The geometry of the KdV equation

In this talk I shall give a fairly geometrical account of the main facts about the KdV equation on the circle, explaining in particular how it is related to the group Diff(S¹), and why it is a completely integrable Hamiltonian system. In §4 I shall describe the theorem of Drinfeld and Sokolov [1] which shows that the KdV system can be regarded as a symplectic quotient of a coadjoint orbit of the loop group of SL₂(R). Finally, in §5 I shall explain how the theory generalizes from second-order equations and SL₂ to n^th-order equations and SL_n: the resulting classical system is the one whose quantization leads to the Zamolodchikov algebra W_n. My aim is purely expository: all the material is well known.