Abstract:  E. Cartan classified conformally flat hypersurfaces in
(n+1)-sphere for n>3, and showed that they are one parameter envelopes
of (n-1)-spheres.  But this is not true for n=3. In this talk, I will
explain some joint work with Neil Donaldson: We show that (i) the
equation for conformally flat hypersurfaces in S^4 is a soliton
equation, (ii) such hypersurfaces are determined by 6 functions of one
variable, (iii) there exists a loop group symmetry and Ribaucour
transformations on the moduli space of these hypersurfaces.