Columbia University

Title: Black Hole Stability

Description: We will give an overlook of recent stability problems in black holes. We will introduce the vectorfield method and how it is used in the analysis of waves in static and rotating black holes. We will then look into the analysis of perturbations of solutions to the Einstein equation, both in linear and nonlinear settings, and give an overview of recent results in this direction.

University of Montpellier

Title: Asymptotic invariants of asymptotically flat and hyperbolic manifolds

Description: The concept of mass and subsequent asymptotic invariants of asymptotically flat manifiolds has been introduced by physicists in general relativity in the 1960’s. These are geometric invariants that have attracted the attention of differential geometers in the 1970s, and much work has been devoted to them during the last 50 years. The subject has undergone a remarkable revival since the beginning of the current century due to (at least) two reasons: the relations between the mass and the geometry of minimal surfaces in asymptotically flat 3-manifolds, and the discovery of a whole family of invariants similar to the mass for a variety of model geometries. These exist not only on asymptotically flat manifolds but also on asymptotically hyperbolic, or complex hyperbolic, manifolds, and variants thereof. This is still a very active area of geometry, and many new ideas have emerged in the last 10 years. In the lectures, I will survey the most important results and try to give a hint on the current questions and most recent works.

University of Vienna

Title: The Penrose inequality in extrinsic geometry

Description: What do minimal surfaces have to do with gravity? The Riemannian Penrose inequality tells us that the total gravitational energy of initial data for a spacetime satisfying the Einstein field equations is at least as large as that of the area of a minimal surface contained within it. This reflects the basic idea that every black hole in the spacetime should make a quantifiable contribution to gravity. Just as minimal surfaces help us understand gravity in relativity, they also reveal important geometric properties of Euclidean 3-space. Precisely, the asymptotic growth of an unbounded support surface in Euclidean 3-space with nonnegative mean curvature turns out to be at least as large as the area of a compact minimal surface that meets this support surface orthogonally. Equality holds if and only if the support surface is a catenoid. This result is known as the Penrose inequality in extrinsic geometry and has only been verified recently.

The goal of this mini-course is to give an overview of the scope and limitations of the techniques employed to prove such energy inequalities, with a focus on the recent resolution of the Penrose inequality in extrinsic geometry. These techniques are very flexible and applicable to a wide range of problems in mathematical relativity and geometric analysis. The proof of the Penrose inequality in extrinsic geometry uses an approach based on so-called minimal capillary surfaces, a class of surfaces that has received much attention in recent years. The mini-course will cover the foundational tools to apply minimal capillary surfaces to a range of modern problems in geometry.

University of Toronto

Title: Introduction to the weak and strong cosmic censorship conjectures

Abstract: Within classical General Relativity, the weak and strong cosmic censorship conjectures are two of the most central open problems, and, one way or another, motivate a tremendous amount of work. In this course we will go over the formulations of these conjectures and survey the state of the art in progress towards resolving them. Among other things, we will discuss the concepts of maximal globally hyperbolic developments, black hole spacetimes, Penrose diagrams, trapped surfaces, Cauchy horizons, and naked singularities.