华罗庚青年数学论坛 学术报告

报告人：Dr. Zhou Zhengyi（IAS, USA）

题  目：Symplectic filling - existence and uniqueness

时  间：2019.08.02（星期五），09:00-10:00

地  点：数学院南楼N224室

摘  要: One natural question in symplectic topology is understanding symplectic fillings of a given contact manifold. One aspect of the question is the existence and uniqueness of symplectic fillings. I will first recall some classical results by Eliashberg, Floer, Gromov, McDuff, etc. Then I will describe recent progress in understanding both topological and symplectic aspects of the uniqueness of symplectic fillings for a large class of contact manifolds, called asymptotically dynamically convex manifolds.  I will also introduce new obstructions to Weinstein fillings and cobordisms as well as quantitative measurement of the complexity of Liouville domains and contact boundaries. No previous knowledge of symplectic topology will be assumed.

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报告人：Dr. Zhou Zhengyi（IAS, USA）

题  目：Vanishing of symplectic cohomology for flexibly fillable contact manifolds

时  间：2019.08.02（星期五），10:30-11:30

地  点：数学院南楼N224室

摘  要: Gromov and McDuff proved exact fillings of the standard contact 3-sphere are uniques based on a Gromov-Witten theoretical argument. Such method was generalized to subcritically fillable manifolds by Oancea -Viterbo and Barth-Geiges-Zehmisch to prove the uniqueness of diffeomorphism type of fillings. I will present a simple proof of similar results for flexibly fillable contact manifolds, which also yields a bit symplectic information, i.e. the symplectic cohomology of fillings of flexibly fillable contact manifolds must vanish.

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报告人：Dr. Zhou Zhengyi（IAS, USA）

题  目：Symplectic fillings of asymptotically dynamically convex (ADC) manifolds

时  间：2019.08.05（星期一），15:00-17:00

地  点：数学院南楼N202室

摘  要: In this talk, I will explain the geometry behind the proof in the previous talk and explain two invariance results for ADC manifolds, which lead to the persistence of both vanishing of symplectic cohomology and existence of symplectic dilation.  Using them, I will prove there are infinitely many exactly fillable, almost Weinstein fillable, but not Weinstein fillable contact manifolds. Then I will explain higher analogs of such structures and their invariance, which leads to a quantitative measurement of the complexity of Liouville domains and contact manifolds.