Mårten Nilsson: On the size of boundary pluripolar sets

Abstract: Boundary pluripolar sets, i.e., sets on the boundary of a domain Ω in \(\mathbf{C}^N\) which are (negative) singularity sets of plurisubharmonic functions on Ω, were recently characterized as the exceptional sets for the Dirichlet problem for the complex Monge–Ampère equation. Furthermore, they exhibit a propagation phenomena, which affects the where the solution to the equation is continuous. In this talk, I will give an overview of this topic and present two theorems, which together roughly says that small sets (in the sense of Hausdorff dimension) must be boundary pluripolar and non-propagating, while large sets (in the sense of topological dimension) must propagate into the interior.