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Geometric study of a sequence of spaces with unbounded dimension

Research Project

Project/Area Number 26400060
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionTohoku University

Principal Investigator

Shioya Takashi  東北大学, 理学研究科, 教授 (90235507)

Project Period (FY) 2014-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords測度の集中現象 / 測度距離空間 / オブザーバブル直径 / オブザーバブル距離 / セパレーション距離 / 等周不等式 / アレクサンドロフ空間 / 負曲率多様体 / グラフ多様体 / エンド / オブザーバブル分散 / 楕円 / ガウス空間 / リーマン多様体 / スティーフェル多様体 / 旗多様体 / 高次元極限 / ピラミッド / 無限次元空間 / 等径定数
Outline of Final Research Achievements

Based on the study of concentration of measure phenomenon, Gromov proposed a new geometric theory for metric measure spaces. One of main motivation to study this theory is to investigate a sequence of spaces with unbounded dimension. In our study, we develop and deepen the theory. The concentration of measure phenomenon can be considered as a geometric variant of the law of large numbers. We study a geometric variant of the central limit theorem, which appears as an analog of phese transition phenomenon. One of examples is the sequence of spheres with unbounded dimension. If the radius of the sphere has small order, then we observe the concentration of measure phenomenon. If the radius has large order, then we observe the dissipation phenomenon. If the radius has the order of the square root of the dimension, then we see that the sphere converges to the infinite-dimensional space with Gaussian measure. We have proved this kind of phenomenon for many other spaces.

Academic Significance and Societal Importance of the Research Achievements

空間列の収束や漸近的挙動を調べた幾何学的な研究である.従来の空間のグロモフ・ハウスドルフ収束の研究では,幾何群論と微分幾何の問題であった.しかし本研究は,幾何学のみならず,確率論,統計学,解析学,統計力学などと関係した非常に興味深い研究であり,国内外で学術的に高く評価されている.従来の研究とは異なった独創的な方法で高次元および無限次元空間へアプローチするものであり,今後の進展が期待されている.

Report

(6 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (30 results)

All 2019 2018 2017 2016 2015 2014

All Journal Article (8 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 8 results,  Open Access: 4 results,  Acknowledgement Compliant: 2 results) Presentation (17 results) (of which Int'l Joint Research: 10 results,  Invited: 16 results) Book (4 results) Funded Workshop (1 results)

  • [Journal Article] Isoperimetric rigidity and distributions of 1-Lipschitz functions2019

    • Author(s)
      Nakajima Hiroki、Shioya Takashi
    • Journal Title

      Advances in Mathematics

      Volume: 349 Pages: 1198-1233

    • DOI

      10.1016/j.aim.2019.04.043

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 測度距離幾何学---高次元および無限次元空間へのアプローチ---2019

    • Author(s)
      塩谷 隆
    • Journal Title

      数学

      Volume: 71 Pages: 159-177

    • NAID

      130008029698

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] High-dimensional metric-measure limit of Stiefel and flag manifolds2018

    • Author(s)
      Shioya Takashi、Takatsu Asuka
    • Journal Title

      Mathematische Zeitschrift

      Volume: - Issue: 3-4 Pages: 1-35

    • DOI

      10.1007/s00209-018-2044-y

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] 測度距離幾何学~高次元および無限次元空間へのアプローチ~2018

    • Author(s)
      塩谷 隆
    • Journal Title

      日本数学会「数学」

      Volume: 印刷中

    • NAID

      130008029698

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Metric measure limits of spheres and complex projective spaces2017

    • Author(s)
      Shioya, Takashi
    • Journal Title

      Measure theory in non-smooth spaces, 261-287, Partial Differ. Equ. Meas. Theory, De Gruyter Open, Warsaw, 2017.

      Volume: - Pages: 261-287

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Limit formulas for metric measure invariants and phase transition property2015

    • Author(s)
      R. Ozawa and T. Shioya
    • Journal Title

      Math. Z

      Volume: 280 Issue: 3-4 Pages: 759-782

    • DOI

      10.1007/s00209-015-1447-2

    • Related Report
      2015 Research-status Report 2014 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Estimate of observable diameter of $l_{p}$-product spaces2015

    • Author(s)
      R. Ozawa and T. Shioya
    • Journal Title

      Manuscripta Math

      Volume: 147 Issue: 3-4 Pages: 501-509

    • DOI

      10.1007/s00229-015-0730-1

    • Related Report
      2015 Research-status Report 2014 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Locally strongly Lipschitz contractibility of Alexandrov spaces2014

    • Author(s)
      A. Mitsuishi and T. Yamaguchi
    • Journal Title

      Pacific J.

