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2011 Fiscal Year Final Research Report

Study of Geometry of a discrete space through randomness

Research Project

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Project/Area Number 20244002
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTohoku University

Principal Investigator

KOTANI Motoko  東北大学, 大学院・理学研究科, 教授 (50230024)

Co-Investigator(Kenkyū-buntansha) SHIOYA Takashi  東北大学, 大学院理学研究科, 教授 (90235507)
Co-Investigator(Renkei-kenkyūsha) ARAI Hitoshi  東京大学, 大学院数理科学研究科, 教授 (10175953)
KUMAGAI Takashi  京都大学, 数理解析研究所, 教授 (90234509)
IZEKI Hiroyasu  慶應義塾大学, 理工学部, 教授 (90244409)
NAYATANI Shin  名古屋大学, 大学院多元数理科学研究科, 教授 (70222180)
TATE Tatsuya  東北大学, 大学院理学研究科, 准教授 (00317299)
ISHIWATA Satoshi  山形大学, 大学院理学部, 准教授 (70375393)
Project Period (FY) 2008 – 2011
Keywordsランダムウォーク / ラプラシアン / 距離測度空間
Research Abstract

Aim of this research proposal is to develop new methods to study geometric objects with singularities, or discrete spaces, which are not accessible by traditional differential geometrical technics. Our idea is to apply probability theory to those geometric objects. Some results are obtained and published from international journals.

  • Research Products

    (34 results)

All 2012 2011 2010 2009 2008

All Journal Article (12 results) (of which Peer Reviewed: 12 results) Presentation (19 results) Book (3 results)

  • [Journal Article] Asymptotic behavior of quantum walks on the line2012

    • Author(s)
      T.Sunada and T.Tate
    • Journal Title

      J.Funct. Anal.

      Volume: 262巻 Pages: 2608-2645

    • DOI

      DOI:10.1016/j.jfa.2011.12.016

    • Peer Reviewed
  • [Journal Article] Geometric analysis of Alexandrov spaces.2011

    • Author(s)
      Takashi Shioya
    • Journal Title

      Su Buqing memorial lectures

      Volume: 1巻 Pages: 31-46

    • Peer Reviewed
  • [Journal Article] Collapsing three-manifolds with a lower curvature bound2011

    • Author(s)
      Takashi Shioya
    • Journal Title

      Tohoku Math. J.

      Volume: 63巻 Pages: 471-487

    • DOI

      DOI:10.2748/tmj/1325886277

    • Peer Reviewed
  • [Journal Article] Global Heat Kernel Estimates for Symmetric Jump Processes2011

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

      Trans. Amer. Math. Soc.

      Volume: 363巻 Pages: 5021-5055

    • DOI

      DOI:10.1090/s0002-9947-2011-05408-5

    • Peer Reviewed
  • [Journal Article] An approach to superrigidity and fixed - point theorems via harmonic maps, Selected Papers on Analysis and Different-ial Equations2010

    • Author(s)
      Shin Nayatani and Hiroyasu Izeki
    • Journal Title

      AMS Translations

      Volume: 230巻 Pages: 135-160

    • Peer Reviewed
  • [Journal Article] A fixed-point property of finitely generated groups and an energy of equivariant maps,in Probabilistic Approach to Geometry2010

    • Author(s)
      H. Izeki
    • Journal Title

      ASPM

      Volume: 57巻 Pages: 171-188

    • Peer Reviewed
  • [Journal Article] Diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice2010

    • Author(s)
      B.M. Hambly、T. Kumagai
    • Journal Title

      Comm. Math. Phys

      Pages: 29-69

    • DOI

      DOI:10.1007/s00220-009-0981-3

    • Peer Reviewed
  • [Journal Article] On heat kernel estimates and parabolic Harnack inequality for jump process on metric measure spaces2009

    • Author(s)
      Z.-Q.Chen、 P.Kim、T. Kumagai
    • Journal Title

      Acta Math. Sin.(Engl.ser.)

