The maximal symmetry rank conjecture for closed, simply-connected, non- negatively curved manifolds of dimension n states that such a manifold admits a torus action of rank < or = 2n/3 and in the case of equality and when the dimension of the manifold is 0 (mod 3), it is diffeomorphic to a product of 3-spheres

In joint work with Fernando Galaz-Garcia, we established the upper bound for manifolds of dimension less than or equal to 9 and proved the conjecture in dimensions less than or equal to 6. In the process of so doing, we also classified 5-manifolds of almost maximal symmetry rank.

I'll discuss work in progress with Christine Escher towards establishing this conjecture in (slightly) higher dimensions as well as understanding the almost maximal symmetry rank case for dimension 6.