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Development of Floer theory and study on symplectic structures

Research Project

Project/Area Number 26247006
Research Category

Grant-in-Aid for Scientific Research (A)

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

Principal Investigator

ONO Kaoru  京都大学, 数理解析研究所, 教授 (20204232)

Co-Investigator(Kenkyū-buntansha) 泉屋 周一  北海道大学, 理学研究院, 名誉教授 (80127422)
石川 剛郎  北海道大学, 理学研究院, 教授 (50176161)
枡田 幹也  大阪市立大学, 大学院理学研究科, 教授 (00143371)
三松 佳彦  中央大学, 理工学部, 教授 (70190725)
秦泉寺 雅夫  北海道大学, 理学研究院, 准教授 (20322795)
松下 大介  北海道大学, 理学研究院, 准教授 (90333591)
Research Collaborator Ogawa Noboru  
Project Period (FY) 2014-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥40,040,000 (Direct Cost: ¥30,800,000、Indirect Cost: ¥9,240,000)
Fiscal Year 2018: ¥7,800,000 (Direct Cost: ¥6,000,000、Indirect Cost: ¥1,800,000)
Fiscal Year 2017: ¥7,800,000 (Direct Cost: ¥6,000,000、Indirect Cost: ¥1,800,000)
Fiscal Year 2016: ¥8,190,000 (Direct Cost: ¥6,300,000、Indirect Cost: ¥1,890,000)
Fiscal Year 2015: ¥7,800,000 (Direct Cost: ¥6,000,000、Indirect Cost: ¥1,800,000)
Fiscal Year 2014: ¥8,450,000 (Direct Cost: ¥6,500,000、Indirect Cost: ¥1,950,000)
KeywordsFloer 理論 / 擬正則曲線 / symplectic 構造 / Lagrange 部分多様体 / A-無限大構造 / トーリック多様体 / ミラー対称性 / 特異点 / シンプレクティック構造 / ラグランジュ部分多様体 / 正則曲線 / A 無限大構造 / 倉西構造と仮想的基本鎖 / スペクトル不変量 / 倉西構造 / ハミルトン微分同相写像 / 正則曲線と量子コホモロジー
Outline of Final Research Achievements

Floer theory and the theory of pseudo-holomorphic curves have provided powerful tools in the study of symplectic geometry and brought many important results. During the academic years 2014-2018, the following achievements on Floer theory and its applications were made public:Lagrangian Floer theory and mirror symmetry on compact toric manifolds (joint paper, Asterisque 376, 2016), Anti-symplectic involution and Floer cohomology (joint paper, Geometry and Topology, 2016).

Academic Significance and Societal Importance of the Research Achievements

symplectic 構造は、古典力学のハミルトン形式に現れる幾何構造で、1980年代に研究対象としての豊かさが認識されて以来、活発な研究が続いている。その時の議論に用いられた擬正則曲線の理論であり、ハミルトン系の周期解を探すための変分法的手法は Floer 理論へと発展した。本研究では、Floer 理論の理論的側面の研究とともに、symplectic 幾何学への応用において成果を挙げた。理論物理学からの示唆に始まるミラー対称性についても、コンパクトケーラートーリック多様体と呼ばれるクラスの空間に対して Lagrangian Floer 理論を用いることで部分的ではあるが検証を行った。

Report

(6 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • Research Products

    (70 results)

All 2020 2019 2018 2017 2016 2015 2014 Other

All Int'l Joint Research (18 results) Journal Article (10 results) (of which Int'l Joint Research: 9 results,  Peer Reviewed: 9 results,  Open Access: 1 results,  Acknowledgement Compliant: 2 results) Presentation (35 results) (of which Int'l Joint Research: 28 results,  Invited: 35 results) Book (2 results) Remarks (2 results) Funded Workshop (3 results)

  • [Int'l Joint Research] Simons Center for Geometry and Physics/Columbia University(米国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] IBS Center for Geometry and Physics(韓国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Sichuan University(中国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Australian National University(オーストラリア)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Czech Academy of Science(チェコ)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Simons Center for Geometry and Physics(米国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] IBS Center for Geometry and Physics(韓国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] 四川大学(中国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Australian National University(オーストラリア)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] University of Nantes(フランス)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research]

