  These notes are adapted from two talks given at the 2004 Clay Institute
Summer School on Floer homology, gauge theory, and low dimensional topology at
the Alfred Renyi Institute. We will quickly 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 then list the techniques used to
date and capture the key features common to all these techniques. We finish
with some approachable questions that further explore the relationship between
these techniques and whose answers may assist in future advances towards a
classification scheme.
