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General Galois theory for differential and difference equations and its applications to dynamical systems

Research Project

Project/Area Number 20540041
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNagoya University

Principal Investigator

UMEMURA Hiroshi  名古屋大学, 大学院・多元数理科学研究科, 名誉教授 (40022678)

Co-Investigator(Kenkyū-buntansha) KAWAHIRA Tomoki  名古屋大学, 大学院・多元数理科学研究科, 准教授 (50377975)
Project Period (FY) 2008 – 2012
Project Status Completed (Fiscal Year 2012)
Budget Amount *help
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2012: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2011: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2009: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2008: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords代数学 / 解析学 / 関数方程式 / 幾何学 / 微分ガロア理論 / 差分ガロア理論 / 可積分系 / 力学系
Research Abstract

(1) Equivalence of Galois theories. We had two general differential Galois theories: (a) Malgrange’s theory proposed in 2001 and (b) ours in 1996. The first is geometric and the second is algebraic. We proved that these two theories are equivalent by algebraic construction of the jet space and thus Lie groupoid of automorphisms.
(2) Construction of general Galois theory for difference equations and its applications to dynamical systems. We succeeded in constructing general difference Galois theory for difference equations. An elegant application of the theory is discrete dynamical systems on compact Riemann surfaces. In fact, we can determine all the discrete dynamical systems on compact Riemann surfaces that have finite dimensional Galois groups. Since the Galois group of these systems are solvable, we may say we have determined all the integrable discrete dynamical systems on compact Riemann surfaces,.
(3) Quantization of differential Galois theory Our student F.Heiderich discovered that we can generalize our Galois theory beyond difference and differenctial framework. He suggests that we can establish a similar theory for functuinal equations with respect to operators, The most fascinating feature of this view point is we may quantizeGalois theory. We started a irst research in this directions exploiting examples.

Report

(7 results)
  • 2012 Annual Research Report   Final Research Report ( PDF )
  • 2011 Annual Research Report
  • 2010 Annual Research Report   Self-evaluation Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (44 results)

All 2012 2011 2010 2009 2008

All Journal Article (20 results) (of which Peer Reviewed: 18 results) Presentation (22 results) Book (2 results)

  • [Journal Article] PIcard-Vessiot theory in general Galois theory, Algebraic Methods in Dynamical Systems, Banach Center of publ. 942011

    • Author(s)
      Umemura, Hiroshi
    • Journal Title

      Polish Acad, Sci.

      Pages: 263-293

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Picard-Vessiot theory in general Galois theory2011

    • Author(s)
      H.Umemura
    • Journal Title

      Banach Center publications

      Volume: 94 Pages: 263-293

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Discrete Burgers equation, Binomial coefficients and Mandala2010

    • Author(s)
      Morikawa Shuji, Saito Katsunori, Takeuchi Taihei, Umemura Hiroshi
    • Journal Title

      Mathematics in computer Science

      Volume: 4 Pages: 151-167

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Topology of the regular part for infinitely renormalizable quadratic polynomials2010

    • Author(s)
      Kawahira Tomoki
    • Journal Title

      Fund. Math

      Volume: 208 Pages: 35-56

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Toppology of the regular part for inifinitely renormalizable quadratic polynomials.2010

    • Author(s)
      Kawahira Tomoki
    • Journal Title

      Fund.Math. 208

      Pages: 35-56

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Discrete Burgers equation, Binomial coefficients and Mandala2010

    • Author(s)
      Morikawa Shuji, Saito Katsunori, Takeuchi Taihei, Umemura Hiroshi
    • Journal Title

      Mathematics in computer Science 4

      Pages: 151-167

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Topology of the regular part for infinitely renormalizable quadratic polynomials2010

    • Author(s)
      T.Kawahira
    • Journal Title

      Fund.Math.

      Volume: 208 Pages: 35-56

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On a general Galois theory of difference equatios 112010

    • Author(s)
      S.Morikawa, H.Umemura
    • Journal Title

      Annales de l'Institut Fourier 59

      Pages: 2733-2771

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On a general difference Galois theory. II2009

    • Author(s)
      Morikawa, Shuji, Umemura Hiroshi
    • Journal Title

      Ann. Inst Fourier (Grenoble)

      Volume: 59 Pages: 2733-2771

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] On the definition of Galois groupoid2009

    • Author(s)
      Umemura Hiroshi
    • Journal Title

      Asterisque

      Volume: 223 Pages: 441-452

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Tessellation and Lyubich-Minsky laminations associated with quadratic maps II.2009

    • Author(s)
      Kawahira Tomoki
    • Journal Title

      Topological structures of 3-laminations, Comform. Geom. Dyn.

