  We confirm a conjecture of Chi Li which says that the minimizer of the
normalized volume function for a klt singularity is unique up to rescaling.
This is achieved by defining stability thresholds for valuations, and then
showing that a valuation is a minimizer if and only if it is K-semistable, and
that K-semistable valuation is unique up to rescaling. As applications, we
prove a finite degree formula for volumes of klt singularities and an effective
bound of the local fundamental group of a klt singularity.

Algebraic Geometry;Commutative Algebra;Differential Geometry