Topology of Symplectic 4-Manifolds

本研究项目研究辛流形的性质。辛结构自然而然地出现在经典力学中，经典力学是描述宏观物体运动的物理学分支，这些数学结构构成了经典和量子物理的基本方程。该项目旨在利用拓扑学、代数几何和微分几何等不同数学分支的各种技术，对辛四维流形的一般形状有新的理解。研究涉及四维拓扑和几何的几个方面。关于闭四维流形的一些工作包括：定义光滑和辛不变量并研究它们的性质；对辛Calabi-Yau四维流形进行分类并对有理流形和规则流形中的拉格朗日曲面和辛曲面的构形进行分类；研究光滑和辛映射类群和有限对称。对于有边界的四流形，该项目研究特殊的辛帽，包括无规则帽和Calabi-Yau帽，以期了解各种类型的辛填充。这项工作的一个新方面是研究凹辛四流形作为独立感兴趣的结构；特别是，该项目旨在证明凹辛四流形与闭辛四流形有许多共同的性质。

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.