Geometry and Topology of Symplectic Four Manifolds

辛四流形的几何与拓扑

• 批准号: 0207488
• 负责人: Tian-Jun Li
• 金额: $ 11.17万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0207488&HistoricalAwards=false
• 关键词: Geometry; Topology; Symplectic; Four; Manifolds

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.

DMS -0207488Tian -Jun Lithe该项目的重点是应用差异拓扑，几何分析和代数几何的方法来研究符合性的四个流形。四方流形可以按四个类别分为四类，根据其Kodaira的kodaira，该维度为-1，差异为-1，1，1，1，1，1，1，1，1，1，1，1和1的分类。研究人员做出了基本的贡献。 Tian-jun Li提议将具有Kodaira尺寸的人分类为0。纤维总和是符号四种流形的最强大的结构，一个人希望以这种方式构造的歧管。天詹·李（Tian-Jun Li）表明，大量的纤维总和确实很少。研究人员认为，他可以使用他在理性表面最小的属问题上使用所有光纤总和来证明这一点。例如，我们所生活的时空世界是四个流派。一个符号四歧管是具有符号结构的四个歧管，这是一个非常基本的结构，最基本的结构是所有经典和量子物理的所有方程。 因此，在数学和物理学中，符合四歧管的核心作用。研究者旨在了解基本问题：对符合性的四个流形进行分类。