Abstract:

Research at the intersection of differential and algebraic geometry has been dominated by the principle that extremal objects in differential geometry correspond to stable objects in algebraic geometry. This remarkably robust principle manifests itself in finite dimensions through Mumford's geometric invariant theory and symplectic reduction, but also underlies a number of significant correspondences: the Hitchin--Kobayashi correspondence between stable bundles and Yang--Mills connections, the Chen--Donaldson--Sun theorem relating K-stable Fano varieties to Kahler--Einstein manifolds, and even the uniformization theorem!
In this talk I will investigate the roots of this principle throughout differential and algebraic geometry, and present a survey of some of the many instances of it in the literature. Key features common to all examples are the existence of a functional whose critical points describe the extremal objects, and the principle can be interpreted as saying this functional is proper and convex (so a critical point exists) if and only if it is convex when measured along rays with rational slope. The latter condition often bares a purely algebro-geometric description, and the interpolation from rational rays to all rays follows from the density of the rational numbers in the real numbers.