Understanding curves on surfaces has become a primary tool for understanding their hyperbolic structures and associated moduli spaces. This talk will be on understanding curves through their intersection with other curves and themselves.
For instance, through classical work of Dehn, simple closed curves can be described using intersection numbers with other simple curves. An underlying theme will be to figure out to what extent you can describe all curves in a similar fashion. More generally, curves are fabulous objects to experience the interplay between the topology and geometry of hyperbolic surfaces.
Part of the talk will be based on joint work with Binbin Xu.