Hilbert schemes and parking functions

We will present a certain projective variety called the flag (or nested) Hilbert scheme. The cohomology of certain sheaves on this variety give rise to the combinatorics of Dyck paths in an $m\times n$ rectangle. This generalizes part of Haiman's work, whose isospectral Hilbert scheme produced incarnations of many objects in symmetric function theory, specifically the shuffle conjecture concerning Dyck paths in an $n\times n$ rectangle. Joint work with Eugene Gorsky.