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Simons Center for Geometry and Physics(米国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] IBS Center for Geometry and Physics(韓国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] University of Nantes(フランス)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] Czech Academy of Science(チェコ)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] IBS Center for Geometry and Physics(韓国)

    • Related Report
      2015 Annual Research Report
  • [Int'l Joint Research] Universite de Nantes(フランス)

    • Related Report
      2015 Annual Research Report
  • [Int'l Joint Research] Czech Academy of Sciences(チェコ)

    • Related Report
      2015 Annual Research Report
  • [Journal Article] Spectral invariants with bulk, quasimorphisms and Lagrangian Floer theory2020

    • Author(s)
      K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono
    • Journal Title

      Memoirs of American Mathematical Society

      Volume: 出版中

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Spectral invariants with bulk, quasimorphisms and Lagrangian Floer theory2020

    • Author(s)
      Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta and Kaoru Ono
    • Journal Title

      Memoir of American Mathematical Society

      Volume: 印刷中

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Construction of Kuranishi structures on the moduli spaces of holomorphic disks, I2018

    • Author(s)
      K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono
    • Journal Title

      Surveys in Differential Geometry

      Volume: 22 Pages: 133-190

    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Journal Article] Antisymplectic involution and Floer cohomology2017

    • Author(s)
      K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono
    • Journal Title

      Geometry and Topology

      Volume: 21 Issue: 1 Pages: 1-106

    • DOI

      10.2140/gt.2017.21.1

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Symplectic fillings of links of quotient surface singularities - corrigendum2017

    • Author(s)
      M. Bhupal, K. Ono
    • Journal Title

      Nagoya Mathematical Journal

      Volume: 225 Pages: 207-212

    • DOI

      10.1017/nmj.2016.42

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Shrinking good coordinate systems associated to Kuranishi structures2017

    • Author(s)
      Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta and Kaoru Ono
    • Journal Title

      Journal of Symplectic Geometry

      Volume: 未定

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Anti-symplectic involutions and Floer cohomology2017

    • Author(s)
      Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta and Kaoru Ono
    • Journal Title

      Geometry and Topology

      Volume: 未定

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] On the fixed point set of a Hamiltonian diffeomorphism in presense of fundamental group2016

    • Author(s)
      Kaoru Ono and Andrei Pajitnov
    • Journal Title

      Essays in Mathematics and its Applicaitons

      Volume: I Pages: 199-228

    • DOI

      10.1007/978-3-319-31338-2_10

    • ISBN
      9783319313368, 9783319313382
    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Shrinking good coordinate systems associated to Kuranishi structures2016

    • Author(s)
      K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono
    • Journal Title

      Journal of Symplectic Geometry

      Volume: 14 Issue: 4 Pages: 1295-1310

    • DOI

      10.4310/jsg.2016.v14.n4.a10

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Some remarks on Lagrangian tori2015

    • Author(s)
      Kaoru Ono
    • Journal Title

      Journal of Fixed Point Theory and Applications

      Volume: 17 Issue: 1 Pages: 221-237

    • DOI

      10.1007/s11784-015-0248-x

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Presentation] Floer-Novikov cohomology, revisited2019

    • Author(s)
      Kaoru Ono
    • Organizer
      Special Session on SYZ mirror symmetry and enumerative geometry, Spring Central and Western Joint Sectional Meeting, AMS
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Floer theory and covering spaces2018

    • Author(s)
      Kaoru Ono
    • Organizer
      Symplectic Geometry Session, Joint Meeting of American Mathematical Society and Chinese Mathematical Society
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Twisted sectors in Lagrangian Floer theory2018

    • Author(s)
      Kaoru Ono
    • Organizer
      Differential Geometry, Geometric Analysis and PDEs
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Examples of superheavy sets2018

    • Author(s)
      Kaoru Ono
    • Organizer
      NCTS Symplectic Expedition: Floer theory and beyond
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Anti-symplectic involution and twisted sectors in Lagrangian Floer theory2018

    • Author(s)
      Kaoru Ono
    • Organizer
      Simons Collaboration Workshop on Homological Mirror Symmetry and Hodge theory
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Twisted sectors in Lagrangian Floer theory2017

