The Geometry of Smooth 4-Manifolds

这个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。这个项目有三个部分。第一部分是关于4-流形的分类问题。虽然最近最小猜想的失败表明光滑4流形倾向于无视分类方案，但还有其他相关猜想，如3/2猜想和简单型猜想，研究者也将试图反驳这些猜想。在第二部分，研究者将继续他与John Morgan的合作，研究Seiberg-Witten不变量是如何沿着h-共体变化的。这个问题与简单型猜想有关，也是项目的第三部分，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。这个方向的长期目标是找到Seiberg-Witten不变量的一般化。单连通四维流形光滑结构的研究是由Simon Donaldson于1984年发起的。在他开创性的工作中，Donaldson在4流形上应用了Yang-Mills方程和相应的Yang-Mills模空间。利用相应的唐纳森不变量解决了四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流形分类中的应用揭示了拓扑学和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. ***