  We develop the theory of spectral invariants in periodic Floer homology (PFH)
of area-preserving surface diffeomorphisms. We use this theory to prove
$C^\infty$ closing lemmas for certain Hamiltonian isotopy classes of
area-preserving surface diffeomorphisms. In particular, we show that for a
$C^\infty$-generic area-preserving diffeomorphism of the torus, the set of
periodic points is dense. Our closing lemmas are quantitative, asserting
roughly speaking that for a given Hamiltonian isotopy, within time $\delta$ a
periodic orbit must appear of period $O(\delta^{-1})$. We also prove a "Weyl
law" describing the asymptotic behavior of PFH spectral invariants.
