The course will start with an introduction to the geometric aspects of compensated compactness and convex integration. The second part will focus on the various exotic laminates found in the literature, namely the Tn configuration and staircase laminates. We will then transition to applications, beginning with the regularity of quasiconformal mappings and solutions to quasilinear equations. The final part of the course will delve into the description of magnetized fluids, such as those found in tokamaks and sunflares. Here, we will explore how the density of magnetic helicity resembles the structure of the Jacobian determinant of quasiconformal maps. To wrap up the course, we will tie everything together by demonstrating how staircase laminates offer critical solutions regarding the conservation of magnetic helicity. 

This course will focus on geometric inequalities — particularly isoperimetric inequalities and inequalities for PDE-driven shape functionals such as the Faber Krahn inequality. We will introduce some quantitative stability estimates, both classical and more contemporary, and discuss how these estimates can be applied in various mathematical settings.

In this mini-course, we shall explain the variational approach to regularity theory for optimal transportation. As a text material, we refer to the lecture notes arXiv:2303.14462. 

The advantage of this approach lies in its robustness, which allows to apply it to the optimal matching problem. For an introduction of the latter, see my recorded lectures under “From combinatorics to partial differential equations”.