Stability theory of singularities

Over the past decade, a local theory of K-stability for Kawamata log terminal (klt) singularities has emerged. This theory studies minimizers of a normalized volume function and associates a K-semistable Fano cone to any given klt singularity. In many cases, this construction significantly generalizes the notion of the metric tangent cone for singular Kähler–Einstein spaces. The lecture is intended for a broad mathematical audience.