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Synthetic research on the relationship among various equivalence relations of singularities

Research Project

Project/Area Number 20540075
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionHyogo University of Teacher Education

Principal Investigator

KOIKE Satoshi  Hyogo University of Teacher Education, 大学院・学校教育研究科, 教授 (60161832)

Co-Investigator(Kenkyū-buntansha) FUKUI Toshizumi  埼玉大学, 理学部, 教授 (90218892)
Co-Investigator(Renkei-kenkyūsha) SHIOTA Masahiro  名古屋大学, 大学院・多元数理科学研究科, 教授 (00027385)
Project Period (FY) 2008 – 2010
Project Status Completed (Fiscal Year 2010)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2008: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Keywords微分トポロジー / 特異点論 / ブロー解析同値 / リプシッツ同値 / 点列選択性性質 / C^1同値 / 実ツリーモデル / 半直線-(SSP)条件 / 曲率集中化問題 / C^r同値 / 点列選択性条件 / 定義可能集合 / オーミニマル構造 / 実閉体 / 層化理論 / Lojasiewicz不等式 / ミルナー半経一様性 / Whitney条件(a)
Research Abstract

T.C.Kuo introduced the notion of blow-analytic equivalence as a desirable equivalence relation for real analytic singularities. In this research, we have proved that C^1 equivalence implies blow-analytic one. This is a negative result to the conjecture given by Kuo and other many real singularitists. In addition, we have proved that the directional dimension of definable sets in o-minimal structure over general real closed field is a Lipschitz invariant, generalizing the Lipschitz invariant I introduced in order to show that blow-analytic equivalence does not always imply Lipschitz one.

Report

(4 results)
  • 2010 Annual Research Report   Final Research Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (35 results)

All 2011 2010 2009 2008 Other

All Journal Article (12 results) (of which Peer Reviewed: 10 results) Presentation (18 results) Book (3 results) Remarks (2 results)

  • [Journal Article] Blow-analytic equivalence of two variable real analytic function germs2010

    • Author(s)
      S. Koike, A.Parusinski
    • Journal Title

      Journal of Algebraic Geometry 19

      Pages: 439-472

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Some questions on the Fukui numerical set for complex function germs2010

    • Author(s)
      S. Koike, A.Parusinski
    • Journal Title

      Demonstratio Mathematica XLIII

      Pages: 285-302

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] 実体上の接方向的性質の実閉体上への一般化2010

    • Author(s)
      小池敏司
    • Journal Title

      数理解析研究所講究録 1707

      Pages: 87-95

    • Related Report
      2010 Final Research Report
  • [Journal Article] Blow-analytic equivalence of two variable real analytic function germs2010

    • Author(s)
      S.Koike, A.Parusinski
    • Journal Title

      Journal of Algebraic Geometry

      Volume: 19 Pages: 439-472

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Some questions on the Fukui numerical set for complex function germs2010

    • Author(s)
      S.Koike, A.Parusinski
    • Journal Title

      Demonstratio Mathematica

      Volume: XLIII Pages: 285-302

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 実体上の接方向的性質の実閉体への一般化2010

    • Author(s)
      小池敏司
    • Journal Title

      RIMS Kokyuroku

      Volume: 1707 Pages: 87-95

    • Related Report
      2010 Annual Research Report
  • [Journal Article] Some questions on the Fukui numerical set for complex function germs2010

    • Author(s)
      S.Koike, A.Parusinski
    • Journal Title

      Demonstratio Mathematica XLIII(掲載確定)

      Pages: 285-302

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Blow-analytic equivalence of two variable real analytic function germs2010

    • Author(s)
      S.Koike, A.Parusinski
    • Journal Title

      Journal of Algebraic Geometry (掲載確定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The directional dimension of subanalytic sets is invariant under bi-Lipschits homeomorphisms2009

