Jerzy Weyman

"Rings of semi-invariants of quivers"

I'll give a general overview of my recently completed work of the last few years on the Hilbert scheme of points in the plane and the applications of its geometric properties to problems in algebraic cominatorics. The geometric results are the Cohen-Macaulay property of the isospectral Hilbert scheme, or "Procesi variety," the isomorphism of the Hilbert scheme with a Hilbert scheme of orbits for the symmetric group, and a strong vanishing theorem for tensors of tautological bundles. The combinatorial applications include proofs of the Macdonald positivity conjecture and the dimension (n+1)^(n-1) conjecture for the diagonal coinvariant ring of the symmetric group.