Mathematical Sciences: Geometric Structures

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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.