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2015 Fiscal Year Annual Research Report

量子スピン系の離散幾何解析学

Research Project

Project/Area Number 24244004
Research InstitutionTohoku University

Principal Investigator

小谷 元子  東北大学, 理学研究科, 教授 (50230024)

Co-Investigator(Kenkyū-buntansha) 熊谷 隆  京都大学, 数理解析研究所, 教授 (90234509)
塩谷 隆  東北大学, 理学研究科, 教授 (90235507)
Project Period (FY) 2012-04-01 – 2017-03-31
Keywords幾何学 / 離散幾何学 / スペクトル解析 / スピン系
Outline of Annual Research Achievements

物性物理における基本的な性質を離散距離空間の幾何学、スペクトル幾何学、非可換幾何学、トポロジーの知識を適用することでより深く理解し、量子スピン系の離散幾何解析学を展開することを目的とした。
研究代表者 小谷を中心に、分担者 熊谷隆、塩谷隆、連携研究者 尾畑伸明、楯辰哉、宮岡礼子と緊密に連絡を取りつつ研究を進めた。量子スピン系の中心的課題である、トポロジカル絶縁体に関わる数学的理解を深め、特にランダムポテンシャルのある場合の界面流を非可換K理論を用いて記述した。さらに非常に一般的な枠組みでのバルク・エッジ対応の機構をK理論のGysin写像により説明した。これらの結果を国際会議の講演及び論文として発表した。また、物性物理学者と数学者が参加する国際研究集会、ミニワークショップ、勉強会を開催した。特に、周期性を仮定しないdisorder系への拡張への方向性を見出したことは有益であった。低次元電子系に関しては、カーボンネットワークの構造と機能の相関を理解するために、カーボンネットワークを3分岐周期グラフととらえ、幾何学的量を定義し、物性との相関関係の評価をおこなった。このために、曲面論の離散可として3分岐グラフの曲面論を構築し、基本形式、面積変分公式、曲率、平均曲率などを定義した。また、離散曲面の背後にある連続曲面を見出すために、ゴールドバーグ・コクセタ細分を用い、与えられた曲面の細分化列の収束理論を構築した。トポロジカル欠陥の周りの収束速度が遅くなることを利用し、トポロジカル欠陥を抽出することができた。量子ウォークは、昨年度に楯氏により一般的な定式化が提案されたが、これを更に発展させ結晶格子の量子ウォークに関する研究、特に局在化の条件の特定を行った。

Current Status of Research Progress
Current Status of Research Progress

1: Research has progressed more than it was originally planned.

Reason

いくつかの研究会や発表会を開催し、物性物理の最近の進展や問題意識についての理解を深めることができた。その結果、特に、非可換幾何学を用いることにより、無秩序系への拡張ができ、数学的にも物性物理にとっても興味深い成果が得られて国際研究集会での招待講演や論文として発表した。 数学だけではなく、材料科学や化学の国際研究集会にキーノート講演者として招待された。また、低次元電子系に関する物性物理から刺激を受け、離散曲面の新理論を構築できた。数学的な問題を提起し、離散幾何解析を展開することになった。数学を用いた物性物理の展開に関する萌芽的論説を発表し、1000以上のViewを得ている。当初予想した以上に、2つの領域の双方に興味ある研究成果が得られた。

Strategy for Future Research Activity

低次元系としてカーボンネットワークの幾何学を確立することができた。離散曲面上のランダムウォークの収束定理(中心極限定理)によって、どのような連続拡散過程が現れるかを調べること、またそのスペクトル理論と第一原理計算による電子物性を比較することは興味深い。これにより、低次元系の電子物性を、第一原理計算などの重たい計算ではなく、幾何学的な特徴量で予測できる手法を開発できれば、物性物理に大きなインパクトを与える。
物質のトポロジー相は、2016年のノーベル物理学賞の受賞対象となるなど、物性物理において重要な課題である。物理では周期系でのみ理論を展開しているが、我々は非可換幾何学を用いてこれを無秩序系の理論へと拡張した。これらの成果を基に、数学的に意義の深い非コンパクト、非線形な設定への拡張へと挑戦したい。

  • Research Products

    (46 results)

All 2016 2015 Other

All Int'l Joint Research (6 results) Journal Article (11 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 11 results) Presentation (26 results) (of which Int'l Joint Research: 20 results,  Invited: 24 results) Book (2 results) Funded Workshop (1 results)

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

    • Country Name
      U.S.A.
    • Counterpart Institution
      Stanford University/University of Washington
  • [Int'l Joint Research] Fujian Normal University/清華大学/北京師範大学(China)

