Hilbert schemes and Combinatorics (4+ lectures)

I'm going to discuss a connection between various combinatorial
objects and results (Catalan numbers, Macdonald polynomials, n! theorem,
parking functions) and the geometry of the Hilbert schemes of points on
the plane. The Hilbert scheme Hilb_n(C^2) is a symplectic smooth algebraic
variety parameterizing codimension n ideals in C[x,y]. In the beginning
of 2000's Mark Haiman constructed a remarkable rank n! vector bundle
on Hilb_n(C^2) called the Procesi bundle. Haiman used it to prove various
combinatorial results such as the n! theorem and its consequence --
Schur positivity for Macdonald polynomials