      Volume: 25巻 Pages: 1067-1086

    • DOI

      DOI:10.1007/s10114-009-8576-7

    • Peer Reviewed
  • [Journal Article] A central limit theorem on modified graphs of nilpotent covering graphs2009

    • Author(s)
      S.Ishiwata
    • Journal Title

      Contemp. Math.

      Volume: 484巻 Pages: 59-72

    • DOI

      DOI:10.1090/conm/484/09465

    • Peer Reviewed
  • [Journal Article] New metallic carbon crystal2009

    • Author(s)
      Masahiro Itoh、 Motoko Kotani、 Hisashi Naito、Toshikazu Sunada、 Yoshiyuki Kawazoe、and Tadafumi Adschiri
    • Journal Title

      Phys. Rev. Lett.

      Volume: 102巻 Pages: 055703-1~4

    • DOI

      DOI:10.1103/PhysRevLett.102.055703

    • Peer Reviewed
  • [Journal Article] Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps2009

    • Author(s)
      M.T. Barlow、 R.F. Bass and T. Kumagai
    • Journal Title

      Math. Z

      Volume: 261巻 Pages: 297-320

    • DOI

      DOI:10.1007/s00209-008-0326-5

    • Peer Reviewed
  • [Journal Article] Random walk on the incipient infinite cluster for oriented percolation in high dimensions2008

    • Author(s)
      M.T. Barlow、 A.A. Tarai、 T. Kumagai and G. Slade
    • Journal Title

      Comm. Math. Phys

      Volume: 278巻

    • DOI

      DOI:385-43110.1007/s00220-007-0410-4

    • Peer Reviewed
  • [Presentation] Heat kernel estimates on connected sums with non-compact joint2012

    • Author(s)
      S. Ishiwata
    • Organizer
      Weekly Seminar
    • Place of Presentation
      香港中文大学, 香港
    • Year and Date
      2012-03-26
  • [Presentation] Quenched invariance principle for random walks and random divergence forms in random media on Cones2012

    • Author(s)
      T. Kumagai
    • Organizer
      The expanding art of expansions
    • Place of Presentation
      Eurandom、オランダ
    • Year and Date
      2012-02-14
  • [Presentation] Convergence of mixing times for sequences of random walks on finite graphs2011

    • Author(s)
      T. Kumagai
    • Organizer
      4th Cornell Conference on Analysis, Probability、and Mathematical Physics on Fractals
    • Place of Presentation
      Cornell University、 アメリカ
    • Year and Date
      2011-09-13
  • [Presentation] Markov chain approximations to non-symmetric diffusions with bounded coefficients2011

    • Author(s)
      T. Kumagai
    • Organizer
      5th international conferences on Stochastic Analysis and its Applications
    • Place of Presentation
      Bonn、ドイツ
    • Year and Date
      2011-09-05
  • [Presentation] Concentration, Ricci curvature, and eigenvalues of Laplacian2011

    • Author(s)
      Takashi Shioya
    • Organizer
      Metric Geometry and Applications
    • Place of Presentation
      The Purple Palace Nanjing, Nanjing、中国
    • Year and Date
      2011-07-28
  • [Presentation] A topological splitting theorem for weighted Alexandrov spaces2011

    • Author(s)
      Takashi Shioya
    • Organizer
      Geometry Seminar
    • Place of Presentation
      Capital Normal University, Beijing、中国
    • Year and Date
      2011-07-12
  • [Presentation] Manifolds with a bottleneck type heat kernel estimate2010

    • Author(s)
      S.Ishiwata
    • Organizer
      Analysis and Geometry of Fractals and Metric Measure spaces
    • Place of Presentation
      香港中文大学、中国
    • Year and Date
      2010-03-20
  • [Presentation] Problems on asymptotic analysis over convex polytopes2009

    • Author(s)
      Tatsuya Tate
    • Organizer
      Inte rnational School on Geometry and Quantization
    • Place of Presentation
      University of Luxembourg、ルクセンブルク
    • Year and Date
      20090901-03-04
  • [Presentation] Geometry and Analysis2009

