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Study on vector bundles, D-modules and harmonic bundles

Research Project

Project/Area Number 22540078
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKyoto University

Principal Investigator

MOCHIZUKI TAKURO  京都大学, 数理解析研究所, 教授 (10315971)

Project Period (FY) 2010-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2014: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2013: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
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: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords調和バンドル / ツイスターD加群 / ホロノミックD加群 / 戸田格子 / インスタントン / ベッチ構造 / 混合ツイスターD加群 / 量子D加群 / 局所ミラー対称性 / Landau-Ginzbugモデル / モノポール / ナーム変換 / ホロノミックD-加群 / ツイスター構造 / 半無限ホッジ構造の変動 / ストークス構造 / 不確定特異点 / 有理型平坦束
Outline of Final Research Achievements

I published a monograph on the study of wild harmonic bundles and pure twistor D-modules. I introduced the concepts of mixed twistor structure and Betti structure on holonomic D-modules, and established their functoriality with respect to various standard operations. I applied the ideas and the results in the theory of wild harmonic bundles to the study of doubly periodic instantons and 2-dimensional Toda equations. I studied the asymptotic behaviour of doubly periodic instantons by regarding them as infinite dimensional harmonic bundles. On the basis of the result, I established that the Nahm transform give an equivalence between wild harmonic bundles on elliptic curve and doubly periodic instantons. As for the Toda equations, I observed an equivalence of solutions of the Toda equations and some type of harmonic bundles, which was useful for the classification of the solutions. I also computed the Stokes structure of the associated meromorphic flat bundles.

Report

(6 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Annual Research Report
  • 2012 Annual Research Report
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • Research Products

    (32 results)

All 2014 2012 2011 2010 Other

All Journal Article (6 results) (of which Peer Reviewed: 6 results) Presentation (24 results) (of which Invited: 14 results) Book (2 results)

  • [Journal Article] armonic bundles and Toda lattices with opposite sign II2014

    • Author(s)
      望月拓郎
    • Journal Title

      Communications in Mathematical Physics

      Volume: 328 Issue: 3 Pages: 1159-1198

    • DOI

      10.1007/s00220-014-1994-0

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic behaviour and the Nahm transform of doubly periodic instantons with square integrable curvature2014

    • Author(s)
      望月拓郎
    • Journal Title

      Geometry & Topology

      Volume: 18 Issue: 5 Pages: 2823-2949

    • DOI

      10.2140/gt.2014.18.2823

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The third cohomology groups of dihedral quandles2011

    • Author(s)
      Takuro Mochizuki
    • Journal Title

      Journal of Knot Theory and Its Ramifications

      Volume: 20 Issue: 07 Pages: 1041-1057

    • DOI

      10.1142/s0218216511009133

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Stokes structure of a good meromorphic flat bundle2011

    • Author(s)
      Takuro Mochizuki
    • Journal Title

      J.Inst.Math.Jussieu

      Volume: 10 Issue: 3 Pages: 675-712

    • DOI

      10.1017/s1474748011000065

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic behavior of variation of pure polarized TERP structure2011

    • Author(s)
      Takuro Mochizuki
    • Journal Title

      Publ.Res.Inst.Math.Sci.

      Volume: 47 Issue: 2 Pages: 419-534

    • DOI

      10.2977/prims/41

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Notes on Stokes structure of Fourier transform2010

    • Author(s)
      Takuro Mochizuki
    • Journal Title

      Acta Math.Vietnam

      Volume: 35 Pages: 107-158

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Presentation] Wild harmonic bundles and twistor D-modules2014

    • Author(s)
      Takuro Mochizuki
    • Organizer
      International Congress of Mathematicians
    • Place of Presentation
      COEX ソウル 韓国
    • Year and Date
      2014-08-20
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Mixed twistor structure and GKZ-hypergeometric systems2014

    • Author(s)
      Takuro Mochizuki
    • Organizer
      The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles
    • Place of Presentation
      シンガポール国立大学(シンガポール)
    • Year and Date
      2014-08-04
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] ホッジからツイスターへ-調和バンドルとその周辺の話題について-2012

    • Author(s)
      望月拓郎
    • Organizer
      日本数学会年会
    • Place of Presentation
      東京理科大学神楽坂キャンパス(招待講演)
    • Year and Date
      2012-03-26
    • Related Report
      2011 Annual Research Report
  • [Presentation] Twistor $D$-module and harmonic bundle2012

    • Author(s)
      Takuro Mochizuki
    • Organizer
      東京大学代数学コロキウム
    • Place of Presentation
      東京大学大学院数理科学研究科(招待講演)
    • Year and Date
      2012-02-22
    • Related Report
      2011 Annual Research Report
  • [Presentation] Twistor $D$-module and harmonic bundle2012

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Symposium on Projective Algebraic Varieties and Moduli
    • Place of Presentation
      ノボテルホテル釜山(韓国)(招待講演)
    • Year and Date
      2012-02-15
    • Related Report
      2011 Annual Research Report
  • [Presentation] turning pointの解消について2011

