"This proposal centers on the special interaction of algebraic and differential geometry which arises from the notion of stability, going back to the celebrated correspondence between polystable vector bundles and Hermite-Einstein connections.A conjecture of Yau, Tian and Donaldson seeks to extend this correspondence to projective manifolds, formulating an algebro-geometric notion of stability (K-polystability) which should be equivalent to the existence of a (unique) Kaehler metric of constant scalar curvature in the first Chern class of an ample line bundle. The necessity of stability is now settled, thanks to the work of Donaldson, myself (Adv. in Math. 2009) and Mabuchi. The existence implication however seems to be out of reach with current techniques. In this project I will motivate the need to go beyond the notion of K-stability, and select some crucial open problems which arise naturally in this context, especially in connection with Donaldson's program for Fano manifolds and his conjecture that a Fano manifold with a Kaehler-Einstein metric is ``birationally stable"". Another surprising application of algebro-geometric stability to differential geometry has recently emerged in the physical work of Gaiotto, Moore and Neitzke. They showed (conjecturally) how to use the stability and wall-crossing of ``BPS states'' to reconstruct the Hitchin hyperkaehler metric on a class of moduli spaces of Higgs bundles. In this project I propose to study some exciting mathematical questions which arise from this theory.This project aims at attacking some central problems which stem from the connection between stability and special metrics, and will be carried out by myself as P.I. and Gabor Szekelyhidi as a Team Member, over a period of four years. We will be supported by two Post-Docs (each position lasting two years) and a graduate student (three years)."