    • Author(s)
      Motoko Kotani
    • Organizer
      International Conference "Geometry and Analysis"
    • Place of Presentation
      Paris 7、フランス
    • Year and Date
      2009-12-21
  • [Presentation] Heat kernel estimates for random walks on random media atcriticality2009

    • Author(s)
      T. Kumagai
    • Organizer
      Above the critical dimension
    • Place of Presentation
      Institut Henri Poincare, Paris、フランス
    • Year and Date
      2009-12-07
  • [Presentation] Geometry of crystal lattices2009

    • Author(s)
      Motoko Kotani
    • Organizer
      Series of Lectures
    • Place of Presentation
      北海道大学
    • Year and Date
      2009-11-02
  • [Presentation] An asymptotic Euler-Maclaurin formula for Delzant polytopes2009

    • Author(s)
      Tatsuya Tate
    • Organizer
      Third International Conference on Geometry and Qunatization
    • Place of Presentation
      Uiversity of Luxembourg、ルクセンブルク
    • Year and Date
      2009-09-10
  • [Presentation] Convergence of discrete Markov chains to jump processes and its application to random conductance models2009

    • Author(s)
      T. Kumagai
    • Organizer
      Scaling Limits in Models of Statistical Mechanics
    • Place of Presentation
      Oberwolfach、ドイツ
    • Year and Date
      2009-08-20
  • [Presentation] Mathematical challenge to a new phase of material sciences2009

    • Author(s)
      Motoko Kotani
    • Organizer
      Workshop "Boundaries"
    • Place of Presentation
      TU Graz、オーストリア
    • Year and Date
      2009-07-01
  • [Presentation] A mathematical challenge to material sciences2009

    • Author(s)
      Motoko Kotani
    • Organizer
      Colloqium talk
    • Place of Presentation
      Technische Universitaet Graz、オーストリア
    • Year and Date
      2009-03-18
  • [Presentation] Collapsing three-manifolds with a lower curvature bound2008

    • Author(s)
      T. Shioya
    • Organizer
      The 4th Geometry Conference for the Friendship of China and Japan
    • Place of Presentation
      南開大学(中国)
    • Year and Date
      2008-12-26
  • [Presentation] Geometric aspect of random walk on a crystal lattice2008

    • Author(s)
      Motoko Kotani
    • Organizer
      Algebras,Operators and Noncommutative Geometry
    • Place of Presentation
      Australian national University Australia、オーストラリア
    • Year and Date
      2008-12-01
  • [Presentation] Geometric aspect of random walk on a crystal lattice2008

    • Author(s)
      Motoko Kotani
    • Organizer
      Perspectives in mathematical sciences
    • Place of Presentation
      東京大学
    • Year and Date
      2008-10-08
  • [Presentation] Bernstein measures on convex polytopes2008

    • Author(s)
      Tatsuya Tate
    • Organizer
      The Mathematical Society of Japan、 Seasonal Institute
    • Place of Presentation
      京都大学百周年時計台記念館 国際交流ホール
    • Year and Date
      2008-07-29
  • [Book] Probabilistic approach to geometry2011

    • Author(s)
      Motoko Kotani、Masanori Hino and Takashi Kumagai
    • Total Pages
      514
    • Publisher
      日本数学会
  • [Book] Advanced Studies in Pure Mathematics 57、Probabilistic Approach to Geometry2010

    • Author(s)
      M. Kotani、M. Hino、T. Kumagai
    • Total Pages
      514
    • Publisher
      Mathematical Society of Japan
  • [Book] Advanced Studies in Pure Mathematics 55、 Noncommutativity and Singularities Proceedings of French-Japanese symposia held at IH.S in 20062009

    • Author(s)
      J. P. Bourguignon、M. Kotani、Y. Mada, N. Tose
    • Total Pages
      363
    • Publisher
      Mathematical Society of Japan

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Published: 2014-08-29  

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