Explore a one-hour mathematics lecture that delves into the theory of Hamiltonian actions of loop groups, presented at the Erwin Schrödinger International Institute's Thematic Programme on "Infinite-dimensional Geometry." Learn about the foundational concepts of loop groups and their central extensions as accessible infinite-dimensional groups in Hamiltonian geometry, tracing their development from the pioneering works of Donaldson and Meinrenken-Woodward. Examine key examples of loop group Hamiltonian spaces, including infinite-dimensional moduli of flat connections, while understanding crucial theoretical results such as reduction and convexity theorems, Duistermaat-Heckman localization, Kirwan surjectivity, and the principle that quantization commutes with reduction ([Q,R]=0).