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Floer cohomology of Lagrangian submanifolds with non-commutative group actions

Research Project

Project/Area Number 16K05120
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

IRIYEH Hiroshi  茨城大学, 理工学研究科(理学野), 准教授 (30385489)

Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
KeywordsFloerコホモロジー / ラグランジュ部分多様体 / 旗多様体 / 対蹠集合 / 実形 / 凸体 / Mahler予想 / Hofer-Zehnder容量 / トポロジー / Floerホモロジー / 幾何学
Outline of Final Research Achievements

We carried out concrete research for symplectic invariants, especially, Lagrangian Floer cohomology. In particular, we explicitly calculated the Lagrangian Floer cohomology of a pair of real forms even with different topological types in a complex flag manifold on which a non-commutative group acts transitively. Furthermore, we affirmatively solved the three dimensional case of Mahler's conjecture concerning the volume of convex bodies. As an application, we obtained new knowledge about the Hamiltonian dynamics on a class of six dimensional convex domains.

Academic Significance and Societal Importance of the Research Achievements

本研究課題の主題であるシンプレクティック不変量を通じて、凸幾何の分野の有名な古典的未解決問題であるMahler予想(1939年)に本質的な寄与ができた。特に、3次元の場合を解決した論文は凸幾何の分野のかなり多くの研究者の興味を喚起し、国際的な波及効果が出ている。Mahler予想は、凸幾何の中でも体積に関わる中心課題の一つであるため、今回の成果の高次元化に向かって、関連する分野の活性化が十分に期待できる。

Report

(5 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (13 results)

All 2020 2019 2018 2017 2016 Other

All Int'l Joint Research (2 results) Journal Article (3 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 3 results,  Open Access: 2 results,  Acknowledgement Compliant: 1 results) Presentation (8 results) (of which Int'l Joint Research: 5 results,  Invited: 6 results)

  • [Int'l Joint Research] 清華大学(中国)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] 国立成功大学(台湾)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Symmetric Mahler’s conjecture for the volume product in the $3$ -dimensional case2020

    • Author(s)
      Iriyeh Hiroshi、Shibata Masataka
    • Journal Title

      Duke Mathematical Journal

      Volume: 169 Issue: 6 Pages: 1077-1134

    • DOI

      10.1215/00127094-2019-0072

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Symplectic topology of Lagrangian submanifolds of CPn with intermediate minimal Maslov numbers2017

    • Author(s)
      Hiroshi Iriyeh
    • Journal Title

      Advances in Geometry

      Volume: 17 Issue: 2 Pages: 247-264

    • DOI

      10.1515/advgeom-2017-0005

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] HAMILTONIAN NON-DISPLACEABILITY OF GAUSS IMAGES OF ISOPARAMETRIC HYPERSURFACES2016

    • Author(s)
      H. Iriyeh, H. Ma, R. Miyaoka, Y. Ohnita
    • Journal Title

      Bull. Lond. Math. Soc.

      Volume: 15 Issue: 12 Pages: 1-11

    • DOI

      10.1112/blms/bdw040

    • URL

      https://ocu-omu.repo.nii.ac.jp/records/2016839

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] A proof of Mahler's conjecture for the volume product of three dimensional convex bodies2019

    • Author(s)
      Hiroshi Iriyeh
    • Organizer
      East Asia Symplectic Conference 2019
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the volume product of three dimensional convex bodies with symmetries of a subgroup of O(3)2019

    • Author(s)
      Hiroshi Iriyeh
    • Organizer
      DGA2019--Differential Geometry and its Applications
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] The volume product of convex bodies with discrete symmetries2019

    • Author(s)
      Hiroshi Iriyeh
    • Organizer
      UK-Japan Winter School 2019 Variational problems in Geometry and Mathematical Physics
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Mahler's conjecture for three dimensional convex bodies and the rigidity of certain symplectic embeddings2018

    • Author(s)
      Hiroshi Iriyeh
    • Organizer
      HAYAMA Symposium on Complex Analysis in Several Variables XX & Pacific Rim Complex-Symplectic Geometry Conference
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 複素旗多様体内の二つの実形のFloerホモロジー2018

    • Author(s)
      井川治、入江博、奥田隆幸、酒井高司、田崎博之
    • Organizer
      日本数学会2018年度年会
    • Related Report
      2017 Research-status Report
  • [Presentation] On Mahler's conjecture in the three dimensional case2017

    • Author(s)
      Hiroshi Iriyeh
    • Organizer
      Singularities Symmetries & Submanifolds (the 18th UK-Japan Winter School in Mathematics)
    • Place of Presentation
      University College London (イギリス)
    • Year and Date
      2017-01-04
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 複素射影空間のLagrange部分多様体のホモロジー的剛性について2017

    • Author(s)
      入江博
    • Organizer
      研究集会「対称空間論とその周辺」
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 凸体に関するMahler予想について2017

    • Author(s)
      入江博
    • Organizer
      2017年度福岡大学微分幾何研究集会
    • Related Report
      2017 Research-status Report
    • Invited

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Published: 2016-04-21   Modified: 2022-02-16  

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