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Geometric study of the hypergeometric function

Research Project

Project/Area Number 14340049
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKYUSHU UNIVERSITY

Principal Investigator

YOSHIDA Masaaki  KYUSHU UNIVERSITY, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (30030787)

Co-Investigator(Kenkyū-buntansha) SASAKI Takeshi  Kobe University, Faculty of Science, Professor, 理学部, 教授 (00022682)
IWASAKI Katsunori  KYUSHU UNIVERSITY, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (00176538)
MIMACHI Katsuhisa  Tokyo Inst of Tech., Faculty of Science and engineering, Professor, 大学院・理工学研究科, 教授 (40211594)
MATSUMO Keiji  Hokkaido University, Faculty of Science, Associate Professor, 大学院・理学研究科, 助教授 (30229546)
CHO Koji  KYUSHU UNIVERSITY, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (10197634)
花村 昌樹  東北大学, 大学院・理学研究科, 教授 (60189587)
Project Period (FY) 2002 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥6,900,000 (Direct Cost: ¥6,900,000)
Fiscal Year 2005: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2004: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2003: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2002: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsSelberg integral / twisted (co)homology / intersection number / hypergeometric / Whitehead-link / Lambda function / Schwarz triangle / hyperbolic Schwarz map / 捩表裏路地群 / 一寸来群 / 背負ってる回路 / 交叉形式 / 黒写像 / 又曲幾何 / ベタ関数 / 三次曲面 / 又曲構造
Research Abstract

I got the following results concerning the hypergeometric functions.
1)Studied the (co)homology groups attached to Selberg-tpe integrals, evaluated the intersection numbers, and discovered a combinatorial properties of the Selberg functions.
2)Presented co-variant function theory. Found the kappa function, and a 3-parameter families of hypergeometric polynomials, which are very different from the classical ones.
3)Found a new infinite-product formula for the elliptic modular function Lambda.
4)studied combinatorial-topologically the shape of the Schwarz triangles when the inner angles are general.
5)Studied the Whitehead-link-complement group, constructed automorphic functions for this group, and embedded the quotient space to a Euclidean space.
6)Studied the behavior of the solutions of the hypergeometric equation when the exponent-diffences are pure-imaginary, and studied the relation between the space of parameters and the Teichmuler space of genus 2 curves.
7)Invented the theory of hyperbolic Schwarz map. The target of the Schwarz map has been the sphere. Our hypergeometric one has the 3-dimensional hyperbolic space as its target. Group theoretically it is more natural
8)Studied the surfaces on which 3-dimensional Lie group acts, especially ones on which SL(2,R) acts.

Report

(5 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (26 results)

All 2006 2005 2004 2003 Other

All Journal Article (14 results) Publications (12 results)

  • [Journal Article] A family of Schottky groups arising from the hypergeometric differential equation2006

    • Author(s)
      T.Ichikawa, Yoshida
    • Journal Title

      Proc AMS

    • Related Report
      2005 Annual Research Report
  • [Journal Article] An infinite product formula for the Lambda function2005

    • Author(s)
      Yoshida
    • Journal Title

      Ramanujan J. 10

      Pages: 341-382

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] A naive-topological study of the contiguity relations of the hypergeometric functions2005

    • Author(s)
      Yoshida
    • Journal Title

      Banach Center Pub. 69

      Pages: 257-268

    • Related Report
      2005 Annual Research Report
  • [Journal Article] An infinite product formula for the Lambda function2005

    • Author(s)
      M.Yoshida
    • Journal Title

      Ramanujan Journal 123

      Pages: 456-789

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On Schottky groups arising from the hypergeometric equation with imaginary exponents2004

    • Author(s)
      Ichikawa, Yoshida
    • Journal Title

      Proc AMS 132

      Pages: 447-454

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Intersection numbers of twisted cycles associated with the Selberg integral and an application to the conformal field theory2004

    • Author(s)
      Mimachi, Yoshida
    • Journal Title

      Commun.Math.Phys. 250

      Pages: 23-45

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Polynomials associated with the hypergeometric functions with finite monodromy groups2004

    • Author(s)
      Ochiai, Yoshida
    • Journal Title

      Internat J Math 15

      Pages: 629-649

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On Schottky groups arising form the hypergeometric equation with imaginary exponents2004

    • Author(s)
      T.Ichikawa, M.Yoshida
    • Journal Title

      Proc AMS 132

      Pages: 447-454

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Intersection numbers of twisted cycles associated with the Selberg integral and an application to the conformal field theory2004

    • Author(s)
      K.Mimachi, M.Yoshida
    • Journal Title

      CMP 250

      Pages: 23-45

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Polynomials associated with the hypergeometric functions with finite monodromy groups2004

    • Author(s)
      H.Ochiai, M.Yoshida
    • Journal Title

      Internat J Math 15

      Pages: 629-649

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Intersection numbers for loaded cycles associated with Selberg-type integrals2004

    • Author(s)
      K.Mimachi, K.Ohara, M.Yoshida
    • Journal Title

      Tohoku J Math 56

      Pages: 531-551

    • NAID

      110001043908

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Intersection numbers of twisted cycles and the correlation functions of the conformal field theory2003

    • Author(s)
      Mimachi, Yoshida
    • Journal Title

      Commun.Math.Phys. 243

      Pages: 339-358

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The kappa function2003

    • Author(s)
      Kaneko, Yoshida
    • Journal Title

      Internat J Math.14(2003)(1003--1013) 14

      Pages: 1003-1013

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The kappa function2003

    • Author(s)
      Kaneko, Yoshida
    • Journal Title

      Internat J Math. 14

      Pages: 1003-1013

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Publications] T.Sasaki, M.Yoshida: "Invariant theory for linear differential systems modeled after Grassmannian"Nagoya J Math. 171. 163-186 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Yoshida: "A geometric interpretation of the Selberg integral"Internat J of modern Physics. 18. 4343-4359 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Kaneko, M.Yoshida: "The kappa function"Internat J Math. 14. 1003-1013 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Ichikawa, M.Yoshida: "On Schottky groups arising from the hypergeometric equation with imaginary exponents"Proc AMS. 132. 447-454 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Mimachi, K.Ohara, M.Yoshida: "Intersection numbers for loaded cycles associated with the Selberg-type integrals"Tohoku J Math. (印刷中). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Mimachi, M.Yoshida: "Intersection numbers of twisted cycles associated with the Selberg integral and an application to the conformal field theory"Commun.Math.Phys. (印刷中). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Matsumoto, M.Yoshida: "Invariants for some real hyperbolic groups"Intn J. Math.. 13. 415-443 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Sasaki, M.Yoshida: "Schwarzian derivatives and uniformization"CRM Proc. And Lect Notes. 32. 271-286 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Sasaki, M.Yoshida: "Invariant theory for linear differential systems modeled after the Grassmannian $Gr(n, 2n)$"Nagoya J. Math.. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Ichikawa, M.Yoshida: "On Schottky groups arising from the hypergeometric equation with imaginary exponents"Proc. AMS. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Mimachi, H.Ochiai, M.Yoshida: "Intersection theory of loaded cycles IV--resonant cases"Math. Nach. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Mimachi, M.Yoshida: "Intersection numbers of twisted cycles and the correlation functions of the conformal field theory"CMP. (to appear).

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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