Geometry and Topology of Symplectic Four Manifolds

DMS-0207488李天军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辛四流形，辛四流形可以根据其科代拉维数分为四类，取值为-1，0，1和2。完成了科代拉维-1的分类，研究者对此作出了重要贡献。李天军建议将那些具有科代拉维度0的分类。纤维和是辛四流形的最强大的构造，人们希望这样构造的流形是最小的。Tian-Jun Li已经证明了一大类纤维和确实是极小的。调查人员认为，他可以证明它的所有纤维和usinghis工作的最小属问题的理性surface.An流形是一个空间，局部看起来像欧几里德空间的维数n。例如，我们生活的时空宇宙是一个四维流形。一个辛四维流形是一个具有辛结构的四维流形，这是一个非常基本的结构，几乎是经典和量子物理学中所有方程的基础。 因此，辛四流形在数学和物理学中起着核心作用。调查员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.