The Geometry of Smooth 4-Manifolds

光滑4流形的几何结构

• 批准号: 9704359
• 负责人: Zoltan Szabo
• 金额: $ 7.87万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704359&HistoricalAwards=false
• 关键词: Geometry; Smooth; Manifolds

9704359 Szabo The main theme of this project is the study of simply-connected smooth closed 4-manifolds and their Seiberg-Witten invariants. The project has three parts. The first part is related to the classification problem for these 4-manifolds. While the recent failure of the Minimal Conjecture shows that smooth 4-manifolds tend to defy classification schemes, there are other related conjectures like the 3/2 Conjecture and the Simple-type Conjecture, and the investigator will try to disprove these conjectures as well. In the second part, the investigator will continue his joint work with John Morgan to study how Seiberg-Witten invariants change along an h-cobordism. This problem is related to the Simple-type Conjecture and also to the third part of the project, in which the investigator will study the relation between the combinatorial presentations (Kirby calculus pictures) and Seiberg-Witten invariants of simply-connected smooth closed 4-manifolds. A long-term goal in this direction is to find generalizations of the Seiberg-Witten invariants. The study of smooth structures of simply-connected 4-dimensional manifolds was initiated in 1984 by Simon Donaldson. In his ground breaking work, Donaldson applied the Yang-Mills equation and the corresponding Yang-Mills moduli spaces over the 4-manifolds. The corresponding Donaldson invariants were used to settle various problems in 4-dimensional topology. The Yang-Mills equation is a partial differential equation that has special importance in theoretical physics, and its use in the classification of smooth 4-manifolds revealed a surprising link between topology and theoretical physics. This link has been underlined by Seiberg and Witten in 1994, when, using ideas coming from theoretical physics, they discovered a pair of new partial differential equations and the corresponding Seiberg-Witten invariants of smooth 4-manifolds. ***

9704359 Szabo该项目的主题是对简单连接的平滑封闭4个manifolds及其Seiberg-intent noftints的研究。 该项目有三个部分。 第一部分与这些4个manifolds的分类问题有关。 虽然最小猜想的最近失败表明平滑的4个manifolds倾向于违抗分类方案，但还有其他相关的猜想，例如3/2猜想和简单型猜想，研究人员也会尝试反驳这些猜想。 在第二部分中，调查人员将继续与约翰·摩根（John Morgan）的联合合作，研究塞伯格（Seiberg-Witten）的不变性人士如何沿着H bobobordism进行变化。 这个问题与简单类型的猜想以及项目的第三部分有关，研究人员将研究组合演示（Kirby conculus图片）和Seiberg-witten的简单连接平滑封闭的4个Manifolds之间的关系。 在这个方向上的长期目标是找到对塞伯格的不变性的概括。 西蒙·唐纳森（Simon Donaldson）于1984年启动了简单连接4维流形的平滑结构的研究。 唐纳森（Donaldson）在他的爆破工作中应用了杨米尔斯方程，并在4个manifolds上使用了相应的阳米尔斯模量空间。 相应的唐纳森不变性用于解决4维拓扑中的各种问题。 Yang-Mills方程是一个部分微分方程，在理论物理学中具有特殊的重要性，并且其在平滑4个manifolds的分类中的使用揭示了拓扑与理论物理学之间的令人惊讶的联系。 Seiberg和Witten在1994年强调了这一链接，当时，当使用理论物理学的想法时，他们发现了一对新的偏微分方程以及相应的Seiberg-witten不变的4个manifolds。 ***