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Study of rigidity of foliations based on global geometry of leaves

Research Project

Project/Area Number 20K03620
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

Nozawa Hiraku  立命館大学, 理工学部, 准教授 (80706557)

Project Period (FY) 2020-04-01 – 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 2023: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2022: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2021: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords葉層構造 / 力学系 / カオス / タイル張り / トポロジー / 群作用 / 情報幾何 / 剛性 / 微分幾何 / グラフ理論 / 調和測度 / 剛性理論 / 十分統計量 / フルネ標構 / 微分位相幾何 / カントール集合 / タイル貼り
Outline of Research at the Start

葉層構造とは,与えられた空間のより次元の低い空間(葉)への分割であって,各点の十分近くでは直積の構造を持つ幾何構造のことである.与えた幾何的条件を満たす葉層構造が特別な対称性を持つとき,それらの葉層構造は剛性を持つという.剛性を持つ葉層構造の知られている例には多くないが,いずれも豊富な幾何を内包する興味深い対象となっている. 本研究では, 葉の大域幾何的性質に着目した上で,剛性を持つ葉層構造の新たな例を構成し,分類問題やタイル貼りの研究に応用する.

Outline of Final Research Achievements

I conducted research on the rigidity of foliations from various viewpoints. We studied the surface group actions on the circle to obtain rigidity results of the harmonic measures of their suspension foliations using Thurston's connection. As an application, we provided alternative proofs for well-known rigidity theorems, such as the Milnor-Wood inequality and the Matsumoto rigidity theorem generalized by Burger-Iozzi-Wienhard to surfaces with cusps. Additionally, I constructed new examples of Delone sets (uniformly scattered point sets) in symmetric spaces of non-compact type by generalizing the cut-and-project method. Furthermore, we constructed new examples of equicontinuous group actions on the Cantor set and investigated the chaotic properties of colored graphs from a topological viewpoint.

Academic Significance and Societal Importance of the Research Achievements

葉層構造は群作用,偏微分方程式などの自然現象をとらえるための理論において現れる幾何的対象である.本研究では,葉層構造の中でも特に対称性の高いものやカオス的な性質を持つものに注目し,様々な研究を行った.これらの葉層構造はリーマン面の幾何学などの古典的数学にも現れ,幾何学,代数学,複素解析や物理学などにおいて研究されている.本研究の結果はこれらの関連分野の研究を深めるために意義があると考えられる.

Report

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

    (32 results)

All 2024 2023 2022 2021 2020 Other

All Int'l Joint Research (13 results) Journal Article (7 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 6 results) Presentation (11 results) (of which Int'l Joint Research: 6 results,  Invited: 5 results) Funded Workshop (1 results)

  • [Int'l Joint Research] マドリッド工科大学/サンティアゴ・デ・コンポステラ大学/ヴィーゴ大学(スペイン)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] マルセイユ大学(フランス)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] ダラム大学(英国)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] ライデン大学(オランダ)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] サンティアゴ・デ・コンポステラ大学/Universidad Politecnica de Madrid/ヴィーゴ大学(スペイン)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] ダラム大学(英国)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] マルセイユ大学(フランス)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] マルセイユ大学(フランス)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] サンティアゴ・デ・コンポステラ大学/ヴィーゴ大学(スペイン)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] 南部ブルターニュ大学(フランス)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] サンティアゴ・デ・コンポステラ大学(スペイン)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] ウィーン大学(オーストリア)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] ダラム大学(英国)

    • Related Report
      2020 Research-status Report
  • [Journal Article] Generalized Bishop frames of regular curves in E^42024

    • Author(s)
      Subaru Nomoto, Hiraku Nozawa
    • Journal Title

      Hokkaido Mathematical Journal

      Volume: 53 Issue: 1 Pages: 71-89

    • DOI

      10.14492/hokmj/2022-611

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] An Ay-Le-Jost-Schwachhofer type characterization of quantitatively weakly sufficient statistics2024

    • Author(s)
      Kaori Yamaguchi, Hiraku Nozawa
    • Journal Title

      Springer Proceedings in Mathematics & Statistics

      Volume: 440 Pages: 249-257

    • DOI

      10.1007/978-3-031-50586-7_13

    • ISBN
      9783031505850, 9783031505867
    • Related Report
      2023 Annual Research Report
  • [Journal Article] Harmonic measures and rigidity for surface group actions on the circle2024