      Volume: 13 Pages: 6-75

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Tessellation and Lyubich-Minsky laminations associated with quadratic maps I.2009

    • Author(s)
      Kawahira Tomoki
    • Journal Title

      Pinching semicomjugacies. Ergodic theory Dynam. Systems

      Volume: 29

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Tessellation and Lyubich-Minsky laminations associated with quadratic maps I. Pinching semicomjugacies.2009

    • Author(s)
      Kawahira Tomoki
    • Journal Title

      Ergodic theory Dynam.Systems 29(有)

      Pages: 2-2

    • Related Report
      2010 Self-evaluation Report
  • [Journal Article] Tessellation and Lyubich-Minsky laminations associated with quadratic maps II. Topological structures of 3-laminations2009

    • Author(s)
      Kawahira Tomoki
    • Journal Title

      Comform.Geom.Dyn. 13

      Pages: 6-75

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] On the definition of Galois groupoid.2009

    • Author(s)
      Umemura Hiroshi
    • Journal Title

      Asterisque 323

      Pages: 441-452

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] On a general difference Galois theory. II.2009

    • Author(s)
      Morikawa, Shuji, Umemura Hiroshi
    • Journal Title

      Ann.Inst Fourier (Grenoble) 59

      Pages: 2733-2771

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] On the definition of Galvis groupoid2009

    • Author(s)
      H. Umemula
    • Journal Title

      Asterisque 323(掲載決定済み)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Sur l'eequivalence des theories de Galois differentielles generals.2008

    • Author(s)
      Umemura Hiroshi
    • Journal Title

      C. R. Math. Acad. Sci. Paris

      Volume: 346

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Sur l' equivalence des theories de Galois differentielles generalles.2008

    • Author(s)
      Umemura Hiroshi
    • Journal Title

      C.R.Math.Acad.Sci.Paris 346(有)

      Pages: 21-22

    • Related Report
      2010 Self-evaluation Report
  • [Journal Article] Sun l'equivalence des theories de Galvis differentielles2008

    • Author(s)
      H. Umemula
    • Journal Title

      C. R. Ac Sa. Sa. l (論文番号(8934)) 346

      Pages: 1155-1158

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Presentation] Can we quantize differential Galois theory?2012

    • Author(s)
      Umemura Hiroshi
    • Place of Presentation
      Kyoto University Kyoto, Japan.
    • Related Report
      2012 Final Research Report
  • [Presentation] Can we quantize differential Galois theory?2012

    • Author(s)
      梅村浩
    • Organizer
      Various aspects around the Painleve equations
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2012 Annual Research Report
  • [Presentation] General differential Galois theory2011

    • Author(s)
      梅村浩
    • Organizer
      Equations differentielles et theorie de Galois
    • Place of Presentation
      IHES, France(招待講演)
    • Year and Date
      2011-10-19
    • Related Report
      2011 Annual Research Report
  • [Presentation] Soliton theory of Sato is abelian2011

    • Author(s)
      梅村浩
    • Organizer
      AADIOS2011
    • Place of Presentation
      Hilton Hotel Houston
    • Year and Date
      2011-07-27
    • Related Report
      2011 Annual Research Report
  • [Presentation] Soloton theory is Abelian2011

    • Author(s)
      Umemura Hiroshi
    • Place of Presentation
      Houston, Texsas U,S,A.
    • Related Report
      2012 Final Research Report
  • [Presentation] General differential Galois theory2011

    • Author(s)
      Umemura Hiroshi
    • Organizer
      Bicentenaire de Galois
    • Place of Presentation
      IHES, Paris France.
    • Related Report
      2012 Final Research Report
  • [Presentation] Picard-Vessiot theory in general differential Galois theory2010

    • Author(s)
      Umemura Hiroshi
    • Place of Presentation
      Barcelona University Barcelona, Spain.
    • Year and Date
      2010-09-08
    • Related Report
      2012 Final Research Report
  • [Presentation]2010

    • Author(s)
      Umemura Hiroshi
    • Organizer
      Picard-Vessiot theory in general differential Galois theory
    • Place of Presentation
      Barcelona University of Barcelona, Spain
    • Year and Date
      2010-09-08
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] Picard-Vessiot theory in general differential Galois theory2010

    • Author(s)
      梅村浩
    • Organizer
      Galois theory and explicit methods
    • Place of Presentation
      Barcelona, Spain
    • Year and Date
      2010-09-08
    • Related Report
      2010 Annual Research Report
  • [Presentation] Soliton theory of Sato is Abelian2010

