Symplectic Geometry and Geometry of Loop Spaces

• Project/Area Number: 18K13407
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kyoto University (2021-2023); The University of Tokyo (2018-2020)
• Principal Investigator: 入江 慶 京都大学, 数理解析研究所, 准教授 (90645467)
• Project Period (FY): 2018-04-01 – 2025-03-31
• Project Status: Granted (Fiscal Year 2023)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: Floerホモロジー / ループ空間 / シンプレクティック容量 / 力学系の周期軌道 / シンプレクティック幾何 / Hamilton力学系 / 周期軌道 / レーブ力学系 / 擬正則曲線 / ストリング・トポロジー

Outline of Annual Research Achievements

R5年度は以下の３つの研究を行った。(A) C^2のトーリック領域のS^1同変シンプレクティック・ホモロジーの計算。(B) 自由ループ空間の新しい鎖複体の構成。(C) 高次元レーブ力学系に対する閉補題。
(A)と(B)はR4年度に行った研究の継続である。進展は見られたがR5年度内に論文をまとめるまでには至らなかった。
(C)については、R3年度にプレプリントとして発表した`Strong closing property of contact forms and action selecting functors'を改訂しアクセプトされた（出版はR6年度）。

Current Status of Research Progress

3: Progress in research has been slightly delayed.

Reason

(A)の研究が年度内に完成しなかったため。

Strategy for Future Research Activity

まずは(A)の研究を完成しプレプリントとして発表することを目指す。
(B)は新たな共同研究者の参加もあり当初より構想が広がっているが、R6年度内に完成の目処をつけたい。
(C)については強い閉補題（strong closing property）を満たす新たなReeb力学系の例を探索する。

Research Products

• [Journal Article] Strong closing property of contact forms and action selecting functors (2024)
  • Author(s): Irie Kei
  • Journal Title: Journal of Fixed Point Theory and Applications Volume: 26 Issue: 2
  • DOI: 10.1007/s11784-024-01102-1
  • Peer Reviewed
• [Journal Article] Capacities of billiard tables and $$S^1$$-equivariant loop space homology (2022)
  • Author(s): Irie Kei
  • Journal Title: Journal of Fixed Point Theory and Applications Volume: 24 Issue: 2
  • DOI: 10.1007/s11784-022-00952-x
  • Peer Reviewed
• [Journal Article] Symplectic homology of fiberwise convex sets and homology of loop spaces (2022)
  • Author(s): Irie Kei
  • Journal Title: Journal of Symplectic Geometry Volume: 20 Issue: 2 Pages: 417-470
  • DOI: 10.4310/jsg.2022.v20.n2.a2
  • Peer Reviewed
• [Journal Article] Remarks about the C∞-closing lemma for 3-dimensional Reeb flows (2021)
  • Author(s): Kei Irie
  • Journal Title: Kyoto Journal of Mathematics Volume: 61 Issue: 2
  • DOI: 10.1215/21562261-2021-0003
  • Peer Reviewed
• [Journal Article] Equidistributed periodic orbits of C∞-generic three-dimensional Reeb flows (2021)
  • Author(s): Kei Irie
  • Journal Title: Journal of Symplectic Geometry Volume: 19 Issue: 3 Pages: 531-566
  • DOI: 10.4310/jsg.2021.v19.n3.a2
  • Peer Reviewed
• [Journal Article] Chain level loop bracket and pseudo-holomorphic disks (2020)
  • Author(s): Kei Irie
  • Journal Title: Journal of Topology Volume: 13 Issue: 2 Pages: 870-938
  • DOI: 10.1112/topo.12140
  • Peer Reviewed
• [Presentation] String closing property in Reeb and Hamiltonian dynamics (2024)
  • Author(s): Kei Irie
  • Organizer: 接触構造、特異点、微分方程式及びその周辺
  • Invited
• [Presentation] Chain level string topology and de Rham chains (2024)
  • Author(s): Kei Irie
  • Organizer: Building-up Differentiable Homotopy Theory
  • Int'l Joint Research / Invited
• [Presentation] A conjectural chain model for positive S1-equivariant symplectic homology of star-shaped toric domains in C2 (2023)
  • Author(s): Kei Irie
  • Organizer: Persistence homology in symplectic and contact topology
  • Int'l Joint Research / Invited
• [Presentation] A conjectural chain model for positive S1-equivariant symplectic homology of star-shaped toric domains in C2 (2023)
  • Author(s): Kei Irie
  • Organizer: Geometry and Topology
  • Invited
• [Presentation] 閉補題とシンプレクティック幾何 (2023)
  • Author(s): 入江慶
  • Organizer: 2023年度日本数学会年会
  • Invited
• [Presentation] S^1-equivariant symplectic capacities of disk cotangent bundles (2021)
  • Author(s): Kei Irie
  • Organizer: From Hamiltonian Dynamics to Symplectic Topology
  • Int'l Joint Research / Invited
• [Presentation] Symplectic homology capacity of convex bodies (2019)
  • Author(s): Kei Irie
  • Organizer: Interactions of symplectic topology and dynamical systems
  • Int'l Joint Research / Invited
• [Presentation] Symplectic容量とHamilton力学系の周期軌道 (2019)
  • Author(s): 入江慶
  • Organizer: 日本数学会秋季総合分科会
  • Invited
• [Presentation] Equidistributed periodic orbits of C-infty-generic three-dimensional Reeb flows (2019)
  • Author(s): Kei Irie
  • Organizer: C^0-aspects of symplectic geometry and Hamiltonian dynamics
  • Int'l Joint Research / Invited
• [Presentation] Equidistributed pedioriodic orbits of C-infinity-generic three-dimensional Reeb flows (2019)
  • Author(s): Kei Irie
  • Organizer: Analytic low-dimensional dynamics
  • Int'l Joint Research / Invited
• [Presentation] Periodic billiard trajectories and Morse theory on loop spaces (2019)
  • Author(s): Kei Irie
  • Organizer: Workshop on mathematical billiards
  • Int'l Joint Research / Invited
• [Presentation] Equidistributed pedioriodic orbits of C-infinity-generic three-dimensional Reeb flows (2019)
  • Author(s): 入江慶
  • Organizer: 葉層構造の幾何学とその応用
  • Invited
• [Presentation] Periodic Reeb orbits and minimal hypersurfaces (2019)
  • Author(s): 入江慶
  • Organizer: UK-Japan winter school in Leeds:variational problems in geometry and mathematical physics
  • Int'l Joint Research / Invited
• [Presentation] Chain level string topology, pseudo-holomorphic disks, and Lagrangian submanifolds (2018)
  • Author(s): 入江慶
  • Organizer: 第６５回トポロジーシンポジウム
  • Invited
• [Presentation] 生成的なRiemann計量における極小閉超曲面の稠 密性 (2018)
  • Author(s): 入江慶
  • Organizer: 第６５回幾何学シンポジウム
  • Invited
• [Presentation] Density of minimal hypersurfaces for generic metrics (2018)
  • Author(s): 入江慶
  • Organizer: 4th China-Japan Geometry Conference
  • Int'l Joint Research / Invited