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Constructing geometric representations of finite groups through equivariant topology

Research Project

Project/Area Number 18K03304
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11020:Geometry-related
Research InstitutionKyushu University

Principal Investigator

Kaji Shizuo  九州大学, マス・フォア・インダストリ研究所, 教授 (00509656)

Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2022: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords同変トポロジー / 計算トポロジー / ワイル群 / 旗多様体 / コンパクトリー群 / 有限群のコホモロジー / 実トーリック多様体 / パーシステントホモロジー / 位相幾何学
Outline of Final Research Achievements

The most interesting result obtained during the research period is the determination of the action of the Weil group on the cohomology of the real toric manifolds associated with root systems. As a byproduct, a topological realisation of a combinatorial object known as the generalised Euler's zigzag numbers were obtained.
Throughout the research period, a number of algorithms for computing concrete examples regarding flag manifolds, real toric manifolds, and Weyl groups were developed.
Also, computer programmes were produced and made publicly available as open-source software, and they have been used in various fields outside of mathematics.

Academic Significance and Societal Importance of the Research Achievements

旗多様体や実トーリック多様体といった空間のトポロジーと組合せ論を協調させて解析する実例を複数提供したことが,同変トポロジー分野における学術的意義といえる.
また,計算アルゴリズムを開発しその実装を公開したことで,今後具体例から新たな知見が得られることが期待される.
応用として,機械学習や画像解析の手法を開発し,こちらもその実装を公開している.実際に医療・材料・データ解析などに利用されており,社会還元がなされている.

Report

(6 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (24 results)

All 2023 2022 2021 2020 2019 2018 Other

All Int'l Joint Research (4 results) Journal Article (9 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 9 results,  Open Access: 3 results) Presentation (6 results) (of which Int'l Joint Research: 5 results,  Invited: 2 results) Remarks (5 results)

  • [Int'l Joint Research] Ajou University(韓国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] University of Southampton(英国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Ajou University(韓国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] University of Ibadan(ナイジェリア)

    • Related Report
      2018 Research-status Report
  • [Journal Article] Alexandrov Puzzle2022

    • Author(s)
      Kodai Takenaga and Shizuo Kaji
    • Journal Title

      Proceedings of Bridges 2022: Mathematics, Art, Music, Architecture, Culture

      Volume: - Pages: 371-374

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Causal inference for empirical dynamical systems based on persistent homology2022

    • Author(s)
      Bando Hiroaki、Kaji Shizuo、Yaguchi Takaharu
    • Journal Title

      JSIAM Letters

      Volume: 14 Issue: 0 Pages: 69-72

    • DOI

      10.14495/jsiaml.14.69

    • ISSN
      1883-0609, 1883-0617
    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A Hyper-surface Arrangement Model of Ranking Distributions2021

    • Author(s)
      Kaji Shizuo、Horiguchi Akira、Abe Takuro、Watanabe Yohsuke
    • Journal Title

      Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining

      Volume: - Pages: 796-804

    • DOI

      10.1145/3447548.3467253

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Nested Subspace Arrangement for Representation of Relational Data2020

    • Author(s)
      Nozomi Hata, Shizuo Kaji, Akihiro Yoshida, and Katsuki Fujisawa
    • Journal Title

      Proceedings of the thirty-seventh International Conference on Machine Learning (ICML2020)

      Volume: 119 Pages: 4127-4137

    • NAID

      120007132124

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Flag Bott manifolds of general Lie type and their equivariant cohomology rings2020

    • Author(s)
      Kaji Shizuo、Kuroki Shintaro、Lee Eunjeong、Suh Dong Youp
    • Journal Title

      Homology, Homotopy and Applications

      Volume: 22 Issue: 1 Pages: 375-390

    • DOI

      10.4310/hha.2020.v22.n1.a21

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Representing a point and the diagonal as zero loci in flag manifolds2019

    • Author(s)
      Kaji Shizuo
    • Journal Title

      Algebraic & Geometric Topology

      Volume: 19 Issue: 4 Pages: 2061-2075

    • DOI

      10.2140/agt.2019.19.2061

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] The Cohomology Groups of Real Toric Varieties Associated with Weyl Chambers of Types C and D2019

    • Author(s)
      Choi Suyoung、Kaji Shizuo、Park Hanchul
    • Journal Title

      Proceedings of the Edinburgh Mathematical Society

      Volume: 62 Issue: 3 Pages: 861-874

    • DOI

      10.1017/s001309151800086x

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Geometric representations of finite groups on real toric spaces2019

    • Author(s)
      Soojin Cho, Suyoung Choi, and Shizuo Kaji
    • Journal Title

      J. Korean Math. Soc.

      Volume: 56-5 Pages: 1265-1283

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Suspension Splittings and Self-maps of Flag Manifolds2019

    • Author(s)
      Kaji Shizuo、Theriault Stephen
    • Journal Title

      Acta Mathematica Sinica, English Series

      Volume: 35 Issue: 4 Pages: 445-462

    • DOI

      10.1007/s10114-019-8051-z

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] The topology and geometry of a closed kinematic linkage2023

    • Author(s)
      Shizuo Kaji
    • Organizer
      The 11th International Career Days at Eastern Mediterranean University
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Introduction to Persistent Homology for Graph Analysis2022

    • Author(s)
      Shizuo Kaji
    • Organizer
      The 5th International Conference on Econometrics and Statistics (EcoSta 2022)
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Modelling preference with hyperplane arrangement2022

    • Author(s)
      Shizuo Kaji
    • Organizer
      Statistics and Mathematical Modelling in Combination
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Tutorial on CubicalRipser and other TDA software using Python2021

    • Author(s)
      S. Kaji
    • Organizer
      POSTECH MINDS & PIAI Workshop on Topological Data Analysis and Machine Learning
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research
  • [Presentation] Geometry of the configuration space of Kaleidocycles2020

    • Author(s)
      Shizuo KAJI
    • Organizer
      African Mathematics Seminar
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 同変ループ積について2018

    • Author(s)
      鍛冶静雄
    • Organizer
      日本数学会2018年度秋季総合分科会
    • Related Report
      2018 Research-status Report
    • Invited
  • [Remarks] Shizuo Kaji's homepage

    • URL

      https://www.skaji.org

    • Related Report
      2022 Annual Research Report
  • [Remarks] Shizuo Kaji's web page

    • URL

      https://www.skaji.org/

    • Related Report
      2021 Research-status Report
  • [Remarks] Shizuo Kaji's Web Page

    • URL

      https://www.skaji.org/

    • Related Report
      2020 Research-status Report
  • [Remarks] Shizuo KAJI's web page

    • URL

      https://www.skaji.org/

    • Related Report
      2019 Research-status Report
  • [Remarks] Shizuo KAJI's homepage

    • URL

      http://www.skaji.org/

    • Related Report
      2018 Research-status Report

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Published: 2018-04-23   Modified: 2024-01-30  

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