Geometric pluripotential theory on Kähler manifolds

Abstract: Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Amp`ere type arising in K\"ahler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of infinite dimensional convex optimization, yielding existence results for solutions of the underlying complex Monge-Amp`ere equations. The purpose of this survey is to describe these developments from basic principles.