An introduction to Hilbert schemes of points on ADE singularities

This paper is based on a talk at the conference `The McKay correspondence, mutation and related topics' from July 2020. We provide an introduction to joint work of the author with S{\o}ren Gammelgaard, \'{A}d\'{a}m Gyenge and Bal\'{a}zs Szendr\H{o}i that constructs the reduced scheme underlying the Hilbert scheme of $n$ points on an ADE singularity as a Nakajima quiver variety for a particular stability parameter. After drawing a parallel with two well-known constructions of the Hilbert scheme of $n$ points in $\mathbb{A}^2$, we summarise results of the author and Gwyn Bellamy before describing the main result by cornering a noncommutative algebra obtained from the preprojective algebra of the framed McKay graph.