Abstract: Given a smooth compact manifold, one would like to equip it with a Riemannian metric which is "as flat as possible," as measured by some natural curvature functional. Dimension 4 turns out to be completely exceptional for this class of problems, and phenomena that only occur in four dimensions play a dominant role in the resulting theory. This lecture will describe some recent results on the geometry of 4-manifolds, and indicate why Dynkin diagrams and representation theory play a key role via gluing in and bubbling off of generalized gravitational instantons.