  Using symplectic topology and the Radon transform, we prove that smooth
4-dimensional projective planes are diffeomorphic to $\mathbb{CP}^2$. We define
the notion of a plane curve in a smooth projective plane, show that plane
curves in high dimensional regular planes are lines, prove that homeomorphisms
preserving plane curves are smooth collineations, and prove a variety of
results analogous to the theory of classical projective planes.
