Topology of Symplectic 4-Manifolds

本课题研究辛流形的性质。辛结构在描述宏观物体运动的物理学分支经典力学中自然出现，这些数学结构构成了经典物理学和量子物理学基本方程的基础。该项目旨在利用来自不同数学分支的各种技术，如拓扑、代数几何和微分几何，对辛四流形的一般形状获得新的理解。该研究涉及四维拓扑和几何的几个方面。关于闭四流形的一些课题是：定义光滑和辛不变量并研究它们的性质；辛Calabi-Yau四流形的分类及有理流形和有规流形中lagrange曲面和辛曲面的组形分类研究光滑辛映射类群和有限对称。对于具有边界的四流形，该项目研究了特殊的辛帽，包括无规帽和Calabi-Yau帽，旨在理解各种类型的辛填充。工作的一个新方面是凹辛四流形作为独立兴趣结构的研究；特别是，该项目旨在证明凹辛四流形与闭辛四流形具有许多相同的性质。

This research project investigates the properties of symplectic manifolds. Symplectic structures arise naturally in classical mechanics, the branch of physics that describes the motion of macroscopic objects, and these mathematical structures underlie the fundamental equations of classical and quantum physics. The project aims to gain new understanding of the general shape of symplectic four-manifolds using a variety of techniques from various branches of mathematics, such as topology, algebraic geometry, and differential geometry.The research concerns several aspects of four-dimensional topology and geometry. Some of the projects on closed four-manifolds are: defining smooth and symplectic invariants and studying their properties; classifying symplectic Calabi-Yau four-manifolds and classifying configurations of Lagrangian and symplectic surfaces in rational and ruled manifolds; and investigating smooth and symplectic mapping class groups and finite symmetries. For four-manifolds with boundary, the project investigates special symplectic caps, including unruled caps and Calabi-Yau caps, with an eye towards understanding various types of symplectic fillings. A new aspect of the work is investigation of concave symplectic four-manifolds as structures of independent interest; in particular, the project aims to show that concave symplectic four-manifolds share many properties with closed symplectic four-manifolds.