Emma Carberry School of Mathematics & Statistics, University of Sydney Constant Mean Curvature Surfaces and Integrable Systems Wednesday 17th May 14:05-14:55pm, Carslaw Building Room 373. 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. The combination of this approach and spectral curve theory has already produced interesting results on two well-known conjectures in differential geometry: Willmore's Conjecture and Lawson's Conjecture, and it is hoped that this new development will lead to progress at least on at least the first of these. However this variety is at present quite mysterious; very little is known about it. I will look at the simplest case, namely constant mean curvature tori in 3-space, where one can prove that the variety is a finite genus algebraic curve and give geometric interpretations of its points (this is joint work with Katrin Leschke and Franz Pedit). This talk will be of a general nature, with an emphasis on the main ideas but few details.