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Research on special functions by relative twisted (co)homology groups

Research Project

Project/Area Number 19K03413
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionHokkaido University

Principal Investigator

Matsumoto Keiji  北海道大学, 理学研究院, 教授 (30229546)

Project Period (FY) 2019-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2021: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords超幾何関数 / 超幾何微分方程式系 / 相対ねじれコホモロジー群 / 相対ねじれホモロジー群 / モノドロミー / ねじれ周期関係式 / 相対ねじれホモロジー / 相対ねじれコホモロジー / 超幾何級数 / モノドロミー表現 / ガウスマニン系 / 超幾何微分方程式 / ねじれホモロジー群 / ねじれコホモロジー群 / 交点形式 / Gauss-Manin 接続
Outline of Research at the Start

解の積分表示を有する種々の超幾何微分方程式系に対して, その局所解空間とあらゆるパラメーターに対して線形同型となる相対ねじれホモロジー群を定義し, その双対空間と標準的に同型となる相対ねじれコホモロジー群を設定する.
相対ねじれ(コ)ホモロジー群とそれらの群に自然に定まる交点形式を用いて, 種々の超幾何微分方程式系のあらゆるパラメーターに対して有効となる公式の発見や既知の公式のパラメーターに関する条件を排除した統一理論の構成を研究目的とする.
その他の積分表示を有する特殊関数に対しても相対ねじれ(コ)ホモロジー群の設定を試み, これらの群による特殊関数に対する新しい研究手法の確立を目指す.

Outline of Final Research Achievements

By introducing relative twisted (co)homology groups. we can study Lauricella's system F_D of hypergeometric differential equations by Euler type integrals even in case some parameters become integers. In fact, we obtain some results on the monodromy representation, Pfaffian systems and twisted preiod relations on this system with integral parameters.

Academic Significance and Societal Importance of the Research Achievements

超幾何微分方程式系の研究では、パラメーターが整数になる場合は除外していた。現時点では、解の積分表示が線積分となる超幾何微分方程式系 F_D だけに対して、相対ねじれ(コ)ホモロジー群が設定されているが、この方法が多重積分についても一般化されることで、パラメーターが整数になる場合を除外する必要がなくなることが期待される。特に統計分野で現れる超幾何関数は、パラメーターが整数になる場合がとても多いので、この方面への応用が期待される。

Report

(6 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (13 results)

All 2024 2023 2022 2020 2019 Other

All Int'l Joint Research (1 results) Journal Article (6 results) (of which Peer Reviewed: 6 results) Presentation (5 results) (of which Int'l Joint Research: 3 results,  Invited: 5 results) Book (1 results)

  • [Int'l Joint Research] Theory Group/Dipartimento di Fisica e Astronomia/Univesity Padova(イタリア)

    • Related Report
      2019 Research-status Report
  • [Journal Article] Relative twisted homology and cohomology groups associated with Lauricella's F_D2024

    • Author(s)
      MATSUMOTO Keiji
    • Journal Title

      Funkcialaj Ekvacioj

      Volume: -

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] An analogy of Jacobi's formula and its applications2023

    • Author(s)
      CHIBA Jun、MATSUMOTO Keiji
    • Journal Title

      Hokkaido Mathematical Journal

      Volume: 52 Issue: 3 Pages: 463-494

    • DOI

      10.14492/hokmj/2021-572

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Schwarz's map for Appell's second hypergeometric system with quarter integer parameters2022

    • Author(s)
      Matsumoto Keiji、Osafune Shohei、Terasoma Tomohide
    • Journal Title

      Tohoku Mathematical Journal

      Volume: 74 Issue: 1 Pages: 109-149

    • DOI

      10.2748/tmj.20201207

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] The structure of a local system associated with a hypergeometric system of rank 92020

    • Author(s)
      J. Kaneko, K. Matsumoto, K. Ohara
    • Journal Title

      International Journal of Mathematics

      Volume: 31 Issue: 03 Pages: 2050021-2050021

    • DOI

      10.1142/s0129167x20500214

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Irreducibility of the monodromy representation of Lauricella's $F_C$2019

    • Author(s)
      GOTO Yoshiaki、MATSUMOTO Keiji
    • Journal Title

      Hokkaido Mathematical Journal

      Volume: 48 Issue: 3 Pages: 489-512

    • DOI

      10.14492/hokmj/1573722015

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Period Map of Triple Coverings of $P^2$ and Mixed Hodge Structures2019

    • Author(s)
      Matsumoto Keiji、Terasoma Tomohide
    • Journal Title

      Publications of the Research Institute for Mathematical Sciences

      Volume: 55 Issue: 3 Pages: 489-529

    • DOI

      10.4171/prims/55-3-2

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Presentation] 超幾何保型形式2022

    • Author(s)
      松本圭司
    • Organizer
      特殊多様体・特殊関数研究会
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Monodromy representations for several hypergeometric systems by the rigidity2020

    • Author(s)
      Matsumoto Keiji
    • Organizer
      Monodromy and Hypergeometric Functions (Galatasaray University, イスタンブール, トルコ)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Monodromy representations for several hypergeometric systems by virtue of intersection forms2020

    • Author(s)
      Matsumoto Keiji
    • Organizer
      Monodromy and Hypergeometric Functions (Galatasaray University, イスタンブール, トルコ)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Schwarz's Map for Appell's F_2(1/2, (1/4, 1/4), (1/2, 1/2)) and Its Inverse2019

    • Author(s)
      Matsumoto Keiji
    • Organizer
      Colloquium at Galatasaray University (Galatasaray University, イスタンブール, トルコ)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] ある2変数超幾何微分方程式系に関 するSchwarz 写像とその逆写像2019

    • Author(s)
      松本圭司
    • Organizer
      千葉大学数学教室代数セミナー (千葉大学, 千葉市)
    • Related Report
      2019 Research-status Report
    • Invited
  • [Book] Encyclopedia of Special Functions: The Askey-Bateman Project Volume II, Chapter 32020

    • Author(s)
      Keiji Matsumoto (Edit by Tom H. Koornwinder, Jasper V. Stokman)
    • Total Pages
      22
    • Publisher
      Cambridge Univesity Press
    • ISBN
      9781107003736
    • Related Report
      2020 Research-status Report

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Published: 2019-04-18   Modified: 2025-01-30  

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