One of the fundamental problems in the classification of complex surfaces is to find a new family of simply connected surfaces with p_g = 0 and K^2 > 0. In this
talk, I will sketch how to construct a new family of simply connected symplectic 4- manifolds using a rational blow-down surgery and how to show that such 4-manifolds
admit a complex structure using a Q-Gorenstein smoothing theory. In particular, I will show explicitly how to construct a simply connected minimal surface of general
type with p_g = 0 and K^2 = 3.

If time allows, I will also sketch how to construct a simply
connected, minimal, symplectic 4-manifold with b_+2 = 1 (equivalently, p_g = 0) and K^2 = 4 using a rational blow-down surgery.