Mathematical Sciences: Geometric Structures

9529369沙利文这项建议旨在为三维流体运动的连续介质模型奠定数学基础。这与研究者早期关于将光滑流形技术应用于拓扑流形的工作有关。这项拟议的研究还涉及新发现的Seiberg和Witten的单极子方程：该提议的目的是确切地确定Seiberg-Witten不变量依赖于什么结构。流体运动动力学可用一组非线性偏微分方程组来描述。传统的偏微分方程法存在困难，因为系统往往不够光滑。拟议的研究旨在引入不要求这些方程组具有任何光滑性的拓扑技术。

9529369 Sullivan This proposal seeks to lay a mathematical foundation for the continuum model of fluid motion in three dimensions. This relates to the investigator's earlier works on applying smooth manifold techniques to topological manifolds. The proposed research also deals with the newly discovered monopole equations of Seiberg and Witten: the proposal aims to determine exactly what structures the Seiberg-Witten invariants depend on. The fluid motion dynamics can be described by a system of nonlinear partial differential equations. There are traditional difficulties with the partial differential equation approach in that the system often is not smooth enough. The proposed research aims to introduce topological techniques not requiring any smoothness to these systems of equations.