    • Author(s)
      S. Koike, L.Paunescu
    • Journal Title

      Annales del'Institut Fourier 59

      Pages: 2445-2467

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms2009

    • Author(s)
      S.Koike, L.Paunescu
    • Journal Title

      Annales de l'Institut Fourier 59

      Pages: 2445-2467

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Blow-analytic equivalence of two variable real analyticfunction germs

    • Author(s)
      S. Koike, A. Parusinski
    • Journal Title

      Journal of Algebraic Geometry 掲載確定

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms

    • Author(s)
      S. Koike, L. Paunescu
    • Journal Title

      Annales de l'Institut Fourier 掲載確定

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Presentation] Tarski-Seidenberg's type theorem in (SSP)2011

    • Author(s)
      小池敏司
    • Organizer
      Geometry Workshop
    • Place of Presentation
      The University of Sydney.
    • Year and Date
      2011-01-05
    • Related Report
      2010 Final Research Report
  • [Presentation] Tarski-Seidenberg's type theorem in (SSP) Geometry2011

    • Author(s)
      S.Koike
    • Organizer
      Workshop on Geometry and Singularities
    • Place of Presentation
      The University of Sydney
    • Year and Date
      2011-01-04
    • Related Report
      2010 Annual Research Report
  • [Presentation] Blow-analytic equivalence of two variable real analytic function germs2010

    • Author(s)
      小池敏司
    • Organizer
      54th Annual Meeting of the Australian Mathematical Society
    • Place of Presentation
      The University of Queensland.
    • Year and Date
      2010-09-27
    • Related Report
      2010 Final Research Report
  • [Presentation] Blow-analytic equivalence of two variable real analytic function germs2010

    • Author(s)
      S.Koike
    • Organizer
      54^<th> Annual Meeting of the Australian Mathematical Society
    • Place of Presentation
      The University of Queensland
    • Year and Date
      2010-09-27
    • Related Report
      2010 Annual Research Report
  • [Presentation] ナッシュ級のイソトピー補題と応用2010

    • Author(s)
      小池敏司
    • Organizer
      福田拓生先生退職記念研究集会「可微分写像の特異点」
    • Place of Presentation
      日本大学
    • Year and Date
      2010-07-21
    • Related Report
      2010 Final Research Report
  • [Presentation] 六級の〓補題と応用2010

    • Author(s)
      小池敏司
    • Organizer
      研究集会「可微分写像の特異点」
    • Place of Presentation
      日本大学文理学部
    • Year and Date
      2010-07-21
    • Related Report
      2010 Annual Research Report
  • [Presentation] Various blow-analytic invariants of real analytic singularities2010

    • Author(s)
      小池敏司
    • Organizer
      第5回代数・解析・幾何セミナー
    • Place of Presentation
      鹿児島大学
    • Year and Date
      2010-02-16
    • Related Report
      2010 Final Research Report
  • [Presentation] Various blow-analytic invariants of real analytic singularities2010

    • Author(s)
      小池敏司
    • Organizer
      第5階代数・解析・幾何セミナー
    • Place of Presentation
      鹿児島大学
    • Year and Date
      2010-02-16
    • Related Report
      2009 Annual Research Report
  • [Presentation] 解析関数のレベル集合の曲率問題2010

    • Author(s)
      小池敏司
    • Organizer
      神戸特異点研究会
    • Place of Presentation
      兵庫教育大学神戸サテライト
    • Year and Date
      2010-01-21
    • Related Report
      2009 Annual Research Report
  • [Presentation] 実閉体上のオーミニマル構造の定義可能集合の接方向的性質2009

    • Author(s)
      小池敏司
    • Organizer
      数理解析研究所研究集会
    • Place of Presentation
      日本大学
    • Year and Date
      2009-12-08
    • Related Report
      2010 Final Research Report 2009 Annual Research Report
  • [Presentation] Equisingularity problem on real algebraic singularities2009

