Appears in collection : Conférence à la mémoire de Jean-Pierre Demailly

One major new birational geometry problem arising in understanding stable degeneration of varieties, which is the algebraic analogue to the compactness of Kähler-Einstein type metrics, is finite generation for valuations of higher rational rank. In the past a few years, we have established finite generation for minimizing valuations of various functionals, by first showing those minimizers are ‘special’; and then proving any special valuation satisfies finite generation. This raises the question on how ‘special’ valuations are distributed, which still remain to be widely open. In this talk, I will report results and questions along this direction. (Based on joint work with Yuchen Liu and Ziquan Zhuang).