Geometric approach to instability phenomena in fluid and mechanical systems

提出的研究的主要目标是发展一种有效的数学语言来处理不稳定和奇点现象，这在物理系统中经常发生。鉴于这些现象的普遍性和无坐标性，这种方法将以几何考虑为基础。特别关注具有复杂界面的流体系统和具有耗散效应的非完整机械系统，其中一些具体的实际问题将使用分析，计算和实验方法进行分析。在更一般的层面上，将探讨场理论（包括流体动力学和许多其他物理理论）的几何处理。这些研究的预期影响将是发展新的几何启发方法，并在这些方法的帮助下解决一些突出的问题。

The major goal of the proposed research is to develop an efficient mathematical language for treating instability and singularity phenomena, which occur frequently in physical systems. In view of universal, coordinate-free, character of these phenomena, the methodology will be based on geometric considerations. Of particular focus are fluid systems with complex interfaces and non-holonomic mechanical systems with effects of dissipation, where a number of concrete practical problems will be analyzed using analytical, computational, and experimental methods. At a more general level, geometric treatment of field theories, which include fluid dynamics and many other physical theories, will be pursued. The anticipated impact of these studies will be a development of new geometry-inspired methods, and solving a number of outstanding problems with the help these methods.