    • Author(s)
      小池敏司
    • Organizer
      The 3rd Australian-Japanese Workshop on Real and Complex Singularities
    • Place of Presentation
      The University of Sydney.
    • Year and Date
      2009-09-18
    • Related Report
      2010 Final Research Report
  • [Presentation] Equisingularity problem on real algebraic singularities2009

    • Author(s)
      小池敏司
    • Organizer
      The 3^<rd> Australian-Japanese Workshop on Real and Complex Singularities
    • Place of Presentation
      University of Sydney(オーストラリア)
    • Year and Date
      2009-09-18
    • Related Report
      2009 Annual Research Report
  • [Presentation] Directional properties of sets definable in o-minimal structure III2009

    • Author(s)
      小池敏司
    • Organizer
      Seminar on Singularities and Topology
    • Place of Presentation
      University of Sydney(オーストラリア)
    • Year and Date
      2009-09-04
    • Related Report
      2009 Annual Research Report
  • [Presentation] Directional properties of sets definable in o-minimal structure II2009

    • Author(s)
      小池敏司
    • Organizer
      Seminar on Singularities and Topology
    • Place of Presentation
      University of Sydney(オーストラリア)
    • Year and Date
      2009-09-02
    • Related Report
      2009 Annual Research Report
  • [Presentation] Directional properties of sets definable in o-minimal structure I2009

    • Author(s)
      小池敏司
    • Organizer
      Seminar on Singularities and Topology
    • Place of Presentation
      University of Sydney (オーストラリア)
    • Year and Date
      2009-08-31
    • Related Report
      2009 Annual Research Report
  • [Presentation] オーミニマル構造の定義可能集合の接方向的性質2009

    • Author(s)
      小池敏司
    • Organizer
      北海道大学特異点論セミナー
    • Place of Presentation
      北海道大学
    • Year and Date
      2009-08-07
    • Related Report
      2009 Annual Research Report
  • [Presentation] 実解析的特異点のブロー解析理論における最近の進展2008

    • Author(s)
      小池敏司
    • Organizer
      第55回トポロジーシンポジウム
    • Place of Presentation
      金沢文化ホール
    • Year and Date
      2008-08-09
    • Related Report
      2010 Final Research Report
  • [Presentation] 実解析的特異点のブロー解析理論における最近の進展2008

    • Author(s)
      小池敏司
    • Organizer
      第55回トポロジーシンポジウム
    • Place of Presentation
      金沢市文化ホール・石川
    • Year and Date
      2008-08-09
    • Related Report
      2008 Annual Research Report
  • [Book] The Japanese-Australian Workshop on Real and Complex Singularities JARCS III2010

    • Author(s)
      L.Paunescu, A.Isaev, A.Harris, T.Fukui, S.Koike(編)
    • Total Pages
      174
    • Publisher
      Australian National University
    • Related Report
      2010 Final Research Report
  • [Book] Proceedings of the Japanese-Australian Workshop on Real and Complex Singularities JARCS III2010

    • Author(s)
      T.Fukui, A.Harris, A.Isaev, S.Koike, L.Paunescu (編)
    • Total Pages
      174
    • Publisher
      Australian National University
    • Related Report
      2010 Annual Research Report
  • [Book] The second Japanese-Australian Workshop on Real and Complex Singularities (RIMS Kokyuroku 1610)2008

    • Author(s)
      S.Koike(編)
    • Total Pages
      128
    • Publisher
      Kyoto University
    • Related Report
      2010 Final Research Report
  • [Remarks] 以下のホームページに、次のプレプリントのPDFファイルを置いている。

    • URL

      http://www.sci.hyogo-u.ac.jp/koike/

    • Related Report
      2010 Final Research Report
  • [Remarks]

    • URL

      http://www.sci.hyogo-u.ac.jp/koike/

    • Related Report
      2010 Annual Research Report

URL: 

Published: 2008-04-01   Modified: 2016-04-21  

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