| There are certain compact 4-manifolds, such as real and complex hyperbolic 4-manifolds, 4-tori, and K3, where we completely understand the moduli space of Einstein metrics. But there are vast numbers of other 4-manifolds where we know that Einstein metrics exist, but cannot currently determine whether or not there are other Einstein metrics on them that are qualitatively different from the currently-known ones. In this lecture, I will first present a characterization of the known Einstein metrics on Del Pezzo surfaces that I discovered several years ago, and then describe a delicate generalization which I obtained only quite recently. |