Vorträge in der Woche 21.01.2008 bis 27.01.2008

Montag, 21.01.2008: Transformations of Constant Mean Curvature Surfaces and an Introduction to Spectral Curves

Prof. Emma Carberry

I will begin by explaining how an integrable system (for example harmonic maps of the plane to a compact Lie group) can be transformed into a LINEAR flow on a complex torus. This torus is the Jacobian of an algebraic curve, called the spectral curve. Rewriting the original differential equation in terms of algebraic curve data allows one to try to describe the space of all solutions, and I will describe several cases in which this has been done. This discussion will be elucidated with the example of constant mean curvature tori; and I will explain how in this case points on the spectral curve can be interpreted geometrically as transformations of the original surface. I will give a number of interesting properties of these transformations, and discuss how they allow one to define a spectral curve for conformal tori, which do not form an integrable system.