      Volume: 270 Issue: 1 Pages: 393-421

    • DOI

      10.1007/s10711-014-0017-9

    • Related Report
      2015 Research-status Report 2014 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] Isoperimetric rigidity and distributions of 1-Lipschitz functions2018

    • Author(s)
      Takashi Shioya
    • Organizer
      第4回日中幾何学研究集会
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Isoperimetric rigidity and distributions of 1-Lipschitz functions2018

    • Author(s)
      Shioya, Takashi
    • Organizer
      Global Differential Geometry Workshop, Tsinghua Sanya International Mathematics Forum, サンヤ(中国)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Isoperimetric rigidity and distributions of 1-Lipschitz functions2018

    • Author(s)
      Shioya, Takashi
    • Organizer
      Conference on "Metrics and Measures", 東北大学
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Soft isoperimetric rigidity2017

    • Author(s)
      Shioya, Takashi
    • Organizer
      Metric Measure Spaces and Ricci Curvature Max Planck Institute for Mathematics, Bonn
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] High-dimensional spaces in metric measure geometry2017

    • Author(s)
      Shioya, Takashi
    • Organizer
      Geometric Analysis on smooth and non-smooth spaces SISSA, トリエステ,イタリア
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Convergence of metric measure spaces2016

    • Author(s)
      Shioya, Takashi
    • Organizer
      Metric Geometry and Its Applications
    • Place of Presentation
      復旦大学,上海,中国
    • Year and Date
      2016-02-22
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] High-dimensional spaces in metric measure geometry2016

    • Author(s)
      Takashi Shioya
    • Organizer
      Workshop on Perspectives in Geometric Analysis (Part A)
    • Place of Presentation
      北京大学,中国
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Metric measure geometry and concentration of measure2016

    • Author(s)
      Takashi Shioya
    • Organizer
      5th Japanese-German University Presidents' Conference (HeKKSaGOn)
    • Place of Presentation
      カールスルーエ工科大学,ドイツ
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] High-dimensional spaces in metric measure geometry2016

    • Author(s)
      Takashi Shioya
    • Organizer
      HeKKSaGOn Mini-Workshop 2016
    • Place of Presentation
      カールスルーエ工科大学,ドイツ
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Convergence of metric measure spaces2015

    • Author(s)
      Shioya, Takashi
    • Organizer
      Workshop on Analysis and Geometry in Metric Spaces
    • Place of Presentation
      ICMAT, Campus de Cantoblanco, Madrid, Spain
    • Year and Date
      2015-06-04
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] What is going on in metric measure geometry?2015

    • Author(s)
      Shioya, Takashi
    • Organizer
      HeKKSaGOn会議
    • Place of Presentation
      東北大学数理科学記念館
    • Year and Date
      2015-04-18
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 閉曲面の等径定数の評価2015

    • Author(s)
      塩谷 隆
    • Organizer
      日本数学会・2015年度年会
    • Place of Presentation
      明治大学駿河台キャンパスリバティタワー
    • Year and Date
      2015-03-24
    • Related Report
      2014 Research-status Report
  • [Presentation] Concentration, convergence and dissipation of spaces2015

    • Author(s)
      T. Shioya
    • Organizer
      Mini-workshop on Differential Geometry
    • Place of Presentation
      東北大学大学院理学研究科 川井ホール(仙台市)
    • Year and Date
      2015-03-19
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Concentration, convergence and dissipation of spaces2015

    • Author(s)
      T. Shioya
    • Organizer
      New Trends in Optimal Transport
    • Place of Presentation
      University of Bonn (ドイツ)
    • Year and Date
      2015-03-03
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Metric measure limits of spheres and complex projective spaces2014

    • Author(s)
      T. Shioya
    • Organizer
      ERC conference on Optimal Transportation and Applications
    • Place of Presentation
      Centro De Giorgi, Scuola Normale Superiore (イタリア)
    • Year and Date
      2014-10-28
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Phase transition in metric measure geometry2014

    • Author(s)
      T. Shioya
    • Organizer
      Geometry Seminar(2回連続)
    • Place of Presentation
      Fudan University (中国)
    • Year and Date
      2014-09-15 – 2014-09-18
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Concentration, convergence, and dissipation of spaces2014

    • Author(s)
      T. Shioya
    • Organizer
      The 10th Geometry Conference for the Friendship between China and Japan
    • Place of Presentation
      Fudan University (中国)
    • Year and Date
      2014-09-07
    • Related Report
      2014 Research-status Report
    • Invited
  • [Book] 最適輸送理論とリッチ曲率2017

    • Author(s)
      桑江一洋,塩谷隆,太田慎一,桑田和正,高津飛鳥
    • Total Pages
      141
    • Publisher
      日本数学会
    • Related Report
      2016 Research-status Report
  • [Book] "Metric measure limits of spheres and complex projective spaces" in "Measure Theory in Non-Smooth Spaces"2017

    • Author(s)
      Takashi Shioya
    • Total Pages
      24
    • Publisher
      De Gruiter Open
    • Related Report
      2016 Research-status Report
  • [Book] Metric Measure Geometry2016

    • Author(s)
      Shioya, Takashi
    • Publisher
      European Mathematical Society
    • Related Report
      2015 Research-status Report
  • [Book] Metric measure geometry2016

    • Author(s)
      T. Shioya
    • Publisher
      the IRMA series of the European Mathematical Society
    • Related Report
      2014 Research-status Report
  • [Funded Workshop] Geometric Analysis on Riemannian and Metric Spaces2016

    • Place of Presentation
      京都大学 数理解析研究所
    • Year and Date
      2016-09-05
    • Related Report
      2016 Research-status Report

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Published: 2014-04-04   Modified: 2020-03-30  

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