It is widely expected that a simply connected closed 4-dimensional Riemannian manifold M with positive sectional curvature must be homeomorphic to the 4-sphere or the complex projective plane.

Using a new take on classical techniques, we prove this to be the case if M is $\delta$-pinched with $\delta=\frac{1}{1+3\sqrt3}\cong 0.161$, that is, if all sectional curvatures of M lie in the interval $(\delta,1]$. We also give new bounds on the Euler characteristic and signature of simply connected $\delta$-pinched $4$-manifolds for any value of $\delta>0$. The main tools used are convex algebro-geometric insights on sets of $\delta$-pinched curvature operators. This is based on joint work with M. Kummer and R. Mendes.