We examine weak solutions of the porous medium equation and establish sharp regularity estimates along their zero-level sets. Our arguments rely on approximation methods and relate the modulus of continuity of the solutions with a proximity regime for the nonlinearity. More precisely, the closer the exponent governing the nonlinearity gets to one, the better the modulus of continuity of the solutions in H\”older spaces. We also comment on generalizations of our findings and extensions of our techniques to the case of more general nonlinearities. To close the talk, we adventure into an exercise on the parallels between our results and recent developments stemming from the realm of gradient flows.
