Positively curved manifolds are notoriously hard to classify. In the early 90's, based on the observation that the few known examples are all highly symmetric, Karsten Grove proposed his "Symmetry Program" which suggests trying to classify such manifolds with the additional hypothesis of symmetry. Much work has been focused on the case of continuous symmetries. In this talk, I'll discuss the case when the isometry group is finite and present some recent results when it is an elementary abelian two-group.
This is joint work with Lee Kennard and Elahe Khalili Samani.