Final Report Summary - RANDOM-KAHLER (Kähler-Einstein metrics, random point processes and variational principles)
In broad terms the project concerns the problem of describing the emergence of coherent structures in large scale complex systems from a geometric and analytic point of view. More specifically, a statistical mechanical scheme is introduced to construct an optimal metric on a given mathematical shape, known as an algebraic manifold. In physical terms this corresponds to constructing a particular solution to Einstein’s equations in cosmology, where the manifold in questions arises as the underlying space-time of the universe. The main achievement of the project is the proof that this scheme indeed works in the case of a negative cosmological constant. This covers a large class of algebraic manifolds. Moreover, for general shapes, known as Kähler manifolds, variational analytic tools have been developed, based on a new “thermodynamical formalism”. This has applications to the Yau-Tian-Donaldson conjecture in current Kähler geometry. In another direction, new surprising links to other current research areas in mathematics and physics have also been explored, such as sampling problems and the problem of optimal transportation.