Holomorphic Curves, Foliations and Dynamics: How a PDE helps to solve an ODE problem

I will give an overview of my thesis problem, some motivating ideas from
contact geometry (including the definition of "contact geometry"), and the
eclectic mix of mathematical machinery that makes such problems tractable.
The most important tools are pseudoholomorphic curves, a generalized
notion of analytic functions which combines complex analysis with heavy
doses of elliptic PDE theory, nonlinear functional analysis, differential
geometry and topology.  For good measure, this toolbox ends up providing
answers to some quite nontrivial problems in Hamiltonian dynamics.  I
shall attempt to summarize this story at the level of the "typical Courant
student who knows what a manifold is."