Moduli spaces parametrize certain kinds of geometric objects, and they are central objects in algebraic geometry. In this talk, we look at two different moduli spaces: one of smooth curves of fixed genus, and the other of one-dimensional sheaves on the projective plane. The former is a classical subject, dating back to Bernhard Riemann, while the latter relatively new, motivated by enumerative geometry, mathematical physics and so on. Despite the different nature of the objects they parametrize, we will show how their ‘intersection rings’ share surprisingly similar features. This talk is based on joint work with Junliang Shen.