The study of the Yang-Mills equations has proven to be extremely fruitful for mathematicians, and the Yang-Mills equations have made appearances in algebraic geometry, geometric analysis, and low dimensional topology. I will talk about the theory in the simplest case when the structure group is U(1), and discuss its relationship with Hodge theory on Riemannian manifolds. The goal will be to prove/explain the following statement:

“The space of solutions (up to gauge equivalence) to Maxwell’s equations on a closed manifold is a torus.”