Manchester Geometry Seminar 2008/2009

Thursday 30 April 2009. The Frank Adams Room (Room 1.212), the Alan Turing Building. 4pm

Geometry and Dynamics of Isothermic Submanifolds

Fran Burstall (University of Bath)

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.