An SO(3)-connection on a 4-manifold is called definite if its curvature is non-zero on every tangent 2-plane. Given such a
connection the corresponding 2-sphere bundle carries a natural symplectic structure. Definite connections carry a sign, + corresponds to
Fano manifolds, - to Calabi-Yaus. The study of definite connections involves both the construction of examples, most notably via
hyperbolic geometry, as well as attempts to understand which 4-manifolds admit definite connections. There is a geometric flow
which attempts to deform a given definite connection into one which solves a certain PDE. Understanding singularity formation in this
flow is an important step forward in understanding which 4-manifolds support definite connections.