The Grove program proposes to classify Riemannian manifolds of positive or non-negative sectional curvature with a "large" isometry group.
The simplest groups that one may consider acting on a manifold are compact abelian Lie groups, i.e. tori. Thanks to the work of several authors fairly complete classification results are available, provided that that either the manifold, or the torus acting upon it, has sufficiently large dimension. For the action of a circle and a 2-torus in dimensions 5 and 6 the usual methods have so far failed to yield topological and equivariant classification results.
Taking as departure point the Grove program and the well-developed theory of cohomological methods for smooth torus actions on smooth manifolds, the project aims, on the one hand, at applying and developing equivariant topological methods in the context of Riemannian manifolds with a lower sectional curvature bound and, on the other hand, at studying closed Alexandrov spaces of cohomogeneity one. The primary goals are, respectively, to obtain a topological and equivariant classification of closed, simply connected 6-manifolds with an effective, isometric 2-torus action, and to classify closed, positively curved Alexandrov spaces of cohomogeneity one.