Geometric approach to instability phenomena in fluid and mechanical systems

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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.