The Yau-Tian-Donaldson conjecture is a central problem in Kähler geometry which relates the existence of a special metric (such as a constant scalar curvature Kähler metric) in a given Kähler class to an algebro-geometric condition of stability of the polarization class. Especially, for Kähler-Einstein metrics, the conjecture was solved affirmatively by Chen-Donaldson-Sun and Tian through the use of cone singularities applying the Cheeger-Colding-Tian theory. However, the conjecture is still open for general polarizations or more generally in extremal Kähler cases.
Keeping this in mind, we study the the variational approach to the conjecture posed by Boucksom et al. ([3], [4]). It is closely related to the non-Archimedian geometry ([1], [2], [4]), and a special emphasis will be put on the reformulation of various concepts from non-Archimedian viewpoints.

Some basic knowledge of Kähler (or algebraic) geometry and algebraic groups

[1] S. Boucksom and M. Jonsson: A non-Archimedean approach to K-stability, arXiv: 1805.11160.
[2] S. Boucksom, C. Favre and M. Jonsson: Singular semipositive metrics in non-Archimedean geometry, J. Algebraic Geom. 25 (2016), 77-139.
[3] S. Boucksom, T. Hisamoto and M. Jonsson: Uniform K-stability and asymptotics of energy functionals in Kähler geometry, arxiv: 1603.01026.
[4] S. Boucksom: Variational and non-Archimedean aspects of the Yau-Tian-Donaldson conjecture, Proc. ICM 2018 Rio de Janeiro, Vol. 1, 589-614.