The Gauss curvature flow – the shape of worn stones

The Gauss curvature flow was introduced to model the shape of worn stones. It is a parabolic Monge-Ampere type flow for convex hypersurfaces, involving regularity issues for fully nonlinear equations and free boundary problems, and involving several entropy. In this talk, we will discuss the free boundary problem concerning worn stone with flat sides, and the optimal regularity of non-concave fully nonlinear equations. When time permits, we also talk about the singularity analysis, especially about the relations to the affine geometry at the critical case, to fully nonlinear equations at the super-critical case, and to convex geometry and spectral analysis at the sub-critical case.