• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Development of the index theorem on foliated manifolds

Research Project

Project/Area Number 25400085
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionNagoya University

Principal Investigator

Moriyoshi Hitoshi  名古屋大学, 多元数理科学研究科, 教授 (00239708)

Co-Investigator(Renkei-kenkyūsha) NATSUME TOSHIKAZU  名古屋工業大学, 工学系研究科, 教授 (00125890)
MAEDA YOSHIAKI  慶應義塾大学, 理工学部, 教授 (40101076)
MITSUMATSU YOSHIHIKO  中央大学, 理工学部, 教授 (70190725)
ONO KAORU  京都大学, 数理解析研究所, 教授 (20204232)
MIYAZAKI NAOYA  慶應義塾大学, 経済学部, 教授 (50315826)
TAKAKURA TATSURU  中央大学, 理工学部, 教授 (30268974)
TATE TETSUYA  名古屋大学, 大学院多元数理科学研究科, 准教授 (00317299)
Project Period (FY) 2013-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords指数定理 / 非可換幾何 / 葉層多様体 / Godbillon-Vey 不変量 / K理論 / 巡回コホモロジー / Godbillon-Vey不変量
Outline of Final Research Achievements

First, we extended the index theorem to fractal sets such as the Cantor set and the Sierpinski gasket. Second, by exploiting the framework of Noncommutative Geometry we generalized the Atiyah-Patodi-Singer index theorem to a Galois covering of compact manifold with boundary, which gives a formula for the pairing between K-group and cyclic cohomology. Third, we clarified the relation of the Dixmier-Douady class and the Godbillon-Vey class, which respectively appears as a characteristic class for Gerbe and foliated circle bundles. It turned out that they are connected via the Cheeger-Chern-Simons invariant. As a byproduct we succeeded to describe the universal central extension of circle diffeomorphism group in terms of the Calabi invariant.

Report

(4 results)
  • 2015 Annual Research Report   Final Research Report ( PDF )
  • 2014 Research-status Report
  • 2013 Research-status Report
  • Research Products

    (17 results)

All 2016 2015 2014 Other

All Int'l Joint Research (2 results) Journal Article (4 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 3 results,  Acknowledgement Compliant: 2 results) Presentation (11 results) (of which Invited: 8 results)

  • [Int'l Joint Research] University of Rome(Italy)

    • Related Report
      2015 Annual Research Report
  • [Int'l Joint Research] University od Colorado(米国)

    • Related Report
      2015 Annual Research Report
  • [Journal Article] A note on the higher Atiyah-Patodi-Singer index theorem on Galois coverings2016

    • Author(s)
      Alexander Gorokhovsky, Hitoshi Moriyoshi and Paolo Piazza
    • Journal Title

      Journal of Noncommutative Geometry

      Volume: 10

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] The Calabi invariant and central extensions of diffeomorphism groups2016

    • Author(s)
      Hitoshi Moriyoshi
    • Journal Title

      Geometry and Topology of Manifolds

      Volume: none

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] カントール集合上の指数定理2015

    • Author(s)
      Hitoshi Moriyoshi
    • Journal Title

      Geometry and Something 報告集

      Volume: none

    • Related Report
      2015 Annual Research Report
  • [Journal Article] A note on the higher Atiyah-Patodi-Singer index theorem on Galois coverings2015

    • Author(s)
      Alexander Gorokhovsky, Hitoshi Moriyoshi and Paolo Piazza,
    • Journal Title

      Journal of Noncommutative Geometry

      Volume: 10

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Presentation] フラクタル集合上の指数定理2016

    • Author(s)
      Hitoshi Moriyoshi
    • Organizer
      香川セミナー
    • Place of Presentation
      香川大学,高松
    • Year and Date
      2016-02-06
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] カントール集合上の指数定理2015

    • Author(s)
      Hitoshi Moriyoshi
    • Organizer
      福岡大学微分幾何研究会
    • Place of Presentation
      福岡大学,福岡
    • Year and Date
      2015-11-01
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] The Calabi invariant and group extensions of diffeomorphisms2015

    • Author(s)
      Hitoshi Moriyoshi
    • Organizer
      Colloquium at Institute of Mathematics, VAST
    • Place of Presentation
      Hanoi, Vietnam
    • Year and Date
      2015-04-02
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] The Godbillon-Vey index theorem in type III operator algebras2014

    • Author(s)
      H. Moriyoshi
    • Organizer
      The Second China-Japan Conference on Noncommutative Geometry and K-Theory
    • Place of Presentation
      Naha, Okinawa, Japan
    • Year and Date
      2014-10-04 – 2014-10-07
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] The Calabi invariant and group extensions of Diffeomorphisms2014

    • Author(s)
      H. Moriyoshi
    • Organizer
      第10回日中友好幾何学研究集会
    • Place of Presentation
      Fudan University, Shanghai, China
    • Year and Date
      2014-09-06 – 2014-09-12
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] The Calabi invariant and extensions of diffeomorphism groups2014

    • Author(s)
      H. Moriyoshi
    • Organizer
      Workshop Geometry and Dynamics of Foliations
    • Place of Presentation
      Madrid, Spain
    • Year and Date
      2014-09-01 – 2014-09-05
    • Related Report
      2014 Research-status Report
  • [Presentation] The Calabi invariant and2014

    • Author(s)
      H. Moriyoshi
    • Organizer
      服部晶夫先生研究報告会
    • Place of Presentation
      東京大学大学院数理科学研究科
    • Year and Date
      2014-06-27 – 2014-06-28
    • Related Report
      2014 Research-status Report
  • [Presentation] Secondary cyclic cocycle for Connes’ Chern character on a Cantor set,

    • Author(s)
      H. Moriyoshi
    • Organizer
      China-Japan Conference on Noncommutative Geometry and K-Theory,
    • Place of Presentation
      Changchun, China
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] On a cyclic volume cocycle in Fractal Geometry,

    • Author(s)
      H. Moriyoshi
    • Organizer
      Further development of Atiyah-Singer index theorem and K-theory,
    • Place of Presentation
      Kyoto, Japan
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Quantum volume form on Cantor sets,

    • Author(s)
      H. Moriyoshi
    • Organizer
      非可換幾何湯谷研究集会
    • Place of Presentation
      愛知県新城市
    • Related Report
      2013 Research-status Report
  • [Presentation] A generalization of the Atiyah-Singer index theorem in Fractal Geometry,

    • Author(s)
      H. Moriyoshi
    • Organizer
      Topological phases in spintronics
    • Place of Presentation
      Sendai, Japan
    • Related Report
      2013 Research-status Report
    • Invited

URL: 

Published: 2014-07-25   Modified: 2022-01-27  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi