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 该项目的主题是研究简单连接的光滑封闭 4 流形及其 Seiberg-Witten 不变量。 该项目分为三个部分。 第一部分与这 4 个流形的分类问题相关。 虽然最小猜想最近的失败表明光滑 4 流形往往不符合分类方案，但还有其他相关猜想，如 3/2 猜想和简单型猜想，研究者也将尝试反驳这些猜想。 在第二部分中，研究者将继续与 John Morgan 合作，研究 Seiberg-Witten 不变量如何沿着 h 配边变化。 这个问题与简单型猜想有关，也与该项目的第三部分有关，其中研究者将研究组合表示（柯比演算图片）与简单连接平滑闭合 4- 的 Seiberg-Witten 不变量之间的关系流形。 这个方向的长期目标是找到 Seiberg-Witten 不变量的概括。 简单连通的 4 维流形的光滑结构的研究由 Simon Donaldson 于 1984 年发起。 在他的开创性工作中，唐纳森在 4 流形上应用了 Yang-Mills 方程和相应的 Yang-Mills 模空间。 相应的唐纳森不变量被用来解决4维拓扑中的各种问题。 Yang-Mills 方程是一个在理论物理中具有特殊重要性的偏微分方程，它在光滑 4 流形分类中的使用揭示了拓扑与理论物理之间令人惊讶的联系。 Seiberg 和 Witten 在 1994 年强调了这种联系，当时他们利用理论物理学的思想，发现了一对新的偏微分方程和相应的光滑 4 流形的 Seiberg-Witten 不变量。 ***