Willmore 2-Spheres in Sn: A Survey

We give an overview of the classification problem of Willmore 2-spheres in S-n, and report the recent progress on this problem when n = 5 ( or even higher). We explain two main ingredients in our work. The first is the adjoint transform of Willmore surfaces introduced by the first author, which generalizes the dual Willmore surface construction. The second is the DPW method applied to Willmore surfaces whose conformal Gaussmap is well-known to be a harmonic map into a non-compact symmetric space (a joint work of Dorfmeister and the second author). We also sketch a possible way to classify all Willmore 2-spheres in S-n.