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Statistics of Random 3-manifolds

Research Project

Project/Area Number 19K14525
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 11020:Geometry-related
Research InstitutionTokyo Institute of Technology

Principal Investigator

Masai Hidetoshi  東京工業大学, 理学院, 助教 (40735734)

Project Period (FY) 2019-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2022: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2021: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywordsタイヒミュラー空間 / 双曲多様体 / 3次元多様体 / 双曲幾何学 / ランダムウォーク / 曲線複体 / 繰り込み体積 / Weil-Peterrson 幾何 / 写像類群
Outline of Research at the Start

写像類群上でランダムウォークを考えることにより、3次元多様体をランダムに生成することができる。ランダムに生成された3次元多様体の様々な不変量の統計と、写像類群の様々な空間への作用から得られる不変量の統計との比較を行う。特に写像類群のタイヒミュラー空間への作用との関係が深い伸縮因子と、ランダムに得られた3次元多様体の体積の比較を行う。伸縮因子の他にも、タイヒミュラー空間上のWeil-Petersson 距離とランダム写像類の関係性などの研究も行う。

Outline of Final Research Achievements

As a major achievement, I wrote the paper "Compactification and Distance on Teichmuller Space via Renormalized Volume." This paper discusses volume, the principal invariant that we aim to study for random 3-manifolds in this research project. By using an invariant closely related to the volume of 3-manifolds, called the renormalized volume, we defined a distance on Teichmuller space. This accomplishment pertains to the relationship of invariants between volume with drifts, the 'random version' of translation distance, which is the goal of this research project. The newly defined distance can be considered the most significant achievement during the research period.

Academic Significance and Societal Importance of the Research Achievements

本研究の主なテーマであるタイヒミュラー空間の研究は、幾何学と複素解析において重要な役割を果たし、特にリーマン面やモジュライ空間の理解に貢献している。この研究は、物理学やエンジニアリング、特に弦理論や情報科学の分野での応用が期待される。さまざまな数理モデルの構築やデータ解析の手法にも応用の可能性がある。さらに、タイヒミュラー空間の概念は画像認識や機械学習の分野でも活用され、データの構造化や解析の効率化に寄与しうるものであり、タイヒミュラー空間の研究は社会的に意義がある。

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

    (38 results)

All 2024 2023 2022 2021 2020 2019 Other

All Int'l Joint Research (3 results) Journal Article (5 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 3 results) Presentation (28 results) (of which Int'l Joint Research: 10 results,  Invited: 26 results) Book (1 results) Remarks (1 results)

  • [Int'l Joint Research] ニューヨーク市立大学(米国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] ウォーリック大学(英国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] l'institut Fourier(フランス)

    • Related Report
      2019 Research-status Report
  • [Journal Article] Quotients of the curve complex2024

    • Author(s)
      Maher Joseph、Masai Hidetoshi、Schleimer Saul
    • Journal Title

      Groups, Geometry, and Dynamics

      Volume: 18 Issue: 2 Pages: 379-405

    • DOI

      10.4171/ggd/768

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Visualizing deformations of complex structures on 4-punctured spheres2023

    • Author(s)
      Hidetoshi Masai
    • Journal Title

      RIMS Kokyuroku

      Volume: 2263 Pages: 1-5

    • Related Report
      2023 Annual Research Report
  • [Journal Article] On systoles and ortho spectrum rigidity2022

    • Author(s)
      Masai Hidetoshi、McShane Greg
    • Journal Title

      Mathematische Annalen

      Volume: 385 Issue: 1-2 Pages: 939-959

    • DOI

      10.1007/s00208-022-02365-0

    • Related Report
      2022 Research-status Report 2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A quick tour to the distance on Teichmuller space via renormalized volume2022

    • Author(s)
      Hidetoshi Masai
    • Journal Title

      RIMS 講究録

      Volume: No. 2219 Pages: 76-83

    • Related Report
      2022 Research-status Report
  • [Journal Article] On continuity of drifts of the mapping class group2021

    • Author(s)
      Masai Hidetoshi
    • Journal Title

      Mathematical Research Letters

      Volume: 28 Issue: 3 Pages: 823-847

    • DOI

      10.4310/mrl.2021.v28.n3.a8

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Presentation] On iso-orthospectrum surfaces.2024

    • Author(s)
      正井秀俊
    • Organizer
      KAIST Geometric Topology Fair 2024
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Visualizing deformations of hyperbolic and complex structures on 4-punctured spheres2023

    • Author(s)
      正井秀俊
    • Organizer
      トポロジーとコンピュータ2023
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A distance on Teichmueller space via renormalized volume2023

