We focus on Riemannian manifolds with one end asymptotic to the hyperbolic space geometry. Such objects arise in general relativity as slices of asymptotically anti-de Sitter (adS) spacetimes. Under an assumption on the decay rate of the metric, a number of authors (Abbott-Deser, Chrusciel-Herzlich, Wang...) have defined global quantities (mass and center of mass) which enjoy “asymptotic invariance” properties that we will review, and for which the group PO(n,1) of isometries of the hyperbolic space plays a central role. We will then see how to construct and classify other such asymptotic invariants when we relax the assumption on the decay rate. They are attached to finite dimensional representations of PO(n,1). We shall finally see how every such invariant is naturally linked to a curvature operator (e.g. the scalar curvature for the classical mass). This is based on a joint work with Mattias Dahl and Romain Gicquaud.

We give an introductory exposition about some surfaces which minimize a nonlocal perimeter functional. These objects naturally arise in the study of the interfaces of phase transitions, when the particles exhibit long-range interactions or when boundary effects are present. In addition, these nonlocal minimal surfaces have concrete applications in several areas, such as image processing and mathematical biology. We will present some results concerning interior regularity and rigidity of nonlocal minimal surfaces, with some quantitative estimates and some qualitative descriptions in several examples. We will describe also the (rather unusual) boundary behavior of these object and their connection with the fractional Allen-Cahn equation.