From complex to non-Archimedean geometry: an approach to the YTD conjecture

These notes expand on talks given by the authors at the 2025 Summer Research Institute in Algebraic Geometry in Fort Collins, Colorado. We discuss the relation between algebraic, analytic, and non-Archimedean geometry over the complex numbers, and sketch a proof of a version of the Yau–Tian–Donaldson conjecture for constant scalar curvature Kähler metrics.