Willmore tori and their spectral curves

Abstract: In this talk we discuss the Willmore functional of surfaces in R^3, S^3, and R^4. It is the integral of the square of mean curvature. Treating 2-tori in particular, we show that a generalisation of the Weierstrass representation of minimal surfaces can be used to define a spectral curve. Spectral curves are algebraic curves that encode the information about a surface. We then consider how to deform these spectral curves, so called Whitham deformations. We finish with what these deformations can tell us about the moduli space of harmonic maps, of surfaces with constant mean curvature, and perhaps of Willmore surfaces.