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Studies on integral representations of GKZ hypergeometric functions

Research Project

Project/Area Number 19K14554
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 12010:Basic analysis-related
Research InstitutionKumamoto University (2022)
Kobe University (2019-2021)

Principal Investigator

Matsubara-Heo Saiei-Jaeyeong  熊本大学, 大学院先端科学研究部(理), 准教授 (70834381)

Project Period (FY) 2019-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2021: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2020: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2019: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
KeywordsGKZ超幾何系 / 交叉理論 / twisted cohomology / 接続問題 / Feynman積分 / GKZ超幾何函数 / likelihood ideal / local cohomology / Mellin変換 / 積分表示 / モノドロミー表現 / 代数的de Rham cohomology / 代数統計 / 超平面配置 / Gauss-Manin系 / 急減少ホモロジー群 / 漸近展開
Outline of Research at the Start

(GKZ)超幾何函数とは、数学のみならず物理、統計学等にも出現する、基本的な数学函数である。(GKZ)超幾何函数の大域挙動を理解するためには、積分論が基本となる。本研究においては積分論を橋として、D加群、ねじれホモロジー群、組み合わせ論といった数学を結び付け、(GKZ)超幾何函数の解析的性質を明らかにすることを目指す。

Outline of Final Research Achievements

In this project, we study integral representations of the so-called GKZ hypergeometric functions. The goal is to understand its global analysis. Moreover, we also seek applications of GKZ systems to sciences.
As a result, we described a formula of analytic continuation in terms of the secondary fan. We also clarified the combinatorial nature of an invariant that integral representations have ((co)homology intersection number). Moreover, we have developed applications for Feynman integrals in quantum field theory.

Academic Significance and Societal Importance of the Research Achievements

函数の性質を理解するうえで,その函数の満たす線形偏微分方程式系を理解することは重要である.しかし,勝手なホロノミー系を考えては具体的な解析は不可能である.GKZ超幾何系は,線形偏微分方程式系の中で大域的な解析が可能と期待される有力なクラスであるとともに,一般論では捨象されている組み合わせ的構造をもつ魅力的な対象である.本研究により,GKZ系の解析接続,積分表示に関する一般論が進展した.また,これらの成果は場の量子論の研究者からも興味を持たれ,共同研究へと繋がった.

Report

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

    (39 results)

All 2023 2022 2021 2020 2019 Other

All Int'l Joint Research (5 results) Journal Article (10 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 8 results,  Open Access: 2 results) Presentation (23 results) (of which Int'l Joint Research: 6 results,  Invited: 14 results) Funded Workshop (1 results)

  • [Int'l Joint Research] Max Planck Institute for MiS(ドイツ)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] Centrum Wiskunde und Informatica(オランダ)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] Padova University(イタリア)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] Padova大学(イタリア)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] University of Seville(スペイン)

    • Related Report
      2019 Research-status Report
  • [Journal Article] Macaulay matrix for Feynman integrals: linear relations and intersection numbers2022

    • Author(s)
      Chestnov Vsevolod、Gasparotto Federico、Mandal Manoj K.、Mastrolia Pierpaolo、Matsubara-Heo Saiei J.、Munch Henrik J.、Takayama Nobuki
    • Journal Title

      Journal of High Energy Physics

      Volume: 2022 Issue: 9

    • DOI

      10.1007/jhep09(2022)187

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] AN ALGORITHM OF COMPUTING COHOMOLOGY INTERSECTION NUMBER OF HYPERGEOMETRIC INTEGRALS2021

    • Author(s)
      MATSUBARA-HEO SAIEI-JAEYEONG、TAKAYAMA NOBUKI
    • Journal Title

      Nagoya Mathematical Journal

      Volume: 未定 Pages: 1-17

    • DOI

      10.1017/nmj.2021.2

    • Related Report
      2021 Research-status Report
  • [Journal Article] Global Analysis of GG Systems2021

    • Author(s)
      Matsubara-Heo Saiei-Jaeyeong
    • Journal Title

      International Mathematics Research Notices

      Volume: 未定 Issue: 19 Pages: 14923-14963

    • DOI

      10.1093/imrn/rnab144

    • Related Report
      2021 Research-status Report
  • [Journal Article] Homology and cohomology intersection numbers of GKZ systems2021

    • Author(s)
      Goto Yoshiaki、Matsubara-Heo Saiei-Jaeyeong
    • Journal Title

      Indagationes Mathematicae

      Volume: 未定 Issue: 3 Pages: 546-580

    • DOI

      10.1016/j.indag.2021.12.002

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Algorithms for Pfaffian systems and cohomology intersection numbers of hypergeometric integrals2020

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo, Nobuki Takayama
    • Journal Title

      Lecture Notes in Computer Science 12097, ICMS 2020

      Volume: 12097 Pages: 73-84

    • DOI

      10.1007/978-3-030-52200-1_7

    • ISBN
      9783030521998, 9783030522001
    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Euler and Laplace integral representations of GKZ hypergeometric functions I2020

