Low Dimensional Topology via Differential Equations

这个项目将继续Mrowka对三维和四维流形(时空的数学模型)的研究。这些研究集中在各种数学物理方程(主要是Yang-Mills方程和Sieberg-Witten方程)的解的性质和解的空间与方程所在的时空的拓扑结构之间的关系。在过去的二十年里，这一策略非常成功，尤、肖恩、陶布斯、唐纳森和乌伦贝克的工作证明了这一点。Mrowka希望通过与Kronheimer的合作来解决一个已有30年历史的关于三维空间拓扑的猜想，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，封闭三维时空的最简单模型的特征是每个闭合环路都可以收缩到一个点，而不是离开时空。

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.***