The Geometry of Smooth 4-Manifolds

9704359 Szabo本项目的主要主题是研究单连通光滑闭4-流形及其Seiberg-Witten不变量。该项目包括三个部分。第一部分是关于这类4-流形的分类问题。虽然最近极小猜想的失败表明光滑的4-流形往往不符合分类模式，但还有其他相关的猜想，如3/2猜想和简单类型猜想，研究者也将试图反驳这些猜想。在第二部分中，调查者将继续他与约翰·摩根的合作，研究Seiberg-Witten不变量是如何沿着h-Coobordism变化的。这个问题与简单类型猜想有关，也与项目的第三部分有关，在第三部分中，研究者将研究单连通光滑闭4-流形的组合表示(Kirby微积分图)与Seiberg-Witten不变量之间的关系。这个方向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 Donaldson于1984年开创的。在他的开创性工作中，Donaldson在4-流形上应用了Yang-Mills方程和相应的Yang-Mills模空间。利用相应的Donaldson不变量解决了四维拓扑中的各种问题。杨-米尔斯方程是一个在理论物理中具有特殊重要性的偏微分方程，它在光滑四维流形的分类中揭示了拓扑学和理论物理之间惊人的联系。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。***

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. ***