  In 1965, S.-S. Chern posed a question concerning the extent to which
fundamental groups of manifolds admitting positive sectional curvature look
like spherical space form groups. The original question was answered in the
negative by Shankar in 1998, but there are a number of positive results in the
presence of symmetry. These classifications fall into categories according to
the strength of their conclusions. We give an overview of these results in the
case of torus symmetry and prove new results in each of these categories.
