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3次元多様体の諸性質のランダムネスから見る特徴付け

Research Project

Project/Area Number 15J08142
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section国内
Research Field Geometry
Research InstitutionThe University of Tokyo

Principal Investigator

正井 秀俊  東京大学, 数理科学研究科, 特別研究員(PD)

Project Period (FY) 2015-04-24 – 2018-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords写像類群 / ランダムウォーク / 3次元多様体 / 位相的エントロピー
Outline of Annual Research Achievements

本年度は,昨年度に引き続きコンパクトな曲面の写像類群上のランダムウォークについて研究した.
ランダムウォークで得られる元をランダム写像類と呼ぶ.以降,ランダム写像類に関する主張は漸近的に確率1で成り立つ場合に,断定の形で述べる.J. Maherにより,ランダム写像類は擬アノソフとなることが知られていた.写像類に対しては写像のべき,持ち上げにより被覆関係が定義される.そのような被覆で定義された被覆関係において,写像類群上に通約関係を定義する事ができる.本年度はランダム写像類の通約性についてまとめた論文「Fibered commensurability and arithmeticity of random mapping tori」がジャーナル「Groups, Geometry, and Dynamics」に受理され,最終校正などをおこなった.
ランダムウォークの曲面上の力学系についても研究を行った.曲面上の微分同相写像に対して,位相的エントロピーと呼ばれる,写像のベキにより区別されうる軌道の数の指数的増大度を測る不変量が定義されている.ランダムウォークに対しても微分同相写像のアナロジーとして位相的エントロピーが定義できる.Thurston は擬アノソフ微分同相写像の位相的エントロピーはタイヒミュラー空間上のtranslation の長さと一致することを示していた.その値は,曲面上の任意リーマン計量における,任意の本質的単純閉曲線の長さの指数的増大度とも一致する事も知られている.本年度の研究では,ランダムウォークに関しても同様の結果が成り立つ事を示した.本結果により,Thurston の一連の結果の``ランダム化”ができたと言える.

Research Progress Status

翌年度、交付申請を辞退するため、記入しない。

Strategy for Future Research Activity

翌年度、交付申請を辞退するため、記入しない。

Report

(2 results)
  • 2016 Annual Research Report
  • 2015 Annual Research Report
  • Research Products

    (26 results)

All 2017 2016 2015

All Journal Article (4 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 4 results,  Acknowledgement Compliant: 3 results) Presentation (22 results) (of which Int'l Joint Research: 8 results,  Invited: 22 results)

  • [Journal Article] Fibered commensurability and arithmeticity of random mapping tori2017

    • Author(s)
      Hidetoshi Masai
    • Journal Title

      Groups, Geometry, and Dynamics

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Topological Entropy of Random Walks on Mapping Class Groups2017

    • Author(s)
      Hidetoshi Masai
    • Journal Title

      International Mathematics Research Notices

      Volume: 印刷中 Pages: rnw252-rnw252

    • DOI

      10.1093/imrn/rnw252

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] On commensurability of quadratic differentials on surfaces2017

    • Author(s)
      Hidetoshi Masai
    • Journal Title

      RIMS Kokyuroku Bessatsu.

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Verified Computations for Hyperbolic 3-Manifolds2016

    • Author(s)
      Neil Hoffman, Kazuhiro Ichihara, Masahide Kashiwagi, Hidetoshi Masai, Shin’ichi Oishi, Akitoshi Takayasu
    • Journal Title

      Experimental Mathematics

      Volume: 25 Issue: 1 Pages: 66-78

    • DOI

      10.1080/10586458.2015.1029599

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] 正則2次微分の通約類と擬アノソフの通約性2017

    • Author(s)
      正井秀俊
    • Organizer
      日大セミナー
    • Place of Presentation
      日本大学( 東京都世田谷区)
    • Year and Date
      2017-02-17
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Symmetry and dynamics of random walks on the mapping class groups II2017

    • Author(s)
      正井秀俊
    • Organizer
      The second Australia-Japan Geometry, Analysis and their Applications
    • Place of Presentation
      京都大学(京都府京都市)
    • Year and Date
      2017-02-03
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Symmetry and dynamics of random walks on the mapping class groups I2017

    • Author(s)
      正井秀俊
    • Organizer
      The second Australia-Japan Geometry, Analysis and their Applications
    • Place of Presentation
      京都大学(京都府京都市)
    • Year and Date
      2017-02-01
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Topological entropy of random walks on the mapping class groups2017