    • Author(s)
      Masanori Adachi, Yoshifumi Matsuda, Hiraku Nozawa
    • Journal Title

      Algebraic & Geometric Topology

      Volume: -

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Generalized Bishop frams of regular curves in E^42022

    • Author(s)
      Subaru Nomoto, Hiraku Nozawa
    • Journal Title

      Hokkaido Mathematical Journal

      Volume: -

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Wild Cantor actions2021

    • Author(s)
      Jesus Antonio Alvarez Lopez, Ramon Barral Lijo;, Olga Lukina, Hiraku Nozawa
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: -

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Coarse distinguishability of graphs with symmetric growth2021

    • Author(s)
      Jesus Antonio Alvarez Lopez, Ramon Barral Lijo, Hiraku Nozawa
    • Journal Title

      Ars Mathematica Contemporanea

      Volume: - Issue: 1 Pages: #P1.06-#P1.06

    • DOI

      10.26493/1855-3974.2354.616

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Chaotic Delone sets2021

    • Author(s)
      Jesus Antonio Alvarez Lopez, Ramon Barral Lijo, John Hunton, Hiraku Nozawa, John R. Parker
    • Journal Title

      Discrete & Continuous Dynamical Systems Series A

      Volume: 41(8) Issue: 8 Pages: 3781-3796

    • DOI

      10.3934/dcds.2021016

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Harmonic measures and rigidity for surface group actions on the circle2024

    • Author(s)
      Masanori Adachi, Yoshifumi Matsuda, Hiraku Nozawa
    • Organizer
      Foliated Spaces, Tilings and Group Actions 2024
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Harmonic measures and rigidity for the Fuchsian actions of surface groups on the circle2024

    • Author(s)
      Masanori Adachi, Yoshifumi Matsuda, Hiraku Nozawa
    • Organizer
      Low dimensional topology and number theory XV
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 曲面群の円周への作用の剛性と調 和測度について(足立真訓(静岡 大),松田能文(青山学院大)と の共同研究2023

    • Author(s)
      足立 正訓、松田能文、野澤 啓
    • Organizer
      第70回トポロジーシンポジウ ム
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Harmonic measures and rigidity for surface group actions on the circle2023

    • Author(s)
      Masanori Adachi, Yoshifumi Matsuda, Hiraku Nozawa
    • Organizer
      the Fourth International Workshop "Differential Geometric Structures and Applications"
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Harmonic measures and rigidity for surface group actions on the circle2023

    • Author(s)
      足立 真訓,松田 能文,野澤 啓
    • Organizer
      日本数学会2023年度年会
    • Related Report
      2022 Research-status Report
  • [Presentation] タイル張りやグラフに現れるカオスについて2022

    • Author(s)
      Jesus Antonio Alvarez Lopez, Ramon Barral Lijo, John Hunton, Hiraku Nozawa, John Parker
    • Organizer
      筑波大学談話会
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Rigidity for transverse foliations of Seifert 3-manifolds2022

    • Author(s)
      足立 真訓,松田 能文,野澤 啓
    • Organizer
      葉層構造の幾何学とその応用
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] カントール空間への同程度連続な群作用2022

    • Author(s)
      Jesus Alvarez Lopez, Ramon Barral Lijo, Olga Lukina, Hiraku Nozawa
    • Organizer
      接触構造、特異点、微分方程式及びその周辺
    • Related Report
      2021 Research-status Report
  • [Presentation] Chaotic colored graphs and foliated spaces2021

    • Author(s)
      ラモン バラル リホ,野澤 啓
    • Organizer
      葉層構造論シンポジウム
    • Related Report
      2021 Research-status Report
  • [Presentation] 重心写像とリー葉層構造2021

    • Author(s)
      Gael Meigniez, Hiraku Nozawa
    • Organizer
      接触構造、特異点、微分方程式及びその周辺
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Presentation] Rigidity of Lie foliations with locally symmetric leaves2020

    • Author(s)
      Gael Meigniez, Hiraku Nozawa
    • Organizer
      Topological Methods in Dynamics and Related Topics - 2020
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Funded Workshop] Foliated spaces, Tilings and Group actions 20222022

    • Related Report
      2022 Research-status Report

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Published: 2020-04-28   Modified: 2025-01-30  

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