Non-negative curvature and torus actions

We prove the Maximal Symmetry Rank conjecture for closed, simply-connected, non-negatively curved Riemannian manifolds in all dimensions under the additional hypothesis that the free rank of the action is roughly one-third of the dimension of the manifold. We also classify closed, simply-connected, non-negatively curved $6$-manifolds of almost maximal symmetry rank and as a corollary of these two results, we show that the Maximal Symmetry Rank conjecture holds for dimensions less than or equal to nine.