    • Country Name
      China
    • Counterpart Institution
      Fujian Normal University/清華大学/北京師範大学
  • [Int'l Joint Research] Free University of Berlin(Germany)

    • Country Name
      Germany
    • Counterpart Institution
      Free University of Berlin
  • [Int'l Joint Research] Seoul National University/NIMS, Korea(韓国)

    • Country Name
      KOREA (REP. OF KOREA)
    • Counterpart Institution
      Seoul National University/NIMS, Korea
  • [Int'l Joint Research] University of Oxford/University of Warwick(United Kingdom)

    • Country Name
      United Kingdom
    • Counterpart Institution
      University of Oxford/University of Warwick
  • [Int'l Joint Research]

    • # of Other Countries
      2
  • [Journal Article] Laws of the iterated logarithm for random walks on Random Conductance Models2016

    • Author(s)
      T. Kumagai and C. Nakamura
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B59 Pages: 141-156.

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Stability and instability of Gaussian heat kernel estimates for random walks among time-dependent conductances2016

    • Author(s)
      R. Huang and T. Kumagai
    • Journal Title

      Electron. Commun. Probab.

      Volume: Vol.21, no.5 Pages: 1-11

    • DOI

      10.1214/15-ECP4347

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Errata of "isoparametric hypersurfaces with (g,m)=(6,2)"2016

    • Author(s)
      Reiko Miyaoka
    • Journal Title

      Annals of Math.

      Volume: 183, 3 Pages: 1057-1071

    • DOI

      10.4007/annals.2016.183.3.7

    • Peer Reviewed
  • [Journal Article] Visualization of topological landscape in shear-flow dynamics of amorphous solids2015

    • Author(s)
      Takeshi Fujita, Koji Ohara, Keiji Miura, Akihiko Hirata, Motoko Kotani, Yasumasa Nishiura and Mingwei Chen
    • Journal Title

      EPL

      Volume: 110,3 Pages: 38002

    • DOI

      10.1209/0295-5075/110/38002

    • Peer Reviewed
  • [Journal Article] The Decomposition Formula of < 001 > Symmetrical Tilt Grain Boundaries2015

    • Author(s)
      Inoue, Kazutoshi; Saito, Mitsuhiro; Wang, Zhongchang; Kotani, Motoko; Ikuhara, Yuichi
    • Journal Title

      MATERIALS TRANSACTIONS

      Volume: 56,12 Pages: 1945-1952

    • DOI

      10.2320/matertrans.M2015277

    • Peer Reviewed
  • [Journal Article] A new direction in mathematics for materials science2015

    • Author(s)
      Ikeda, Susumu, Kotani, Motoko
    • Journal Title

      Springer Briefs in the Mathematics of Materials

      Volume: vol.1 Pages: 1-86

    • DOI

      10.1007/978-4-431-55864-4

    • Peer Reviewed
  • [Journal Article] On the periodicity of < 001 > symmetrical tilt grain boundaries2015

    • Author(s)
      K. Inoue, M. Saito, Z. Wang, M. Kotani, Y. Ikuhara
    • Journal Title

      Mater. Trans.

      Volume: 56, 3 Pages: 281-287

    • DOI

      10.2320/matertrans.M2014394

    • Peer Reviewed
  • [Journal Article] Harnack inequalities and local central limit theorem for the polynomial lower tail random conductance model2015

    • Author(s)
      O. Boukhadra, T. Kumagai and P. Mathieu
    • Journal Title

      J. Math. Soc. Japan

      Volume: 67, no. 4 Pages: 1413--1448.

    • DOI

      10.2969/jmsj/06741413

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Estimate of observable diameter of lp -product spaces2015

    • Author(s)
      Ozawa, Ryunosuke ; Shioya, Takashi
    • Journal Title

      Manuscripta Math

      Volume: 147, no. 3-4 Pages: 501-509

    • DOI

      10.1007/s00229-015-0730-1

    • Peer Reviewed
  • [Journal Article] Moment map description of the Cartan-Munzner polynomials of degree four2015

    • Author(s)
      R. Miyaoka
    • Journal Title

      Progress in Math.

      Volume: 308 Pages: 437--447

    • DOI

      10.1007/978-3-319-11523-8_19

    • Peer Reviewed
  • [Journal Article] Remarks on "the Dorfmeister--Neher theorem on isoparametric hypersurfaces''2015

    • Author(s)
      R. Miyaoka
    • Journal Title

      Osaka J. Math.

