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Application of Optimal Transport and Information geometry to Metric Measure Spaces

Research Project

Project/Area Number 15K17536
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Geometry
Research InstitutionTokyo Metropolitan University

Principal Investigator

TAKATSU Asuka  首都大学東京, 理学研究科, 准教授 (90623554)

Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords最適輸送理論 / 情報幾何 / 測度距離空間 / 熱流 / エントロピー / 凸性 / 曲率 / 測度の集中 / 測度の集中現象
Outline of Final Research Achievements

The aim of this research is to combine Wasserstein geometry with Information geometry and application of the both geometry to geometric analysis by taking account of the fact that the asymptotic behavior of the heat equation plays an important role in geometric analysis. In the joint work with Kazuhiro Ishige (the university of Tokyo) and Paolo Salani (university of Florence), we proved that the logarithmic concavity is not the strongest concavity preserved by the heat equation. Moreover we clarified what is the strongest concavity preserved by the heat equation. From the nature of the strongest concavity, it is expected to generalize the theory of Wasserstein geometry and Information geometry together by using the strongest concavity. In this way, this research made some contributions to a combining Wasserstein geometry with Information geometry.

Academic Significance and Societal Importance of the Research Achievements

H.J.BrascampとE.H.Liebが1976年に打ち出した不等式以降、対数凹性は熱流に最適とされ、熱流の漸近解析や形状解析などの大域解析の軸になっていた。よって上記で述べた熱流で保たれる最強の凹性が対数凹性ではないという事実、および熱流で保たれる最強の凹性を決定した事実は、熱流の大域解析に新機軸を与える。特に熱流におけるこの最強の凹性の保存則を導くエントロピーの凸性、およびそのエントロピーの凸性から得られる幾何学的条件を明らかにすることは、熱流の大域挙動に関わる幾何学、曲率の条件をかした幾何解析に新展開を持ち込むことが期待できる。

Report

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

    (18 results)

All 2019 2018 2017 2016 2015 Other

All Journal Article (4 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 4 results,  Open Access: 3 results) Presentation (11 results) (of which Int'l Joint Research: 7 results,  Invited: 8 results) Book (1 results) Remarks (2 results)

  • [Journal Article] Convergence of combinatorial Ricci flows to degenerate circle patterns2019

    • Author(s)
      Asuka Takatsu
    • Journal Title

      Transactions of the American Mathematical Society

      Volume: 印刷中 Issue: 11 Pages: 7597-7617

    • DOI

      10.1090/tran/7778

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] To logconcavity and beyond2019

    • Author(s)
      Kazuhiro Ishige, Paolo Salani, Asuka Takatsu
    • Journal Title

      Communications in Contemporary Mathematics

      Volume: 印刷中 Issue: 02 Pages: 1950009-1950009

    • DOI

      10.1142/s0219199719500093

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [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] 書評 C. Villani: Optimal Transport-Old and New2015

    • Author(s)
      高津 飛鳥
    • Journal Title

      数学

      Volume: 67

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] Convergence of combinatorial Ricci flows to degenerate circle patterns2019

    • Author(s)
      Asuka TAKATSU
    • Organizer
      AIMR Workshop on Pure and Applied Mathematics,
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 退化した Circle Pattern に収束するトーラス上の組合せリッチ流2018

    • Author(s)
      高津飛鳥
    • Organizer
      2018年度秋季総合分科会
    • Related Report
      2018 Annual Research Report
  • [Presentation] To logconcavity and beyond2018

    • Author(s)
      Asuka TAKATSU
    • Organizer
      Workshop on barycenters, convexity on metric spaces and positive operators
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] High-dimensional metric-measure limit of Stiefel and Grassmann manifolds2017

    • Author(s)
      Asuka TAKATSU
    • Organizer
      Dirichlet forms and their geometry
    • Place of Presentation
      東北大学
    • Year and Date
      2017-03-18
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Riemannian Wasserstein geometry on the space of Gaussian measures over the Wiener space2016

    • Author(s)
      Asuka TAKATSU
    • Organizer
      第1回日本-台湾微分幾何研究集会
    • Place of Presentation
      早稲田大学
    • Year and Date
      2016-12-17
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Curvature Dimension condition from the viewpoint of Information geometry2016

    • Author(s)
      Asuka TAKATSU
    • Organizer
      Follow-up Workshop to JTP "Optimal Transportation"
    • Place of Presentation
      ボン(ドイツ)
    • Year and Date
      2016-09-02
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Wasserstein/Information geometry and its applications2016

    • Author(s)
      Asuka TAKATSU
    • Organizer
      The 7th Pacific RIM Conference on Mathematics 2016
    • Place of Presentation
      ソウル(韓国)
    • Year and Date
      2016-06-28
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Wasserstein/Information geometry and its applications2016

    • Author(s)
      高津飛鳥
    • Organizer
      日本数学会2016年年会
    • Place of Presentation
      筑波大学
    • Year and Date
      2016-03-19
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Behaviors of varphi-Gaussian measures in Wasserstein geometry2015

    • Author(s)
      Asuka TAKATSU
    • Organizer
      Computational information geometry for image and signal processing
    • Place of Presentation
      ICMS(Edinburgh)
    • Year and Date
      2015-09-24
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 測度距離空間としての Stiefel 多様体族の極限2015

    • Author(s)
      高津飛鳥
    • Organizer
      日本数学会2015年度秋季総合分科会
    • Place of Presentation
      京都産業大学
    • Year and Date
      2015-09-16
    • Related Report
      2015 Research-status Report
  • [Presentation] 抽象 Wiener 空間上の Gauss 測度族の Wasserstein 幾何2015

    • Author(s)
      高津飛鳥
    • Organizer
      日本数学会2015年度秋季総合分科会,
    • Place of Presentation
      京都産業大学
    • Year and Date
      2015-09-16
    • Related Report
      2015 Research-status Report
  • [Book] 数学メモアール 第8巻 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; 高津飛鳥; 桒田和正
    • Total Pages
      141
    • Publisher
      日本数学会
    • Related Report
      2016 Research-status Report
  • [Remarks] Asuka Takatsu's Home Page

    • URL

      https://sites.google.com/site/asukatakatsu/

    • Related Report
      2018 Annual Research Report 2017 Research-status Report
  • [Remarks] Asuka Takatsu's Home Page

    • URL

      https://sites.google.com/site/asukatakatsu/home

    • Related Report
      2016 Research-status Report 2015 Research-status Report

URL: 

Published: 2015-04-16   Modified: 2020-03-30  

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