Abstract: We discuss a classic critical estimate for conformally parametrized surfaces with $L^2$ curvature. We present local and global properties, and applications to compactness results and existence of Willmore minimizers.

Lecture notes: Link 

Abstract: In Sobolev spaces of maps to a manifold, all the familiar properties of Sobolev spaces fail. For example, in general, one cannot approximate such maps, in the Sobolev norm, with smooth maps to the manifold. I will first present some significant results in several directions, such as lifting, strong approximation, topological invariants, or singularities, with special focus on spaces with fractional regularity. The lecture is mostly based on the lecture notes https://hal.science/hal-04101540 and the preprint https://hal.science/hal-05050996

Additional resources: https://hal.science/hal-01819040 

https://hal.science/hal-05110119 

Abstract: In this lecture series I will review some recent results on nonlocal interaction problems. The focus will be on interaction kernels that are anisotropic variants of the classical Riesz kernel. In other words, while preserving the same singularity at zero of the Riesz kernel, they present preferred directions of interaction. For kernels of this kind I will prove existence and uniqueness for the corresponding minimization problems. I will then show how to characterize explicitly their minimizers in the case of some specific confinements. If time permits, I will discuss some related results and open questions.

Lecture notes: Link