Vector Bundles with Connections on Curves

Vector bundles with connections on complex algebraic curves recently received lots of attention. For example, they come in the formulation of the (recently proved) geometric Langlands correspondence for GL_n as well as of the (recently proved) P=W Conjecture.

I will review the definition and discuss the equivalence with representations of fundamental groups known as Riemann-Hilbert correspondence. Time permits, I will generalize vector bundles with connections to principal bundles with connections. I will only assume basic knowledge of algebraic or complex geometry.