symplectic geometry has recently emerged as a key tool in the study of low-dimensional topology. one approach, championed by arnol'd, is to examine the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. i'll describe a way to use this approach to construct a rather powerful invariant of knots called "knot contact homology", and discuss its properties. if time permits, i'll also outline a surprising connection to string theory and mirror symmetry.