It is an open problem in Riemannian geometry whether every simply connected, closed manifold admitting non-negative sectional curvature also admits positive sectional curvature. One conjectured obstruction is due to Hopf: In even dimensions, positive curvature implies positive Euler characteristic. I will discuss joint work with Burkhard Wilking that confirms this conjecture in the case where the isometry group has rank at least five. The proof relies on structure results for torus representations all of whose isotropy groups are connected.