    • Author(s)
      望月拓郎
    • Organizer
      アクセサリーパラメーター研究会
    • Place of Presentation
      東京大学玉原国際セミナーハウス(招待講演)
    • Year and Date
      2011-10-09
    • Related Report
      2011 Annual Research Report
  • [Presentation] Kobayashi-Hitchin correspondence for D-modules2011

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Doctoral Forum Fudan-Kyoto
    • Place of Presentation
      京都大学(招待講演)
    • Year and Date
      2011-03-11
    • Related Report
      2010 Annual Research Report
  • [Presentation] 調和バンドルについて2011

    • Author(s)
      望月拓郎
    • Organizer
      北海道大学数論幾何セミナー
    • Place of Presentation
      北海道大学(招待講演)
    • Related Report
      2010 Annual Research Report
  • [Presentation] From Hodge toward Twistor2010

    • Author(s)
      望月拓郎
    • Organizer
      トポロジーの多様性
    • Place of Presentation
      城崎大会議館(招待講演)
    • Year and Date
      2010-12-03
    • Related Report
      2010 Annual Research Report
  • [Presentation] Kobayashi-Hitchin correspondence for D-modules2010

    • Author(s)
      Takuro Mochizuki
    • Organizer
      PRIMP Conference on Geometric Analysis
    • Place of Presentation
      ブリティッシュコロンビア大学(USA)(招待講演)
    • Year and Date
      2010-07-29
    • Related Report
      2010 Annual Research Report
  • [Presentation] Wild Harmonic Bundles2010

    • Author(s)
      Takuro Mochizuki
    • Organizer
      NCTS/TPE & TIMS Joint Seminar in Algebraic Geometry
    • Place of Presentation
      台湾国立大学(台湾)(招待講演)
    • Year and Date
      2010-06-10
    • Related Report
      2010 Annual Research Report
  • [Presentation] From Hodge toward Twistor2010

    • Author(s)
      Takuro Mochizuki
    • Organizer
      International Conference on Noncommutative Geometry and Physics
    • Place of Presentation
      京都大学(招待講演)
    • Related Report
      2010 Annual Research Report
  • [Presentation] Nahm transforms for harmonic bundles

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Development of Moduli Theory: Conference
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Wild harmonic bundles and some applications

    • Author(s)
      Takuro Mochizuki
    • Organizer
      The Asian Mathematical Conference 2013
    • Place of Presentation
      Convention Halls in BEXCO, Busan, Korea
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] The asymptotic behaviour of doubly periodic instantons and Stokes structure

    • Author(s)
      Takuro Mochizuki
    • Organizer
      国際研究集会Spring School on Algebraic Microlocal Analysis
    • Place of Presentation
      Northwestern大学(アメリカ合衆国)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Harmonic bundle and twistor D-module

    • Author(s)
      Takuro Mochizuki
    • Organizer
      国際研究集会 Singularities of differential equations in algebraic geometry
    • Place of Presentation
      CIRM(フランス)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] The asymptotic behaviour of doubly periodic instantons and Stokes structure

    • Author(s)
      望月拓郎
    • Organizer
      微分トポロジーセミナー
    • Place of Presentation
      京都大学
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] 調和バンドルと戸田格子について

    • Author(s)
      望月拓郎
    • Organizer
      豊田中研コロキウム
    • Place of Presentation
      豊田中央研究所
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] 調和バンドルと戸田格子

    • Author(s)
      望月拓郎
    • Organizer
      京都大学大談話会
    • Place of Presentation
      京都大学
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Harmonic bundle and Toda lattice with opposite sign

    • Author(s)
      Takuro Mochizuk
    • Organizer
      国際研究集会 Various Aspects on the Painleve Equations
    • Place of Presentation
      京都大学
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] 調和バンドルと戸田格子

    • Author(s)
      望月拓郎
    • Organizer
      研究集会「代数幾何とその周辺」
    • Place of Presentation
      高知大学
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] 調和バンドルと戸田格子

    • Author(s)
      望月拓郎
    • Organizer
      大阪大学幾何セミナー
    • Place of Presentation
      大阪大学
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Harmonic bundles and Toda lattices with opposite sign

    • Author(s)
      Takuro Mochizuki
    • Organizer
      Kyoto University and SNU joint symposium on mathematics
    • Place of Presentation
      ソウル国立大学(韓国)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Harmonic bundles and Toda lattices with opposite sign

    • Author(s)
      Takuro Mochizuki
    • Organizer
      国際研究集会 "Recent trends in Algebraic Analysis"
    • Place of Presentation
      パドヴァ大学(イタリア)
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Book] Holonomic D-modules with Betti structure2014

    • Author(s)
      Takuro Mochizuki
    • Total Pages
      205
    • Publisher
      Societe Mathematique de France
    • Related Report
      2014 Annual Research Report
  • [Book] Wild Harmonic Bundles and Wild Pure Twistor D-modules2011

    • Author(s)
      Takuro Mochizuki
    • Total Pages
      607
    • Publisher
      Societe Mathematique de France
    • Related Report
      2011 Annual Research Report

URL: 

Published: 2010-08-23   Modified: 2019-07-29  

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