Department of MathematicsAn important question in the geometry of 4-manifolds is as follows: "Given M a smooth manifold, what are the best metrics it can support? When do they exist?"
Usually, the best metrics are those with the least amount of curvature. After a brief overview, in this talk we will concentrate on obstructions for the existence of almost-scalar-flat anti-self-dual metrics on 4-manifolds, which are a subclass of 'best' metrics.