The n! Conjecture and all that: What did Haiman prove, anyway?

Abstract:

In a celebrated body of work, Mark Haiman set out to prove a combinatorial conjecture of Macdonald about his eponymous symmetric functions, and to do so wound up proving some geometric theorems about the Hilbert scheme of points in the plane. This will be a high-level and idiosyncratic overview of this work – we won’t assume any knowledge of symmetric functions, but we will assume basic knowledge of the Hilbert scheme of points on the level of my talks last semester.