Fran Burstall (University of Bath)
feb@maths.bath.ac.uk
This talk will have three parts: in the first, I will describe the beautiful classical theory of isothermic surfaces in the 3-sphere due to Christoffel, Darboux, Bianchi and others. Then I will indicate how the 3-sphere may be replaced by any symmetric R-space (a conjugacy class of real parabolic subalgebras with abelian nilradicals) with essentially no loss of integrable structure. Finally, I shall show how dynamics of the simplest examples (curves in the real projective space) provide a geometric interpretation of the KdV equation, its relation to the mKdV equation via the Miura transform and the Bäcklund transformations of KdV discovered by Walhquist--Estabrook.