Geometry and Topology of Symplectic Four Manifolds

本课题的重点是应用微分拓扑、几何分析和代数几何的方法研究辛四流形。辛四流形可以根据其Kodaira维数分为四类，分别取值为- 1,0,1和2。实现了Kodaira维数-1的分类，研究者对此做出了重要贡献。李天军提出对Kodaira维数为0的数据进行分类。纤维和是辛四流形最强大的构造，人们希望用这种方式构造的流形是最小的。李天军已经证明，一大类纤维的总和确实是最小的。研究者相信他可以用他在有理曲面的最小属问题上的工作证明它适用于所有的纤维和。一个n流形是一个局部看起来像n维欧几里得空间的空间。例如，我们生活的时空宇宙是一个4流形。辛四流形是一种具有辛结构的四流形，这是一种非常基本的结构，几乎是所有经典和量子物理方程的基础。因此辛四流形在数学和物理学中起着中心作用。研究者的目的是获得一些基本问题的理解：分类辛四流形。

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.