    • Author(s)
      正井秀俊
    • Organizer
      2016年度「リーマン面・不連続群論」研究集会
    • Place of Presentation
      東北大学片平キャンパス(宮城県仙台市)
    • Year and Date
      2017-01-08
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 低次元トポロジーと幾何2016

    • Author(s)
      正井秀俊
    • Organizer
      日本応用数理学会三部会連携「応用数理セミナー」
    • Place of Presentation
      東京大学本郷キャンパス(東京都文京区)
    • Year and Date
      2016-12-27
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] ハンドル体結び目補空間の双曲構造入門2016

    • Author(s)
      正井秀俊
    • Organizer
      ハンドル体結び目とその周辺 IX
    • Place of Presentation
      近畿大学東大阪キャンパス(大阪府東大阪市)
    • Year and Date
      2016-10-09
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] 写像類群上のランダムウォーク2016

    • Author(s)
      正井秀俊
    • Organizer
      トポロジーシンポジウム2016
    • Place of Presentation
      神戸大学(兵庫県神戸市)
    • Year and Date
      2016-07-07
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Some dynamics of random walks on the mapping class groups2016

    • Author(s)
      正井秀俊
    • Organizer
      火曜トポロジーセミナー
    • Place of Presentation
      東京大学(東京都目黒区)
    • Year and Date
      2016-05-17
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Closed random mapping tori are asymmetric2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      Geometry Colloquium
    • Place of Presentation
      University of Tokyo(東京都,目黒区)
    • Year and Date
      2015-11-13
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Experiments on random braids2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      Topology and Computer 2015
    • Place of Presentation
      Nihon University(東京都,世田谷区)
    • Year and Date
      2015-11-06
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Closed random mapping tori are asymmetric2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      Hiroshima University Topology-Geometry Seminar
    • Place of Presentation
      Hiroshima University(広島県,東広島市)
    • Year and Date
      2015-10-20
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] HIKMOT, hyperbolicity verifier, and its applications2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      Topology Friday Seminar
    • Place of Presentation
      Kyushu University(福岡県,福岡市)
    • Year and Date
      2015-10-16
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] How to prove hyperbolicity of given 3-manifolds by computer2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      計算代数システムによる新しい数学の開拓と発展
    • Place of Presentation
      Kyoto University(京都府,京都市)
    • Year and Date
      2015-10-01
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Fibered commensurability of random mapping classes2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      リーマン面に関連する位相幾何学
    • Place of Presentation
      University of Tokyo(東京都,目黒区)
    • Year and Date
      2015-08-27
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] ランダムな3次元多様体について2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      Tsukuba
    • Place of Presentation
      Tsukuba University(茨城県,つくば市)
    • Year and Date
      2015-07-18
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Fibered commensurability of random mapping classes2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      Friday seminar
    • Place of Presentation
      Osaka city university(大阪府,大阪市)
    • Year and Date
      2015-07-10
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] HIKMOT, hyperbolicity verifier, and its applications2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      Topology seminar
    • Place of Presentation
      University of Grenoble(フランス,グルノーブル)
    • Year and Date
      2015-06-26
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Harmonic measure of the set of lifted projective foliations2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      Lyon-Tokyo symposium
    • Place of Presentation
      Unite de Mathematiques Pures et Appliquees de l'ENS Lyon(フランス,Lyon)
    • Year and Date
      2015-06-24
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Finite coverings and random walks on the mapping class groups2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      Analysis and Geometry of Discrete Groups and Hyperbolic Spaces
    • Place of Presentation
      Kyoto University(京都府,京都市)
    • Year and Date
      2015-06-22
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Fibered commensurability of random mapping classes2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      早稲田大学双曲幾何幾何学的群論セミナー
    • Place of Presentation
      Waseda University(東京都,新宿区)
    • Year and Date
      2015-06-12
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Harmonic measure of the set of lifted projective measured foliations2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      Topology, Geometry and Algebra of low-dimensional manifolds
    • Place of Presentation
      KKR Numazu Hamayu(静岡県,沼津市)
    • Year and Date
      2015-05-28
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Verify hyperbolicity by computer, and its applications2015

    • Author(s)
      Hidetoshi Masai
    • Organizer
      Tuesday Seminar on Topology
    • Place of Presentation
      University of Tokyo(東京都,目黒区)
    • Year and Date
      2015-04-28
    • Related Report
      2015 Annual Research Report
    • Invited

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Published: 2015-11-26   Modified: 2024-03-26  

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