Title: Geometric regularity theory for (nonlinear) partial differential equations.
Abstract: In this course we examine the regularity theory for partial differential equations, both in the linear and nonlinear settings. The main goal of the course is to expose the audience to a class of methods and techniques entitled regularity transmission by approximation methods. In this framework, we approximate a given problem of interest by an auxiliary one, for which a richer theory is available. Then, by intrinsic geometric methods, we import information from the latter to the former one. This program contemplates important examples, such as fully nonlinear equations (including degenerate models), non-convex problems (e.g. the Isaacs equation), roughly degenerate diffusions, double divergence operators and perturbations of the porous medium equation.