Stability conditions for varieties

Enormous progress in algebraic geometry has been achieved through linking with differential geometry and geometric analysis. A modern example of this is the "Yau-Tian-Donaldson conjecture", which relates the algebro-geometric notion of K-stability of a projective variety to the existence of solutions to a special PDE on the variety. In this setting, the PDE is the "constant scalar curvature equation", which can be thought of as giving the variety a canonical choice of metric. I will describe a general framework associating geometric PDEs on projective varieties to notions of algegbro-geometric stability, and will sketch a proof showing that existence of solutions is equivalent to stability in a model case. The framework can be seen as a loose analogue in the setting of varieties of Bridgeland's stability conditions.