p-harmonic approximations of mass-type invariants in mathematical GR

In this talk, I will present recent developments in the study of mass notions in mathematical general relativity. Starting with the ADM mass, I will recall its role in fundamental results such as the positive mass theorem and the Riemannian Penrose inequality, while also pointing out its limitations in the study of more general geometric settings. These limitations motivate the introduction of isocapacitary masses, a family of invariants modeled on nonlinear elliptic equations involving the p-Laplacian. I will describe the main properties of this class of invariants in the asymptotically flat case, together with their relation with the ADM mass. I will then conclude by outlining possible extensions and open questions in other settings.