Abstract: Singularities appear naturally in many areas of mathematics, from the studies of partial differential equations (PDEs) to moduli spaces in algebraic geometry. Remarkably, in recent years, advances in the study of the Einstein equation (a fundamental geometric PDE) on complex spaces (certain important classes of algebraic varieties) have shown the emergence of non-trivial links between singularities in these two apparently distant worlds. The analytic multiscale aspects of singularities of PDEs, such as tangent cones, asymptotics and bubbling analysis, enhance the understanding of algebraic degenerations, e.g., leading to the construction of K-moduli spaces of Fano varieties. Vice versa, more concrete algebro-geometric tools can be used to “explicitly” describe solutions to this geometric PDE. In this talk, I will explain some of such relations, focusing on my work in the area, as well as discuss open problems and expectations for future developments.