Geometry of diffeomorphism groups and Hamiltonian systems

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Ocean currents, fluid motions in rivers, atmospheric flows, and magnetic fields in tokamaks and stars are mathematically modelled by the hydrodynamical Euler equation and its relatives. While Euler-type equations arise in a broad range of physical phenomena, they share many similarities: Hamiltonian structure, a variety of conservation laws, geometric formulation, and symmetries related to particle relabelling (i.e. diffeomorphism action). The goal of the proposed research is to contribute toward understanding of two key problems of fluid dynamics: singularity formation and appearance of turbulence by exploring the geometric and group-theoretical approach to fluids. It is aimed at understanding of the robustness, accuracy, and predicting power of geometric infinite-dimensional mathematical models for wide range of hydrodynamical phenomena, from how reliable are weather forecasts to appearance of shock waves and behaviour of ocean currents.