A celebrated conjecture of Yau states that the existence of a Kähler metric of constant scalar curvature on a projective manifold should be equivalent to a purely algebraic stability condition. Much progress have been done on this conjecture, which culminated in what is now called the Yau-Tian-Donaldson program. In this talk, I will explain the key role played by quantization methods in this program, and how they can be improved by a semiclassical study of the quantum noise of Berezin-Toeplitz quantization.

This is partly based on joint works in collaboration with Victoria Kaminker, Leonid Polterovich and Dor Shmoish.