The space of cycles in a compact Riemannian manifold has very rich topological structure. The space of hypercycles, for instance, taken with coefficients modulo two, is weakly homotopically equivalent to the infinite dimensional real projective space. We will explain how to use this structure, together with Lusternik-Schnirelman theory and work of Gromov and Guth, to prove that every compact Riemannian manifold of positive Ricci curvature contains infinitely many embedded minimal hypersurfaces. Then we will discuss a proof of a Weyl's law conjectured by Gromov (joint work with Liokumovich and Neves) and more recent work in which Andre and I prove the first Morse index bounds of the theory. The main difficulty comes from the problem of multiplicity, which we are able to settle in the classical one-parameter case.