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Quantization of Anosov foliations and noncommutative geometry

Research Project

Project/Area Number 15540203
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionNagoya Institute of Technology

Principal Investigator

NATSUME Toshikazu  Nagoya Institute of Technology, Graduate School of Engineering, Professor of Mathematics, 大学院・工学研究科, 教授 (00125890)

Co-Investigator(Kenkyū-buntansha) ADACHI Toshiaki  Nagoya Institute of Technology, Graduate School of Engineering, Professor of Mathematics, 大学院・工学研究科, 教授 (60191855)
NAKAMURA Yoshihiro  Nagoya Institute of Technology, Graduate School of Engineering, Associate Professor of Mathematics, 大学院・工学研究科, 助教授 (50155868)
MORIYOSHI Hitoshi  Keio University, Faculty of Science and Engineering, Associate Professor of Mathematics, 理工学部, 助教授 (00239708)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2004: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2003: ¥2,000,000 (Direct Cost: ¥2,000,000)
KeywordsAnosov foliations / C^*-algebra / noncommutative geometry / K-theory / cyclic cohomology / quantization / C*-環 / 非可換幾何
Research Abstract

The purpose of this project is to obtain a quantum version of the results in "The Godbillon-Vey cyclic cocycle and longitudinal Dirac operators (with the investigator Hitoshi Moriyoshi)" and "Topological approach to quantum surfaces( with Ryszard Nest of the University of Copenhagen)", more precisely to construct noncommutative Anosov foliations on "the unit tangent bundles" over noncommutative Riemann surfaces. This noncommutative Anosov foliations are regarded as quantizations of the (commutative) Anosov foliations associated with geodesic flows on the unit tangent bundles. The ultimate goal of the project is to prove the foliation index theorem of A. Connes, for the noncommutative Anosov foliations.
In a joint project with Nest (unpublished) we constructed noncommutative 3-manifolds as strict quantizations of unit circle bundles of closed Riemann surfaces of genus greater than 1.These noncommutaive 3-manifolds were constructed in such a way that the relationship between the Riemann surface and its unit tangent bundle is kept intact through a suitable group action. Moreover we constructed a foliation on the noncommutaive 3-manifold as a certain C^*-algebra in the spirit of A. Connes's noncommutative geometry. We are preparing a paper "Noncommutaive Anosov foliations (tentative title)". We are currently working on detail. As one expects, on view of commutative case, the C^*-algebra representing a "leaf of the noncommutaive foliation is a covering space. We developed some idea how to lift the Dirac operator on the quantized Riemann surface to a longitudinal elliptic operator for the noncommutative Anosov foliation.
Unfortunately we were unable to complete the project. However, we certainly continue to work on the project, as we now have a clear idea how to achieve the goal.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (15 results)

All 2005 2004 2003 Other

All Journal Article (8 results) Book (3 results) Publications (4 results)

  • [Journal Article] Geometry of ordinary helices in a complex projective space2004

    • Author(s)
      T.Adachi, S.Maeda, S.Udagawa
    • Journal Title

      Hokkaido Journal of Mathematics 33

      Pages: 233-246

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Geometry of ordinary helics in a complex projective space2004

    • Author(s)
      T.Adachi, S.Maeda, S.Udagawa
    • Journal Title

      Hokkaido Journal of Mathematics 33

      Pages: 233-246

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A new family of nnoncommutative 2-spheres2003

    • Author(s)
      T.Natsume, C.L.Olsen
    • Journal Title

      Journal of Functional Analysis 202

      Pages: 363-391

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Strict quantization of symplectic manifolds2003

    • Author(s)
      T.Natsume, R.Nest, I.Peter
    • Journal Title

      Letters in Mathematical Pysics 66

      Pages: 73-89

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Lamination of moduli space of circles and their length spectrum for a non-flat complex form2003

    • Author(s)
      T.Adachi
    • Journal Title

      Osaka Journal of Mathematics 40

      Pages: 895-916

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Quaternionic distribution of curvature-adapted real hypersurfaces in quaternionic hyperbolic space2003

    • Author(s)
      T.Adachi, S.Maeda
    • Journal Title

      Journal of Geometry 75

      Pages: 1-14

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A new family of noncommutative 2-spheres2003

    • Author(s)
      T.Natsume, C.L.Olsen
    • Journal Title

      Journal of Functional Analysis 202

      Pages: 363-391

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Lamination of moduli space of crcles and their length spectrum for a nonflat complex form2003

    • Author(s)
      T.Adachi
    • Journal Title

      Osaka Journal of Mathematics 40

      Pages: 895-916

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Book] 数理物理への誘い5(河東泰之編)2005

    • Author(s)
      夏目 利一
    • Publisher
      遊星社
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Book] A strange world of nonccommutative geometry(in Japanese), An invitation to mathematical physics 5(Edit. : Y.Kawahigashi)2005

    • Author(s)
      T.Natsume
    • Publisher
      Yuseisha
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Book] 数理物理への誘い5(河東泰之編)2005

    • Author(s)
      夏目 利
    • Publisher
      遊星社
    • Related Report
      2004 Annual Research Report
  • [Publications] T.Natsume, R.Nest: "Strict quantization of symplectic manifolds"Letters in Mathematical Physics. 66. 73-89 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Natsume, C.L.Olsen: "A new family of noncommutative 2-spheres"Journal of Functional Analysis. 202. 363-391 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Adachi, S.Maeda: "Lamination of moduli space of circles and their length spectrum for a Non-flat complex space form"Osaka Journal of Mathematics. 40. 895-916 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Adachi, S.Maeda: "Circles and hypersurfaces in space forms"Memoirs of Faculty of Science and Engineering, Shimane University, Series B, Mathematical Science. 1-9 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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