    • Author(s)
      正井秀俊
    • Organizer
      The 14th MSJ-SI: New Aspects of Teichmuller theory
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Visualizing deformations of hyperbolic and complex structures on 4-punctured spheres2023

    • Author(s)
      正井秀俊
    • Organizer
      Intelligence of Low-dimensional Topology
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] ビッグデータの利用と可視化, web アンケートデータを用いた水産練り製品の消費動向の解析2023

    • Author(s)
      正井秀俊
    • Organizer
      web アンケートデータを用いた水産練り製品の消費動向の解析
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] On distances on distances2023

    • Author(s)
      正井秀俊
    • Organizer
      福岡大学トポロジーセミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Visualizing deformations of hyperbolic and complex structures on 4-punctured spheres2023

    • Author(s)
      正井秀俊
    • Organizer
      ウプサラ・東工大シンポジウム
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Visualizing deformations of hyperbolic and complex structures on 4-punctured spheres.2023

    • Author(s)
      正井秀俊
    • Organizer
      Friday Seminar on Knot Theory
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] A distance on Teichmueller space via renormalized volume2022

    • Author(s)
      正井秀俊
    • Organizer
      YMSC Topology Seminar
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] 4 点穴あき球面のタイヒミュラー測地線に沿った複素変形2022

    • Author(s)
      正井秀俊
    • Organizer
      日本数学会
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] 繰り込み体積とタイヒミュラー空間上の距離2022

    • Author(s)
      正井秀俊
    • Organizer
      第65回 函数論シンポジウム
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] 距離のレシピとそのタイヒミュラー理論への応用2022

    • Author(s)
      正井秀俊
    • Organizer
      東北大学幾何セミナー
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] 写像類群上のランダムウォークについて2022

    • Author(s)
      正井秀俊
    • Organizer
      Random topics on Teichmuller theory
    • Related Report
      2022 Research-status Report
  • [Presentation] A note on distances and horofunctions2022

    • Author(s)
      正井秀俊
    • Organizer
      Geometry in Low dimensions 2022
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Quotients of the curve complex2021

    • Author(s)
      正井秀俊
    • Organizer
      Geometry of discrete groups and hyperbolic spaces
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] タイヒミュラー空間のホロ境界2021

    • Author(s)
      正井秀俊
    • Organizer
      トポロジーシンポジウム
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Continuity of the drift of mapping class group2021

    • Author(s)
      正井秀俊
    • Organizer
      Geometric Group Theory in East Asia(GGTEA)
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Complete exceptional surgeries on two-bridge links2021

    • Author(s)
      正井秀俊
    • Organizer
      The 16th East Asian Conference on Geometric Topology
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Presentation] 曲線複体の商空間2021

    • Author(s)
      正井秀俊
    • Organizer
      2020年度「リーマン面・不連続群論」研究集会
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] 3次元多様体の(非)双曲性の計算機による判定2021

    • Author(s)
      正井秀俊
    • Organizer
      数学ソフトウェアとフリードキュメント 32
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] On systoles and ortho spectrum rigidity2020

    • Author(s)
      正井秀俊
    • Organizer
      阪大トポロジーセミナー
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Quotients of curve complex2020

    • Author(s)
      正井秀俊
    • Organizer
      京都大学微分トポロジーセミナー
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] On systoles and ortho spectrum rigidity2020

    • Author(s)
      正井秀俊
    • Organizer
      NUS Topology and Geometry Seminar
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On systoles and ortho spectrum rigidity2020

    • Author(s)
      正井秀俊
    • Organizer
      2019年度「リーマン面・不連続群論」研究集会
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] On continuity of drifts of the mapping class group2019

    • Author(s)
      正井秀俊
    • Organizer
      首都大学幾何セミナー
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] On the Continuity of Drifts of Mapping Class Groups2019

    • Author(s)
      正井秀俊
    • Organizer
      Geometric Topology Fair 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On systoles and ortho spectrum rigidity2019

    • Author(s)
      正井秀俊
    • Organizer
      Seminar on Geometry
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On rigidity of length-related spectra2019

    • Author(s)
      正井秀俊
    • Organizer
      The 6th Tokyo Tech - Uppsala University Joint Symposium
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Book] 群と幾何をみる : 無限の彼方から2023

    • Author(s)
      正井 秀俊
    • Total Pages
      192
    • Publisher
      日本評論社
    • ISBN
      4535789894
    • Related Report
      2023 Annual Research Report
  • [Remarks]

    • URL

      https://sites.google.com/view/hidetoshimasai/home

    • Related Report
      2020 Research-status Report

URL: 

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

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