      Volume: 52 Pages: 373-377

    • Peer Reviewed
  • [Presentation] Figure Materials2016

    • Author(s)
      Motoko Kotani
    • Organizer
      16th International Conference Computational and Mathematical Methods in Science and Engineering
    • Place of Presentation
      BARCELO COSTA BALLENA HOTEL ( Cadiz, Spain)
    • Year and Date
      2016-07-05
    • Int'l Joint Research / Invited
  • [Presentation] Stability of heat kernel estimates and parabolic Harnack inequalities for jump processes on metric measure spaces2016

    • Author(s)
      Takashi Kumagai
    • Organizer
      The 1st Hong Kong/Kyoto Workshop on Fractal Geometry and Related Areas
    • Place of Presentation
      Hong Kong University of Science and Technology (香港)
    • Year and Date
      2016-03-22
    • Int'l Joint Research / Invited
  • [Presentation] Mathematical Challenge to a New Phase of Materials Science2016

    • Author(s)
      Motoko Kotani
    • Organizer
      Japanese Swiss Energy Materials Workshop
    • Place of Presentation
      EMPA (スイス)
    • Year and Date
      2016-03-08
    • Int'l Joint Research / Invited
  • [Presentation] recent topics on random conductance model2016

    • Author(s)
      Takashi Kumagai
    • Organizer
      2016 Spring Probability Workshop at Academia Sinica
    • Place of Presentation
      Academia Sinica(台湾)
    • Year and Date
      2016-03-07
    • Int'l Joint Research / Invited
  • [Presentation] Stability of heat kernel estimates and parabolic Harnack inequalities for jump processes on metric measure spaces2016

    • Author(s)
      Takashi Kumagai
    • Organizer
      Probability Seminar
    • Place of Presentation
      Stanford University(California, US)
    • Year and Date
      2016-02-22
    • Int'l Joint Research / Invited
  • [Presentation] Convergence of metric measure spaces2016

    • Author(s)
      Takashi Shioya
    • Organizer
      Metric Geometry and Its Applications
    • Place of Presentation
      復旦大学(上海)
    • Year and Date
      2016-02-22
    • Int'l Joint Research / Invited
  • [Presentation] 等径超曲面のガウス像のハミルトン交叉とフレアホモロジー2016

    • Author(s)
      宮岡 礼子
    • Organizer
      日本数学会東北支部会
    • Place of Presentation
      弘前大学(青森県)
    • Year and Date
      2016-02-13
    • Invited
  • [Presentation] マルチスケール材料開発のための幾何学的手法2016

    • Author(s)
      小谷 元子
    • Organizer
      先端化学・材料技術部会 コンピュータケミストリ分科会講演会「マルチスケール材料開発のための幾何学的手法」
    • Place of Presentation
      (公社)新化学技術推進協会(東京)
    • Year and Date
      2016-01-14
    • Invited
  • [Presentation] Localization for periodic unitary transition operators2016

    • Author(s)
      Tatsuya Tate
    • Organizer
      2016 Joint Mathematics Meetings, AMS Special Session on "Quantum Walks, Quantum Markov Chains, Quantum Computation and Related Topics''
    • Place of Presentation
      Washington state convention center, Seattle, USA
    • Year and Date
      2016-01-08
    • Int'l Joint Research
  • [Presentation] Hamiltonian non-displaceability of Gauss images of isoparametric2015

    • Author(s)
      宮岡 礼子
    • Organizer
      中央大学数学科談話会
    • Place of Presentation
      中央大学(東京)
    • Year and Date
      2015-12-18
    • Invited
  • [Presentation] Mathematical challenge to structural understanding of materials2015

    • Author(s)
      Motoko Kotani
    • Organizer
      YHMA(Yau’s High School Mathematical Award)
    • Place of Presentation
      Tsinghua University(北京,中国)
    • Year and Date
      2015-12-12
    • Int'l Joint Research / Invited
  • [Presentation] Mathematical challenge to structural understanding of materials2015

    • Author(s)
      Motoko Kotani
    • Organizer
      2nd International Symposium on Frontiers in Materials Science (FMS)
    • Place of Presentation
      Waseda University, Tokyo, Japan
    • Year and Date
      2015-11-21
    • Int'l Joint Research / Invited
  • [Presentation] 等径超曲面のガウス像のHamilton non-displaceabilityについて2015

    • Author(s)
      宮岡礼子
    • Organizer
      部分多様体論・湯沢2015
    • Place of Presentation
      湯沢町・湯沢グランドホテル(新潟県)
    • Year and Date
      2015-11-20
  • [Presentation] Anomalous random walks and diffusions in random media2015

    • Author(s)
      Takashi Kumagai
    • Organizer
      International Workshop on the Multi-Phase Flow; Analysis, Modeling and Numerics
    • Place of Presentation
      早稲田大学(東京都)
    • Year and Date
      2015-11-12
    • Int'l Joint Research / Invited
  • [Presentation] Mathematical challenge to structural understanding of Materials2015

