Title: The smooth closing lemma for area-preserving maps of surfaces abstract: I will talk about some recent joint work showing that a generic smooth area-preserving diffeomorphism of a closed surface has a dense set of periodic points. In the C^1 topology, these kind of results were proved by Pugh and Robinson and are generally called “closing lemmas”; finding closing lemmas in higher regularity is the subject of Smale’s 10th problem. A kind of Weyl law recovering the average rotation through the actions of periodic points plays a central role in the proof.