Abstract: The classification of Riemannian manifolds with
positive and non-negative sectional curvature is a long-standing
problem in Riemannian geometry. In this talk I will give an overview
of known results and summarize joint work with Zheting Dong and
Catherine Searle on the classification of closed, simply-connected,
non-negatively curved Riemannian manifolds admitting an isometric,
effective, almost isotropy-maximal torus action. This classification
has many applications, in particular the Maximal Symmetry Rank
conjecture holds for this class of manifolds.