Stony Brook University

of the Dissertation Self-Dual Metrics on 4-Manifolds by Mustafa Kalafat Doctor of Philosophy in Mathematics Stony Brook University 2007 Under a vanishing hypothesis, Donaldson and Friedman proved that the connected sum of two self-dual Riemannian 4-Manifolds is again self-dual. Here we prove that the same result can be extended over to the positive scalar curvature case. Secondly we give an example of a 4-manifold with b = 0 admitting a scalar-flat anti-self-dual metric. Finally we apply the Geometric Invariant Theory(GIT) for Toric Varieties to the Einstein-Weyl Geometry and obtain a partial result.