Topology and Geometry of Symplectic Four Manifolds

该提案涉及四维拓扑和几何的几个方面。拓扑学方面的一些工作包括：定义光滑不变量和辛不变量并研究它们的性质，分类辛Calabi-Yau4-流形和有理流形和规则流形中的拉格朗日曲面和辛曲面，研究光滑和辛映射类群。而在几何方面，PI是在4维流形上比较驯服的、相容的和可积的几乎复杂的结构。n维流形是局部看起来像n维的欧几里得空间的大尺度空间。特别地，我们生活的时空宇宙是四维流形。在偶数维流形上，辛结构是一种几何结构，它支撑着经典物理和量子物理的一些基本方程。因此，辛四流形在数学和物理中都扮演着中心角色。本研究旨在利用拓扑学、代数几何和微分几何的多种方法，对辛四流形的一般形状有一些新的认识。

The proposal concerns several aspects of four dimensional topology and geometry. Some of the projects on the topological side are: defining smooth and symplectic invariants and studying their properties, classifying symplectic Calabi-Yau 4-manifolds and classifying Lagrangian and symplectic surfaces in rational and ruled manifolds, investigating smooth and symplectic mapping class groups. And on the geometric side, the PI is comparing tamed, compatible and integrable almost complex structures on 4-manifolds.An n dimensional manifold is a large scale space that locally looks like the Euclidean space of dimension n. In particular, the space-time universe we live in is a four dimensional manifold. On an even dimensional manifold, a symplectic structure is a geometric structure that underlies some fundamental equations of classical and quantum physics. Thus symplectic four manifolds play a central role in both mathematics and physics. This research project aims to gain some new understandings of the general shape of symplectic four manifolds, using a variety of techniques from topology, algebraic geometry and differential geometry.