An action of a finitely generated group G on a manifold M is called "geometric" if it comes from an embedding of G as a lattice in a Lie group acting transitively on M. In this talk, I'll explain new joint work with Maxime Wolff that characterizes geometric actions of surface groups on the circle by topological rigidity, in the dynamical sense -- from a purely dynamical hypothesis, we can reconstruct a geometric structure. This talk will be self-contained, but I will give motivation and context for this problem (and related ones) at my talk at the Fields on Wednesday the 14th at 2:30pm. I recommend that talk as a prequel!