    • Author(s)
      Kaoru Ono
    • Organizer
      Holomorphic curves and symplectic topology, Institute Mittag-Leffler
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Twisted sectors in Lagrangian Floer theory2017

    • Author(s)
      Kaoru Ono
    • Organizer
      The second BICMR & IBS-CGP Joint Symplectic Geometry Workshop
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Twisted sectors in Lagrangian Floer theory2017

    • Author(s)
      Kaoru Ono
    • Organizer
      Algebraic Geometry and Symplectic Geometry at Crossroads
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Twisted sectors in Lagrangian Floer theory2017

    • Author(s)
      Kaoru Ono
    • Organizer
      Seminar on symplectic and contact geometry at Univ. Cologne
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Lagrangian Floer theory2017

    • Author(s)
      Kaoru Ono
    • Organizer
      Colloquim at University of Amsterdam
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Twisted sectors in Lagrangian Floer theory2017

    • Author(s)
      Kaoru Ono
    • Organizer
      Mini-workshop on Symplectic Geometry at University of Amsterdam
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Anti-symplectic involution and twisted sectors in Lagrangian Floer theory2017

    • Author(s)
      Kaoru Ono
    • Organizer
      Mirror symmetry and related topics, 2017
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Lagrangian Floer theory - around generating criterion for Fukaya category2017

    • Author(s)
      Kaoru Ono
    • Organizer
      UK-Japan Winter School, University Collgege London
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Floer theory and its applications2017

    • Author(s)
      Kaoru Ono
    • Organizer
      Differential Geometry Seminar, University of California, Irvine
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Some results on (super) heaviness for subsets in symplectic manifolds2016

    • Author(s)
      Kaoru Ono
    • Organizer
      International Conference ``Gromov-Witten theory, Gauge theory and Dualities''
    • Place of Presentation
      Australian National University, Kioloa Campus, Kioloa, Australia
    • Year and Date
      2016-01-12
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Symplectic Floer theory and its applications2016

    • Author(s)
      Kaoru Ono
    • Organizer
      Introductory minicourse ``Gromov-Witten theory, Gauge theory and Dualities''
    • Place of Presentation
      Australian National University, Canberra, Australia
    • Year and Date
      2016-01-06
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Floer theory and covering spaces2016

    • Author(s)
      Kaoru Ono
    • Organizer
      Mathematical Institute Seminar, San Yet Sun University
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Floer theory and covering spaces2016

    • Author(s)
      Kaoru Ono
    • Organizer
      Low Dimensional Topology and Symplectic Geomertry (7th Pacific Rim Mathematics Conference)
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Lagrangian Floer theory and Generation Criterion for Fukaya category2016

    • Author(s)
      Kaoru Ono
    • Organizer
      Trends in Modern Geomertry
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Non-displaceable Lagrangian submanifold2016

    • Author(s)
      Kaoru Ono
    • Organizer
      Differential Geometry and Symplectic Topology Seminar, University of Minnesota
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Lagrangian Floer theory - Generating criterion for Fukaya category and related topics2016

    • Author(s)
      Kaoru Ono
    • Organizer
      Yamabe Memorial Symposium, University of Minnesota
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Lagrangian Floer theory - Generating criterion for Fukaya category2016

    • Author(s)
      Kaoru Ono
    • Organizer
      KMS-MSJ Joint Meeting 2016
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Generatiion criterion for Fukaya category and related topics2016

    • Author(s)
      Kaoru Ono
    • Organizer
      Workshop on mirror symmetry and related topics
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Symplectic Floer 理論の話題から2016

    • Author(s)
      小野 薫
    • Organizer
      トポロジーシンポジウム
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Holomorphic curve technique in symplectic geometry - Lagrangian Floer theory2015

    • Author(s)
      Kaoru Ono
    • Organizer
      Conference on Geometry, Topology and Physics
    • Place of Presentation
      国立清華大学、台湾 新竹市
    • Year and Date
      2015-12-27
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Some results on (super)heaviness for subsets in symplectic manifolds2015

    • Author(s)
      Kaoru Ono
    • Organizer
      Workshop ``Symplectic Geometry and Topology''
    • Place of Presentation
      Uppsala University, Uppsala, Sweden
    • Year and Date
      2015-09-21
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] シンプレクティック幾何学での Floer 理論とその応用2015