    • Author(s)
      Umemura Hiroshi
    • Organizer
      Algebraic methods in Dynamical systems
    • Place of Presentation
      Institute of Mathematics Conference centre, Bedlewo, Poland.
    • Year and Date
      2010-05-20
    • Related Report
      2012 Final Research Report
  • [Presentation] Soliton theory of Sato is Abelian2010

    • Author(s)
      Umemura Hiroshi
    • Organizer
      Algebraic methods in Dynamical systems
    • Place of Presentation
      Institute of Mathematics Conference centre, Bedlewo, Poland
    • Year and Date
      2010-05-20
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] Soliton theory of Sato is Abelian2010

    • Author(s)
      梅村浩
    • Organizer
      Algebraic methods in Dynamical systems
    • Place of Presentation
      Bedlewo, Poland
    • Year and Date
      2010-05-20
    • Related Report
      2010 Annual Research Report
  • [Presentation] Theorie de Galois differentielle generale; Compariasons et applications2009

    • Author(s)
      Umemura Hiroshi
    • Organizer
      Journee Galois differential non-lineaire
    • Place of Presentation
      Institut de Mathematiwue a Jussieu. Universite de Paris, Paris France.
    • Year and Date
      2009-11-27
    • Related Report
      2012 Final Research Report
  • [Presentation] Theorie de Galois differentielle generale ; Comparaisons et applications2009

    • Author(s)
      Umemura Hiroshi
    • Organizer
      Journee Galois differential non-lineaire
    • Place of Presentation
      Institut de Mathematique a Jussieu. Universite de Paris, Paris France
    • Year and Date
      2009-11-27
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] Theories de Galois differentielles, comparaisons et applications2009

    • Author(s)
      H.Umemura
    • Organizer
      Universite de Paris VI
    • Place of Presentation
      パリ, フランス
    • Year and Date
      2009-11-27
    • Related Report
      2009 Annual Research Report
  • [Presentation] Discrete Burgers' equation, Binomial coefficients and Mandala2009

    • Author(s)
      Umemura Hiroshi
    • Organizer
      Applications of Computer Algebra
    • Place of Presentation
      Ecole Polytechnique,Montreal Canada.
    • Year and Date
      2009-06-26
    • Related Report
      2012 Final Research Report
  • [Presentation] Discrete Burgers' equation, Binomial coefficients and Mandala2009

    • Author(s)
      Umemura Hiroshi
    • Organizer
      Applications of Computer Algebra.
    • Place of Presentation
      Ecole Polytechnique,Montreal Canada
    • Year and Date
      2009-06-26
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] Discrete Burgers' equaton, Binomial coefficients and Mandala2009

    • Author(s)
      H.Umemura
    • Organizer
      Ecole Polytechnique a Montreal
    • Place of Presentation
      モントリオール, カナダ
    • Year and Date
      2009-06-26
    • Related Report
      2009 Annual Research Report
  • [Presentation] On a general principle of Galois theory2009

    • Author(s)
      H.Umemura
    • Organizer
      Algebraic theory of difference equations
    • Place of Presentation
      Leeds大学, イギリス
    • Year and Date
      2009-05-11
    • Related Report
      2009 Annual Research Report
  • [Presentation] Examples in differential Galvis theory2008

    • Author(s)
      H. Umemula
    • Organizer
      Portul differential quationo and diff. Galvis Th.
    • Place of Presentation
      CIRM, France
    • Year and Date
      2008-10-06
    • Related Report
      2008 Annual Research Report
  • [Presentation] General differential Galois theory2008

    • Author(s)
      Umemura Hiroshi
    • Organizer
      Symbolic Analysis Workshop
    • Place of Presentation
      Hong Kong City University, Hong Kong, China.
    • Year and Date
      2008-06-26
    • Related Report
      2012 Final Research Report
  • [Presentation] General differential Galois theory2008

    • Author(s)
      Umemura Hiroshi
    • Organizer
      Symbolic Analysis Workshop
    • Place of Presentation
      Hong Kong City University, Hong Kong, China
    • Year and Date
      2008-06-26
    • Related Report
      2010 Self-evaluation Report
  • [Book] ガロア 偉大なる曖昧さの理論2011

    • Author(s)
      梅村 浩
    • Publisher
      現代数学社
    • Related Report
      2012 Final Research Report
  • [Book] ガロア、偉大なる曖昧さの理論2011

    • Author(s)
      梅村浩
    • Total Pages
      255
    • Publisher
      現代数学社
    • Related Report
      2011 Annual Research Report

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Published: 2008-04-01   Modified: 2019-07-29  

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