  Despite spectacular advances in defining invariants for simply connected
smooth and symplectic 4-dimensional manifolds and the discovery of effective
surgical techniques, we still have been unable to classify simply connected
smooth manifolds up to diffeomorphism. In these notes, adapted from six
lectures given at the 2006 Park City Mathematics Institute Graduate Summer
School on Low Dimensional Topology, we will review what we do and do not know
about the existence and uniqueness of smooth and symplectic structures on
closed, simply connected 4-manifolds. We will focus on those surgical
techniques that have been effective in altering smooth and symplectic
structures and the Seiberg-Witten invariants that are used to distinguish them.
In the last lecture we will then pose a possible classification scheme and test
it on a few examples.
