We prove that, on a closed surface with a $C^\infty$-generic Riemannian metric, the union of nonconstant closed geodesics is dense.   This result follows from a more general result about Reeb dynamics on contact three-manifolds.

The proof uses embedded contact homology (ECH), a version of Floer homology defined for contact three-manifolds (developed by Hutchings and Taubes), and the asymptotic formula for ECH spectral numbers, which was proved by Cristofaro-Gardiner, Hutchings, and Ramos.  We will also discuss a recent denseness result of minimal hypersurfaces for generic metrics (joint work with Marques and Neves), which was obtained by applying a similar idea to the asymptotic formula for volume spectrum, which was proved by Liokumovich, Marques, and Neves.