We survey several problems concerning Riemannian manifolds with positive curvature of one form or another. We describe the PIC1 notion of positive curvature and argue that it is often the sharp notion of positive curvature to consider. Finally we explain how recent Ricci flow theory is particularly well adapted to solve these problems.
Some new examples with almost positive curvature
Obstructions to positive curvature and symmetry
Almost positive curvature on $T^1\mathbb{K}P^n$ and other manifolds
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