A dynamical reinterpretation of the L^2 mass conservation law for solutions of the KdV equation has led to a general procedure for proving "almost conservation laws" for solutions of nonlinear Hamiltonian PDE. The almost conserved quantities may be used to globalize the available local-in-time well-posedness results for various equations and provide insights into the long-time behavior of solutions. These general ideas will be described in the context of the KdV equation. Applications to other equations may also be presented. This talk concerns joint work with M. Keel, G. Staffilani, H. Takaoka and T. Tao.
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