Please note different day and time.  In this lecture I will explain the moment-weight inequality, and its role in the proof of the Hilbert-Mumford numerical criterion for μ-stability. The setting is Hamiltonian group actions on closed Kaehler manifolds. The major ingredients are the moment map μ and the finite dimensional analogues of the Mabuchi functional and the Futaki invariant. This is joint work with Valentina Georgoulas and Joel Robbin, based on conversations with Xiuxiong Chen, Song Sun, and Sean Paul.