Emergent complex geometry

Abstract

This is a double exposure of the probabilistic construction of Kähler–Einstein metrics on a complex projective algebraic variety $X$ – where the Kähler–Einstein metric emerges from a canonical random point process on $X$ – and the variational approach to the Yau–Tian–Donaldson conjecture, highlighting their connections. The final section is a report on joint work in progress with Sébastien Boucksom and Mattias Jonsson on how the non-Archimedean geometry of $X$ (with respect to the trivial absolute value) also emerges from the probabilistic framework.