The Geometry of Smooth 4-Manifolds

小行星9704359 本项目的主要内容是研究单连通光滑闭4-流形及其Seiberg-Witten不变量。 该项目有三个部分。 第一部分是关于这类四维流形的分类问题。 虽然最近最小猜想的失败表明光滑4-流形倾向于违反分类方案，但还有其他相关的猜想，如3/2猜想和单纯型猜想，研究者也将试图反驳这些猜想。 在第二部分中，调查员将继续他的联合工作与约翰摩根研究如何塞伯格-威滕不变量改变沿着一个h-协边。 这个问题与单纯型猜想有关，也与项目的第三部分有关，在该部分中，研究者将研究简单连通光滑闭4-流形的组合表示（Kirby演算图）与Seiberg-Witten不变量之间的关系。 在这个方向上的一个长期目标是找到Seiberg-Witten不变量的推广。 单连通四维流形的光滑结构的研究是由Simon唐纳森于1984年开始的。 在他开创性的工作，唐纳森适用于杨米尔斯方程和相应的杨米尔斯模空间的4流形。 相应的唐纳森不变量被用来解决四维拓扑中的各种问题。 杨-米尔斯方程是一个偏微分方程，在理论物理中具有特殊的重要性，它在光滑4-流形分类中的应用揭示了拓扑学和理论物理之间令人惊讶的联系。 Seiberg和维滕在1994年强调了这一联系，当时，他们利用来自理论物理的想法，发现了一对新的偏微分方程和相应的光滑四维流形的Seiberg-维滕不变量。 ***

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. ***