Mathematical Sciences: Symplectic Methods in Bifuration Theory, Hamiltonian Dynamics, and Lie Theory

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9503273 Ratiu The proposed research lies in the area of dynamical systems and geometric mechanics. Specifically, the following four topics are proposed: convexity and invariants, approximation in Hamiltonian dynamics, symmetry and bifurcations, and conserving numerical algorithms on nonlinear manifolds. The proposed research is in part motivated by large scale fluid flow problems arising in ocean dynamics. For example, the Euler equations in thin domains can be used to model shallow water waves. Also, numerical simulations of conservative systems may lead to a more accurate modeling of various weather phenomena.