Geometric tangential analysis and sharp regularity for degenerate pdes

We provide a broad overview on qualitative versus quantitative regularity estimates in the theory of degenerate parabolic pdes. The former relates to DiBenedetto's revolutionary method of intrinsic scaling, while the latter is achieved by means of what has been termed geometric tangential analysis. We discuss, in particular, sharp estimates for the parabolic p-Poisson equation, for the porous medium equation and for the doubly nonlinear equation.