Hilbert scheme of points and moduli space of instantons

Abstract

The two main topics of this thesis are Hilbert schemes of points and moduli spaces. We will see how a set of ideals can be given the structure of a projective variety. With this structure we will use group actions to determine the Euler characteristic of the Hilbert schemes of points on a surface. Subsequently, we will explore connections on R4 and see how the Yang-Mills equations arise from minimizing a quantity known as action. We will then construct all solutions to these equations and see how the geometrization of these solutions gives rise to the moduli space of framed instantons. We will see how this space is some sense equal to the Hilbert scheme of points.