Hilbert schemes of points and Fulton-MacPherson compactifications

The two spaces appearing in the title provide distinct compactifications of unordered configuration spaces of points on a complex variety. I will describe several ways in which they are related. In particular, in dimension two, I will explain how the latter can be used to prove the Hilbert-Chow crepant resolution conjecture, proposed by Ruan, which asserts an isomorphism between the quantum cohomology of Hilbert schemes and the orbifold cohomology of symmetric products.