Low Dimensional Topology via Differential Equations

小行星9803166 这个项目将继续Mrowka的调查研究三维和四维流形（时空的数学模型）。 这些调查中心的有关性质的解决方案和空间的解决方案的各种方程ofmathematical物理（主要是杨米尔斯和Sieberg-Wittenequations）的拓扑结构的时空进行方程。 这一策略在过去的二十年里非常成功，丘、舍恩、陶伯斯、唐纳森和乌伦贝克的工作就是明证。 Mrowka希望与Kronheimer共同解决一个关于三维空间拓扑结构的30年前的猜想，即P性质猜想，这是迈向庞加莱猜想的重要一步，庞加莱猜想认为，最简单的封闭三维时空模型的特征是每个闭合环都可以收缩到一个点而不离开时空。

9803166Mrowka This project will continue Mrowka's investigations into the studyof three- and four-dimensional manifolds (mathematical models forspace-time). These investigations center on relating properties ofsolutions and spaces of solutions of the various equations ofmathematical physics (primarily the Yang-Mills and Sieberg-Wittenequations) to the topology of the space-times that carry theequations. This strategy has been very successful in the past twentyyears, as evidenced by work of Yau, Schoen, Taubes, Donaldson, andUhlenbeck. Mrowka hopes in joint work with Kronheimer to resolve a30-year-old conjecture about the topology of three-dimensional spacesknown as the Property P conjecture, an important step towardsPoincare's conjecture that the simplest model of a closedthree-dimensional space-time is characterized by the property thatevery closed loop is contractible to a point without leaving thespace-time.***