Lecture for the Gårding Prize of 2023

Robert Berman (GU/Chalmers)

Title: 

Emergent complex geometry

Abstract: 

A recurrent theme in geometry is the quest for canonical metrics on a given manifold X. The prototypical case is when X is a compact orientable two-dimensional surface. Such a manifold can be endowed with a metric of constant curvature, which is uniquely determined by a fixing a complex structure on X. However, from a physical point of view, geometrical shapes - as we know them from everyday experience - are, of course, not fundamental physical entities. They merely arise as macroscopic emergent features of ensembles of microscopic point particles in the limit as the number N of particles tends to infinity. This leads one to wonder if there is a canonical random point process on a given complex manifold X, from which a canonical metrics emerges as the number N of points tends to infinity? This is, indeed, the case, when X is a complex algebraic hypersurface of any dimension, as explained in the present talk. In this case the emerging metrics in question have constant Ricci curvature. More precisely, they are Kähler-Einstein metrics. The talk is aimed to be non-technical and no previous background in complex geometry is required.