Geometry and Topology of Symplectic Four Manifolds

DMS-0207488Tian-Jun li本项目的重点是应用微分拓扑学、几何分析和代数几何的方法来研究四种辛流形。根据Kodaira维度将四种辛流形分为四类，取值分别为-1、0、1和2。实现了对Kodaira维度为-1的流形的分类，这是研究者做出的重要贡献。李田军建议对Kodaira维度为0的人进行分类。纤维和是四个辛流形的最强结构，人们希望以这种方式构造的流形是最小的。李田俊已经证明了一大类纤维和确实是极小的。这位研究者相信，他可以用他在有理曲面的极小亏格问题上的工作来证明所有的纤维和。流形是一个局部看起来像n维欧几里德空间的空间。例如，我们生活的时空宇宙是一个四维流形。辛四流形是具有辛结构的四流形，这是一种非常基本的结构，几乎涵盖了经典和量子物理的所有方程。因此，辛四流形在数学和物理中起着核心作用。研究人员的目的是对基本问题有一些了解：对辛四流形进行分类。

DMS-0207488Tian-Jun LiThe focus of this project is to apply methods from differential topology, geometric analysis and algebraic geometry to study symplectic four manifolds.Symplectic four manifolds can be divided into four categories according to their Kodaira dimensions, which take values -1, 0, 1 and 2. The classification of those with Kodaira dimenision -1has been achieved, to which the investigator has made essential contribution. Tian-Jun Li proposes to classify those with Kodaira dimension 0. Fiber sum is the most powerful construction of symplectic four manifolds, and one would like manifolds constructed this way to be minimal. Tian-Jun Li has shown that a large class of fiber sums are indeed minimal. The investigator believes that he can prove it for all fiber sums usinghis work on the minimal genus problem for rational surfaces.An n manifold is a space that locally looks like the Euclidean space of dimension n. For example, the space-time universe we live in is a four manifold. A symplectic four manifold is a four manifold witha symplectic structure, a very basic structure that underliesalmost all the equations of classical and quantum physics. Thus symplectic four manifoldsplay a central role in mathematics and physics. The investigator aims to gain some understanding of the fundamental problem: classifying symplectic four manifolds.