Abstract: (joint work with R. Bettiol and M. Kummer) A wide-open conjecture states that simply-connected closed 4-dimensional manifolds with positive sectional curvature are diffeomorphic to the sphere or to the complex projective plane. One class of partial results involves the additional hypothesis, called delta-pinching, that the sectional curvatures lie in the interval [delta,1], where delta>0. I will present our recent result for delta>0.161, and discuss the proof. Standard ingredients are the Freedman-Donaldson work on the topology of 4-manifolds, integral formulas for the Euler characteristic and signature, and the Bochner technique. The new ingredients are improved point-wise estimates for the components of the curvature tensor obtained via convex optimization, the applicability of which stems from the structure of the set of pinched 4D curvature operators as a spectrahedral shadow.