The Yang-Mills equations and Uhlenbeck Compactness

This talk is a brief introduction to the analysis of Donaldson theory, a branch of gauge theory. Roughly, this is an area of differential topology that aims to extract smooth structure invariants from the geometry of the space of solutions (moduli space) to a system of partial differential equations: the Yang-Mills equations.

I will start by discussing the differential geometric background required to talk about Yang-Mills connections. This will involve introducing the concepts of principal fibre bundles, connections and curvature. In the second half of the talk I will attempt to convey the flavour of the mathematics used to address technical issues in gauge theory. I plan to do this by presenting a sketch of the proof of Uhlenbeck's compactness theorem, the main technical tool involved in the compactification of the moduli space.