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This paper gives an essentially self contained exposition (except for an appeal to the Lojasiewicz gradient inequality) of geometric invariant theory from a differential geometric viewpoint. Central ingredients are the moment-weight inequality (relating the Mumford numerical invariants to the norm of the moment map), the negative gradient flow of the moment map squared, and the Kempf-Ness function. Contents 1 Introduction 2 The moment map 3 The moment map squared 4 The Kempf-Ness function 5 µ-Weights 6 The moment-weight inequality 7 Stability in symplectic geometry 8 The Kempf-Ness theorem generalized 9 Stability in algebraic geometry 10 Rationality 11 The dominant µ-weight 12 Torus actions

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