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Study of Analysis and Geometry of complex spaces

Research Project

Project/Area Number 21H00989
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Review Section Basic Section 12010:Basic analysis-related
Research InstitutionKyoto University

Principal Investigator

KIGAMI JUN  京都大学, 情報学研究科, 教授 (90202035)

Co-Investigator(Kenkyū-buntansha) 白石 大典  京都大学, 情報学研究科, 准教授 (00647323)
相川 弘明  中部大学, 工学部, 教授 (20137889)
角 大輝  京都大学, 人間・環境学研究科, 教授 (40313324)
秋山 茂樹  筑波大学, 数理物質系, 教授 (60212445)
宍倉 光広  京都大学, 理学研究科, 教授 (70192606)
熊谷 隆  早稲田大学, 理工学術院, 教授 (90234509)
梶野 直孝  京都大学, 数理解析研究所, 准教授 (90700352)
Project Period (FY) 2021-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥17,290,000 (Direct Cost: ¥13,300,000、Indirect Cost: ¥3,990,000)
Fiscal Year 2023: ¥5,330,000 (Direct Cost: ¥4,100,000、Indirect Cost: ¥1,230,000)
Fiscal Year 2022: ¥5,330,000 (Direct Cost: ¥4,100,000、Indirect Cost: ¥1,230,000)
Fiscal Year 2021: ¥6,630,000 (Direct Cost: ¥5,100,000、Indirect Cost: ¥1,530,000)
Keywordsフラクタル / 拡散過程 / ラプラシアン / 熱核
Outline of Research at the Start

ユークリッド空間では空間の自然な幾何構造(ユークリッドの距離)から、微分の概念が定義され、微積分学を基盤とした解析学が所与の幾何構造のもとで展開されてきた。一方、フラクタルに代表される複雑な空間では、解析と幾何構造の間の関係は自明ではなくなる。本研究においては、自己相似集合、力学系に表れる不変集合・タイリングなどの複雑な空間において、まず空間のグラフによる離散近似などを用いて拡散過程などの解析的構造の構成し、更にその解析的構造を表現するのに適切な幾何的構造を見出すことを目指す。具体的には、まず、対称性の弱いSiperpinski carpet や Julia 集合などを対象とする。

Outline of Final Research Achievements

To construct a diffusion process(or a Dirichlet form) on a metric space, we approximate the space by an infinite sequence of discrete graphs and consider when the natural discrete Dirichlet forms with certain scaling on those graphs converge to a local regular Dirichlet form on the original space. As a result, we find that a generalization of the "Knight move" condition, which was found by Barlow-Bass when they constructed the Brownian motion on the Sierpinski carpet, is a sufficient condition for the success of the above strategy. Moreover, we have found a new class of self-similar sets where we can construct a diffusion process by the above mentioned approach.

Academic Significance and Societal Importance of the Research Achievements

マンデルプローによって自然界の物体の適切なモデルとして提案されたフラクタル上では、その複雑な形状により通常の微分を基本とする解析学は適用できない。従って、自然界のモデルとしてのこのような複雑な空間で、物理現象を記述するためには、新しい解析学の理論が必要となる。本研究は、複雑な空間の幾何と解析の係わりの研究を通じて、複雑な空間上の拡散現象や波動現象を記述するための基本理論を確立し、さらに複雑な空間と従来の滑らかな空間上の物理現象の本質的な違いを明らかにすることに貢献している。

Report

(4 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Annual Research Report
  • 2021 Annual Research Report
  • Research Products

    (20 results)

All 2024 2023 2022 Other

All Int'l Joint Research (5 results) Journal Article (6 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 6 results,  Open Access: 4 results) Presentation (8 results) (of which Int'l Joint Research: 8 results,  Invited: 8 results) Funded Workshop (1 results)

  • [Int'l Joint Research] University of Washington(米国)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] University of British Columbia(カナダ)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] University of Washington(米国)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] University of Helsinki(フィンランド)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] Universiy of Paris, Est/ENS, Lyon(フランス)

    • Related Report
      2022 Annual Research Report
  • [Journal Article] “The Sierpinski gasket minus its bottom line” as a tree of Sierpinski gaskets2024

    • Author(s)
      J. Kigami & K. Takahashi
    • Journal Title

      Math. Z.

      Volume: 28 Issue: 2

    • DOI

      10.1007/s00209-023-03416-1

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Width Deviation of Convex Polygons2023

    • Author(s)
      Akiyama Shigeki、Kamae Teturo
    • Journal Title

      Discrete & Computational Geometry

      Volume: - Issue: 4 Pages: 1403-1428

    • DOI

      10.1007/s00454-023-00545-6

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Scaling limit for random walk on the range of random walk in four dimensions2023

    • Author(s)
      D. A. Croydon and D. Shiraishi
    • Journal Title

      Annales de l'institut Henri Poincare (B) Probabilites et Statistiques

      Volume: 59 Issue: 1 Pages: 166-184

    • DOI

      10.1214/22-aihp1243

    • Related Report
      2023 Annual Research Report 2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Conductive homogeneity of compact metric spaces and construction of p-energy2023

    • Author(s)
      Jun Kigami
    • Journal Title

      Memoirs of the European Mathematical Society

      Volume: -

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] On the conformal walk dimension: quasisymmetric uniformization for symmetric diffusions2023

    • Author(s)
      N. Kajino and M. Murugan
    • Journal Title

      Invent. Math.

      Volume: 231 Issue: 1 Pages: 263-405

    • DOI

      10.1007/s00222-022-01148-3

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Heat kernels for reflected diffusions with jumps on inner uniform domains2022

    • Author(s)
      Z.-Q. Chen, P. Kim, T. Kumagai and J. Wang
    • Journal Title

      Trans. Amer. Math. Soc

      Volume: 375 Issue: 10 Pages: 6797-6841

    • DOI

      10.1090/tran/8678

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Conductive homogeneity of locally symmetric polygon-based self-similar sets2024

    • Author(s)
      J. Kigami
    • Organizer
      CIRM workshop "Analysis on fractals and networks, and applications"
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Construction of Sobolev spaces on metric spaces2024

    • Author(s)
      J. Kigami
    • Organizer
      Fractals, quantum graphs and applications in pure and applied sciences
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Yet another construction of "Sobolev spaces" on metric spaces2023

    • Author(s)
      J. Kigami
    • Organizer
      Quasiworld workshop
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Random Dynamical Systems of Polynomial Automorphisms on \Bbb{C}^{2}2023

    • Author(s)
      H, Sumi
    • Organizer
      The 10th Visegrad Conference on Dynamical Systems
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Loop-erased random walk in three dimensions2023

    • Author(s)
      D. Shiraishi
    • Organizer
      Random Interacting Systems, Scaling Limits, and Universality
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Conductive homogeneity of compact metric spaces and construction of p-energy2022

    • Author(s)
      Jun Kigami
    • Organizer
      2nd Joint Congress of Mathematics of AMS-EMS-SMF
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Yet another construction of ``Sobolev spaces'' on metric spaces2022

    • Author(s)
      Jun Kigami
    • Organizer
      Smooth Functions on Rough Spaces and Fractals with Connections to Curvature Functional Inequalities
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Conductive homogeneity of compact metric spacees and construction of p-energies2022

    • Author(s)
      Jun Kigami
    • Organizer
      7th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Geometric and Stochastic analysis on metric spaces2023

    • Related Report
      2022 Annual Research Report

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Published: 2021-04-28   Modified: 2025-01-30  

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