In this talk, I will discuss recent joint work with D. Cristofaro-Gardiner and B. Zhang showing that a generic area-preserving diffeomorphism of a closed surface has a dense set of periodic points. This follows from a result called a “smooth closing lemma” for area-preserving surface diffeomorphisms; this answers in the affirmative Smale’s 10th problem in the setting of area-preserving surface diffeomorphisms. The proof uses quantitative analysis of spectral invariants from periodic Floer homology via various estimates in Seiberg-Witten theory.