [Presentation] Holonomic systems with singularities along free divisors (2011)
  • Author(s): J. Sekiguchi
  • Organizer: Local Analytic Geometry, ESI, University of Wien
  • Place of Presentation: Wien, Austria
  • Year and Date: 2011-12-01
• [Presentation] Holonomic systems related to free divisors and their solutions (2011)
  • Author(s): J. Sekiguchi
  • Organizer: The 4th Japanese-Australian Workshop on real and complex singularities
  • Place of Presentation: Kobe Satellite of Hyogo University of Teacher Education
  • Year and Date: 2011-11-23
• [Presentation] 二面体群に関連する超楕円積分 (2011)
  • Author(s): 関口次郎
  • Organizer: 保型形式の整数論月例セミナー
  • Place of Presentation: 東京大学
  • Year and Date: 2011-11-19
• [Presentation] The discriminant of the reflection group of type D4 and holonomic systems related to its zero locus (2011)
  • Author(s): J. Sekiguchi
  • Organizer: Workshop on Accessory Parameters
  • Place of Presentation: 東京大学玉原国際セミナーハウス
  • Year and Date: 2011-10-09
• [Presentation] Holonomic systems related with free divisors of rank three (2011)
  • Author(s): J. Sekiguchi
  • Organizer: Workshop on the Theory of Weyl Groups and Root Systems
  • Place of Presentation: University of Tokyo
  • Year and Date: 2011-09-21
• [Presentation] A Schwarz map of Appell's F2 whose monodromy group is related to the reflection group of type D4 (2011)
  • Author(s): J. Sekiguchi
  • Organizer: 古典解析セミナー
  • Place of Presentation: 東京大学
  • Year and Date: 2011-06-24
• [Presentation] Saito free divisors and uniformization equations (2011)
  • Author(s): J. Sekiguchi
  • Organizer: Warwick Workshop on Free Divisors
  • Place of Presentation: University of Warwick (Warwick, England)
  • Year and Date: 2011-06-01
• [Presentation] Free divisors and Painleve equation after M. Kato (2011)
  • Author(s): J. Sekiguchi
  • Organizer: Mathematics Seminar
  • Place of Presentation: Galatasaray University, Istanbul, Turkey
  • Year and Date: 2011-03-20
• [Presentation] Saito free divisors and uniformization equations (2011)
  • Author(s): Sekiguchi, Jiro
  • Organizer: Warwick Workshop on free divisors(招待講演)
  • Place of Presentation: University of Warwick, England
• [Presentation] Holonomic systems related to free divisors of rank three (2011)
  • Author(s): 関口次郎
  • Organizer: Workshop ''Topics in the theory of Weyl groups and root systems''(招待講演)
  • Place of Presentation: 東京大学大学院数理科学研究科
• [Presentation] The discriminant of the reflection group of type D_4 and holonomic systems related with its zero locus (2011)
  • Author(s): 関口次郎
  • Organizer: Workshop on accessory parameters(招待講演)
  • Place of Presentation: 東京大学玉原セミナーハウス
• [Presentation] 二面体群に関連する超楕円積分 (2011)
  • Author(s): 関口次郎
  • Organizer: 保型形式の整数論月例セミナー(招待講演)
  • Place of Presentation: 東京大学大学院数理科学研究科
• [Presentation] Holonomic systems related with free divisors and their solutions (2011)
  • Author(s): Sekiguchi, Jiro
  • Organizer: The 4th Japanese-Australian Workshop on real and complex singularities(招待講演)
  • Place of Presentation: 兵庫教育大学神戸サテライト
• [Presentation] Holonomic systems with singularities along free divisors (2011)
  • Author(s): Sekiguchi, Jiro
  • Organizer: Algebraic versus Analytic Geometry(招待講演)
  • Place of Presentation: ESI, University of Wien, Austria
• [Presentation] Reflection groups of type D_4 and holonomic systems (2011)
  • Author(s): Sekiguchi, Jiro
  • Organizer: The 19th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications(招待講演)
  • Place of Presentation: 広島アスタープラザ
• [Presentation] A Schwarz map of Appell's F_2 whose monodromy group is related to the reflection group of type D_4 (2011)
  • Author(s): 関口次郎
  • Organizer: 東京大学古典解析セミナー
  • Place of Presentation: 東京大学大学院数理科学研究科