Irregular connections, Hitchin systems and Kac-Moody root systems

I will give a brief survey of `wild nonabelian Hodge theory on curves'
and describe some first steps towards classifying the class of
hyperkahler manifolds that arise in this way (i.e. as moduli spaces of
meromorphic connections or Higgs bundles).
In particular we will see there is a rich class of examples that may
be thought of as ``more transcendental versions'' of certain
(Nakajima) quiver varieties.
Said differently we may attach a graph to certain meromorphic Hitchin
moduli spaces, and these graphs play the role of Dynkin diagrams for
such spaces.