The action of Hamiltonian homeomorphisms on surfaces and its applications
报告题目(Title)：The action of Hamiltonian homeomorphisms on surfaces and its applications
报告人(Speaker)：王俭 研究员 巴西理论数学与应用数学研究所（IMPA）
地点(Place)：腾讯会议：993 268 592
时间(Time)：2020年10月09日 (星期五) 16: 00-17:00
邀请人(Inviter)：卢广存，龚文敏
报告摘要
The famous Gromov-Eliashberg Theorem, that the group of symplectic diffeomorphisms is $C^0$-closed in the full group of diffeomorphisms, makes us interested in defining a symplectic homeomorphism as a homeomorphism which is a $C^0$-limit of symplectic diffeomorphisms. This becomes a central theme of what is now called ``$C^0$-symplectic topology”.
In symplectic geometry, the action function is a classical object defined on the set of contractible fixed points of the time-one map of a Hamiltonian isotopy. Under a weaker boundedness condition (WB for short), we can generalise the classical action function to the case of Hamiltonian homeomorphisms on surfaces. Through studying the properties of the generalised action function, we can generalise several classical results from the smooth world to the $C^0$ world, e.g., the $C^0$-Schwarz's theorem (that is, the existence of two actions of a non-trivial Hamiltonian homeomorphism), the existence of three actions of a non-trivial Hamiltonian homeomorphism under the WB and a natural topological hypothesis (which is a strengthening of the $C^0$-Arnold Conjecture on surfaces). Moreover, we answer a variant of a question posed by Buhovsky-Humiliere-Seyfaddini in [Invent. Math., 213 (2018), no. 2, 759–809] on dimension 2.
