(Quasi)-Plurisubharmonic functions are a key notion in complex geometry. The study of their singularity (in terms -for example- of integrability properties or smoothing procedures) is conceived to develop analytic techniques in order to solve problems in complex and algebraic geometry. In this talk we study the space of all possible singularity types of quasi-plurisubharmonic functions and we introduce a natural (pseudo)-distance on it. As applications we present a stability result for complex Monge-Ampère equations with prescribed singularity and a semicontinuity result for multiplier ideal sheaves associated to singularity types. This is a joint work with T. Darvas and C. Lu.

In mathematical general relativity, Riemannian manifolds with non-negative scalar curvature are particularly important, as they correspond to a class of physically relevant solutions to the Einstein field equations (via the initial data formulation). In this talk, we will review a procedure due to Mantoulidis and Schoen to glue together Riemannian manifolds with non-negative scalar curvature by first attaching a small "collar" to the manifold. We will then review some applications of this construction to problems related to quasi-local mass and a geometric inequality in general relativity. This talk is partially based on joint work with Armando Cabrera Pacheco, Carla Cederbaum, and Pengzi Miao.

I will explain what the Willmore Morse Index of unbranched Willmore spheres in Euclidean three-space is and how to compute it. A consequence of that computation is that all unbranched Willmore spheres are unstable (except for the round sphere). This talk is based on work with Jonas Hirsch.