    • Author(s)
      Motoko Kotani
    • Organizer
      Geometric analysis in geometry and topology 2015
    • Place of Presentation
      東京理科大学(神楽坂),森戸記念館
    • Year and Date
      2015-11-10
    • Int'l Joint Research / Invited
  • [Presentation] Quantum walks in one dimension2015

    • Author(s)
      Tatsuya Tate
    • Organizer
      Colloquium at Shanghai Mathematical Center
    • Place of Presentation
      Fudan University, Shanghai, China
    • Year and Date
      2015-11-05
    • Invited
  • [Presentation] Hamiltonian non-displaceability of the Gauss image of isoparametric hypersurfaces2015

    • Author(s)
      宮岡礼子
    • Organizer
      学習院・早稲田 幾何学セミナー(GWGS)
    • Place of Presentation
      早稲田大学(東京)
    • Year and Date
      2015-10-26
    • Invited
  • [Presentation] Harnack inequalities and local CLT for the polynomial lower tail random conductance model2015

    • Author(s)
      Takashi Kumagai
    • Organizer
      RIMS workshop “Stochastic Analysis”
    • Place of Presentation
      京都大学(京都市)
    • Year and Date
      2015-09-10
    • Int'l Joint Research / Invited
  • [Presentation] Stability of heat kernel estimates and parabolic Harnack inequalities for jump processes on metric measure spaces2015

    • Author(s)
      Takashi Kumagai
    • Organizer
      International Conference on Stochastic Analysis and Related Topics
    • Place of Presentation
      Wuhan University (China)
    • Year and Date
      2015-08-05
    • Int'l Joint Research / Invited
  • [Presentation] Stability of heat kernel estimates and parabolic Harnack inequalities for jump processes on metric measure spaces2015

    • Author(s)
      Takashi Kumagai
    • Organizer
      SPA 2015 Oxford
    • Place of Presentation
      University of Oxford(UK)
    • Year and Date
      2015-07-15
    • Int'l Joint Research / Invited
  • [Presentation] Quantum walks in one dimension2015

    • Author(s)
      Tatsuya Tate
    • Organizer
      XXXIV Workshop on Geometric Method in Physics
    • Place of Presentation
      Bialowieza, Poland
    • Year and Date
      2015-07-01
    • Int'l Joint Research / Invited
  • [Presentation] Heat kernel estimates and local CLT for random walk among random conductances with a power-law tail near zero2015

    • Author(s)
      Takashi Kumagai
    • Organizer
      University of Bath Probability Seminar
    • Place of Presentation
      University of BATH (Bath, UK)
    • Year and Date
      2015-06-08
    • Int'l Joint Research / Invited
  • [Presentation] Heat kernel estimates and local CLT for random walk among random conductances with a power-law tail near zero2015

    • Author(s)
      Takashi Kumagai
    • Organizer
      University of Bristol Probability seminar
    • Place of Presentation
      University of BRISTOL(Bristol, UK)
    • Year and Date
      2015-06-05
    • Int'l Joint Research / Invited
  • [Presentation] Convergence of metric measure spaces2015

    • Author(s)
      Takashi Shioya
    • Organizer
      Workshop on Analysis and Geometry in Metric Spaces
    • Place of Presentation
      ICMAT, Campus de Cantoblanco (Madrid, Spain)
    • Year and Date
      2015-06-04
    • Int'l Joint Research / Invited
  • [Presentation] Mathematical Challenge to Structural Understandingof Materials2015

    • Author(s)
      Motoko Kotani
    • Organizer
      Mathematics of NovelMaterials
    • Place of Presentation
      Mittag-Leffler Institute, Djursholm, Sweden
    • Year and Date
      2015-06-01
    • Int'l Joint Research / Invited
  • [Presentation] Mathematical Challenge to understanding of Materials2015

    • Author(s)
      Motoko Kotani
    • Organizer
      WE-Heraeus Seminar
    • Place of Presentation
      Physikzentrum Bad Honnef (ドイツ)
    • Year and Date
      2015-04-28
    • Int'l Joint Research / Invited
  • [Book] Gromov's theory of convergence and concentration of metrics and measures2016

    • Author(s)
      Shioya, Takashi
    • Total Pages
      194ページ
    • Publisher
      EMS Publishing House
  • [Book] 現代幾何学への招待2016

    • Author(s)
      宮岡礼子
    • Total Pages
      144ページ
    • Publisher
      サイエンス社
  • [Funded Workshop] RIMS workshop Stochastic Analysis2015

    • Place of Presentation
      京都大学理学研究科(京都市)
    • Year and Date
      2015-09-07 – 2015-09-11

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Published: 2018-01-16   Modified: 2022-01-31  

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