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo
    • Journal Title

      Proc. Japan Acad. Ser. A

      Volume: 96 Issue: 9 Pages: 75-78

    • DOI

      10.3792/pjaa.96.014

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Euler and Laplace integral representations of GKZ hypergeometric functions II2020

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo
    • Journal Title

      Proc. Japan Acad. Ser. A

      Volume: 96 Issue: 9 Pages: 79-82

    • DOI

      10.3792/pjaa.96.015

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] A tree expansion formula of a homology intersection number on the configuration space $M_{0,n}$2020

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo
    • Journal Title

      Annales of Henri Poincare

      Volume: 未定(online first) Issue: 8 Pages: 2831-2852

    • DOI

      10.1007/s00023-021-01041-4

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] On Mellin-Barnes integral representations of GKZ hypergeometric functions2020

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo
    • Journal Title

      Kyushu Journal of Mathematics

      Volume: 74

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] GG system and its application to connection problem of GKZ hypergeometric functions2020

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo
    • Journal Title

      Josai Mathematical Monographs

      Volume: 12 Pages: 69-80

    • NAID

      120006839341

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] The signature of the invariant hermitian form for a regular holonomic GKZ system2023

    • Author(s)
      松原宰栄,後藤良彰
    • Organizer
      日本数学会年会
    • Related Report
      2022 Annual Research Report
  • [Presentation] Twisted Cohomology and Likelihood Ideals2023

    • Author(s)
      松原宰栄
    • Organizer
      代数解析日大研究集会
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Twisted Cohomology and Likelihood Ideals2023

    • Author(s)
      松原宰栄
    • Organizer
      超幾何研究集会2023
    • Related Report
      2022 Annual Research Report
  • [Presentation] Hypergeometric system of contingency table2022

    • Author(s)
      松原宰栄
    • Organizer
      RIMS Symposium on Prospects in microlocal analysis and asymptotic analysis
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Euler積分をめぐって2022

    • Author(s)
      松原宰栄
    • Organizer
      函数方程式論サマーセミナー2022
    • Related Report
      2022 Annual Research Report
  • [Presentation] Restrictions of integrable connection and hypergeometric system of contingency table2022

    • Author(s)
      松原宰栄
    • Organizer
      Seminar on Nonlinear Algebra at MPIMiS
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] GKZ 系と計算代数解析2022

    • Author(s)
      松原 宰栄
    • Organizer
      超幾何方程式研究会 2022
    • Related Report
      2021 Research-status Report
  • [Presentation] 分割表の超幾何函数について2022

    • Author(s)
      松原 宰栄
    • Organizer
      Recent topics in algebraic analysis
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] 分割表の超幾何系について2022

    • Author(s)
      松原 宰栄
    • Organizer
      シミュレーションとモデリングのための計 算代数 2022
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] 超幾何交叉形式と large parameter2021

    • Author(s)
      松原 宰栄
    • Organizer
      函数方程式論サマーセミナー 2021
    • Related Report
      2021 Research-status Report
  • [Presentation] Combinatorics of a point conguration space on Riemann sphere and homology intersection numbers2021

    • Author(s)
      松原宰栄
    • Organizer
      超幾何方程式研究会2021
    • Related Report
      2020 Research-status Report
  • [Presentation] cohomology 交叉数と計算代数, その1, その22021

    • Author(s)
      松原宰栄
    • Organizer
      モデリングとシミュレーションのための計算代数
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Computing cohomology intersection numbers of GKZ systems2020

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo
    • Organizer
      ICMS2020
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Presentation] Signature of the monodromy invariant hermitian form of GKZ system2020

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo
    • Organizer
      GADEPs
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Computing cohomology intersection numbers of GKZ systems2020

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo
    • Organizer
      A-hypergeometric conference
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Computing cohomology intersection numbers of GKZ systems2020

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo
    • Organizer
      Limoges Computer Algebra Seminar
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] ombinatorics of a point conguration space on Riemann sphere and homology intersection numbers2020

    • Author(s)
      松原宰栄
    • Organizer
      多変数函数論冬セミナー
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] 二次扇の合流について2020

    • Author(s)
      松原宰栄
    • Organizer
      超幾何方程式研究会2020
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Intersection numbers of Euler-Laplace integrals2020

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo
    • Organizer
      Monodromy and Hypergeometric Functions
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] GKZ系の境界値問題とモノドロミー不変部分空間2019

    • Author(s)
      松原宰栄
    • Organizer
      函数方程式論サマー セミナー2019
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Connection problem of GKZ hypergeometric functions2019

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo
    • Organizer
      Orthogonal Polynomials, Special Functions and its Applications (OPSFA2019)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Boundary value problem and GKZ systems2019

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo
    • Organizer
      Formal and Asymptotic Solutions of Functional Equations (FASFE19)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Computing cohomology intersection numbers of GKZ hypergeometric systems2019

    • Author(s)
      Saiei-Jaeyeong Matsubara-Heo
    • Organizer
      Mathemamplitude
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Workshop on Accessory Parameters, Haraoka 652023

    • Related Report
      2022 Annual Research Report

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Published: 2019-04-18   Modified: 2024-01-30  

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