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Emma Carberry : Conformal Surface Geometry: an algebro-geometric approach.

A number of classical integrable systems, for example harmonic maps of the plane to a compact Lie group or symmetric space, can be transformed into a \{\\em linear\} flow on a complex torus. This torus is the Jacobian of an algebraic curve, called the spectral curve. Recently several authors have produced an analogous one-dimensional analytic variety for conformal 2-tori in $S4$ (which are not in general integrable!) using the geometry of the quaternions. It is hoped that this new development will lead to progress on the Willmore conjecture for reasons that I will explain. However this variety is at present quite mysterious; very little is known about it. I will discuss the simplest case, namely constant mean curvature tori in $\mathbb{R}3$. I will demonstrate that in this case the variety is not at all mysterious and interpret its points geometrically in terms of transformations generalising the classical transform of Darboux. This is joint work with Katrin Leschke and Franz Pedit.

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