Gauge theory is a subject that has emerged from theoretical physics. It has deep links with many areas of mathematics (including partial differential equations, representation theory, algebraic geometry, differential geometry and topology). Most of the mathematical work on gauge theory has focused on low dimensions, where one can exploit the anti-self-dual Yang-Mills equation and the analytic difficulties are still quite tractable.
In this talk I will discuss three concrete questions that arise in gauge theory in higher dimensions. First, I will discuss gauge theory on Kähler manifolds, with a particular focus on singular Hermitian Yang-Mills connections. Afterwards, I will move on to the more exotic topic of gauge theory on G2-manifolds. I will discuss a method to construct solutions of the Yang-Mills equation on a class of G2-manifolds called twisted connected sums. Finally, I will talk about the prospects of defining enumerate gauge theoretical invariants for G2-manifolds and the difficulties arising from codimension four bubbling.