Pseudo-holomorphic curves and periodic orbits in Hamiltonian dynamics

2016 Fiscal Year Final Research Report

• Project/Area Number: 25800041
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Kyoto University
• Principal Investigator: Irie Kei 京都大学, 数理解析研究所, 助教 (90645467)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: 擬正則曲線 / ハミルトン力学系 / 周期軌道

Outline of Final Research Achievements

I studied applications of pseudo-holomorphic curve theory in symplectic geometry to the study of periodic orbits of Hamiltonian systems.
Main achievements are: (1). Computation of symplectic capacity of unit disk cotangent bundle of a Riemannian manifold with boundary via geometry of free loop space. As an application, a good estimate of the shortest length of periodic billiard trajectory was obtained. (2). Construction of chain-level algebraic structures in string topology, which conjecturally correspond to higher products in Floer homology of cotangent bundles. (3). Proof of C-infinity closing lemma for three-dimensional Reeb flows and two-dimensional Hamiltonian diffeomorphisms using embedded contact homology.

Free Research Field

シンプレクティック幾何学