1 Introduction
2 Preliminaries
2.1 Gauge Theory
2.2 Parabolic Subgroups
3 Algebraic and Symplectic Stability
4 The Yang-Mills Flow and Symplectic Stability
4.1 Analytical Foundations
4.2 Uniqueness of Yang-Mills Connections
4.3 Yang-Mills Characterization of $\mu_{\tau}$-Stability
5 Maximal Weights
5.1 Finite Weights
5.2 Weights and Algebraic Stability
5.3 The Moment Weight Inequality
6 The Kempf-Ness Functional
6.1 The Generalized Kempf-Ness Theorem
6.2 The Narasimhan-Seshadri-Ramanathan Theorem
7 The Dominant Weight Theorem