Abstract: Over the past period of a decade and more, mathematicians in the PDE community have developed techniques for the phase space analysis of the dynamics of many model nonlinear Hamiltonian PDEs. These results include constructions of KAM invariant tori, Birkhoff normal forms and Nekhoroshev stability theorems, and constructions of cascade orbits. In this talk I will describe some applications and extensions of these ideas to a problem in fluid dynamics concerning the interaction of two near-parallel vortex filaments in three dimensions. In addition, as well as generalizations of this problem, I will speculate about further applications of the techniques of Hamiltonian PDEs to other nonlinear systems of fluid dynamics, in the form of a class of conservative nonlinear evolution problems of physical significance.