Singularities of fully nonlinear geometric flows

We will discuss the evolution of hypersurfaces by fully nonlinear geometric flows. These are cousins of the mean curvature flow which can be tailored to preserve different features of the underlying hypersurface geometry. Solutions often form singularities. I will present new classification results for blow-ups of singularities which confirm the expectation that these are highly symmetric and hence rigid. I will explain how this work fits into a broader program aimed at characterising Riemannian manifolds with positively curved boundaries.