Geodesic Currents and the Marked Length Spectrum

Given a surface, the marked length spectrum (MLS) is the collection of the lengths of the closed geodesics, with each length marked by the free homotopy class it belongs to. Under certain conditions, information on most of the MLS is enough to completely determine the metric of the surface. This is called partial MLS rigidity. The proof heavily relies on the notion of geodesic currents. Geodesic currents are also an interesting geometric and dynamical tool on their own. I plan to introduce geodesic currents and show how they tie into the proof of partial MLS rigidity.