    • Author(s)
      小野 薫
    • Organizer
      第62回幾何学シンポジウム
    • Place of Presentation
      東京理科大学 神楽坂キャンパス 東京都新宿区神楽坂
    • Year and Date
      2015-08-28
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Idempotents in quantum cohomology and non-displaceable Lagrangian submanifolds2015

    • Author(s)
      Kaoru Ono
    • Organizer
      International Conference on Symplectic Geometry and Mathematical Physics
    • Place of Presentation
      四川大学、中国 成都市
    • Year and Date
      2015-06-30
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Kuranishi structures2015

    • Author(s)
      Kaoru Ono
    • Organizer
      International Conference on Symplectic Geometry and Mathematical Physics
    • Place of Presentation
      四川大学、中国 成都市
    • Year and Date
      2015-06-22
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Idempotents in quantum cohomology and non-displaceable Lagrangian submanifolds2015

    • Author(s)
      Kaoru Ono
    • Organizer
      Symplectix Seminar
    • Place of Presentation
      College de France, Paris, France
    • Year and Date
      2015-05-22
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Floer theory and mirror symmetry for compact toric manifolds2015

    • Author(s)
      Kaoru Ono
    • Organizer
      Minicourse
    • Place of Presentation
      Institut de Mathematiques Jussieu-Paris Rive Gauche, Paris, France
    • Year and Date
      2015-05-19
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Idempotents in quantum cohomology and non-displaceable Lagrangian submanifolds2014

    • Author(s)
      Kaoru Ono
    • Organizer
      Primitive Forms, Mirror Symmetry and related topics, 2014
    • Place of Presentation
      京都大学大学院理学研究科数学教室
    • Year and Date
      2014-12-24
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Non-displaceable Lagrangian submanifolds, partial symplectic quasi-states and generation criterion for Fukaya category2014

    • Author(s)
      Kaoru Ono
    • Organizer
      Mathematics inspired by string theory
    • Place of Presentation
      Tsinghua Sanya International Mathematics Forum, Sanya, China
    • Year and Date
      2014-12-08
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Floer complex and covering spaces2014

    • Author(s)
      Kaoru Ono
    • Organizer
      Trends in Modern Geometry
    • Place of Presentation
      東京大学大学院数理科学研究科
    • Year and Date
      2014-07-11
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Floer complex and covering spaces2014

    • Author(s)
      Kaoru Ono
    • Organizer
      Mathematics New Goals
    • Place of Presentation
      National Center for Theoretical Sciences (South), Natinal Cheng Kung University, Tainan, Taiwan
    • Year and Date
      2014-07-01
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Book] Lagrangian Floer theory and mirror symmetry on compact toric manifolds2016

    • Author(s)
      Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta and Kaoru Ono
    • Total Pages
      340
    • Publisher
      Societe Mathematique de France
    • ISBN
      9782856298251
    • Related Report
      2016 Annual Research Report
  • [Book] Lagrangian Floer theory and mirror symmetry on compact topic manifolds2016

    • Author(s)
      Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta and Kaoru Ono
    • Total Pages
      340
    • Publisher
      Societe Mathematiques de France
    • Related Report
      2015 Annual Research Report
  • [Remarks]

    • URL

      http://www.kurims.kyoto-u.ac.jp/ja/list/ono.html

    • Related Report
      2017 Annual Research Report
  • [Remarks]

    • URL

      http://www.kurims.kyoto-u.ac.jp/ja/list/ono.html

    • Related Report
      2016 Annual Research Report
  • [Funded Workshop] Geometry and Everything - Fukaya 602019

    • Related Report
      2018 Annual Research Report
  • [Funded Workshop] Current Trends in Symplectic Geometry and Topology2017

    • Related Report
      2017 Annual Research Report
  • [Funded Workshop] Mirror Symmetry and Symplectic Geometry, Kyoto 20152015

    • Place of Presentation
      京都大学 理学研究科
    • Year and Date
      2015-12-10
    • Related Report
      2015 Annual Research Report

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Published: 2014-04-04   Modified: 2022-05-18  

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