Harmonic maps, Toda frames and extended Dynkin diagrams
I shall discuss harmonic maps from surfaces into homogeneous spaces G/T where G is any simple real Lie group (not necessarily compact) and T is a Cartan subgroup. All immersions of a genus one surface into G/T possessing a Toda frame can be constructed by integrating a pair of commuting vector fields on a finite dimensional Lie algebra. I shall provide necessary and sufficient conditions for the existence of a Toda frame and describe those G/T to which the theory applies in terms of involutions of extended Dynkin diagrams. Applications will be given to harmonic maps into de Sitter spaces and to Willmore tori in S^3. This is joint work with Katharine Turner (University of Chicago).