L3

Two classes of solutions of the Einstein equations with symmetry which

are frequently studied are the Bianchi and Gowdy models. The aim of this

talk is to explain certain relations between these two classes of

spacetimes which can provide insights into the dynamics of both. In

particular it is explained that the special case of the Gowdy models known as circular loop spacetimes are Bianchi models in disguise. Generalizations of Gowdy spacetimes which can be thought of as inhomogeneous perturbations of some of the Bianchi models are introduced.

Results concerning their dynamics are presented.

The modern approach to integrability proceeds via a Riemann surface, the spectral curve.

In many applications this curve is specified by transcendental constraints in terms of periods. I will highlight some of the problems this leads to in the context of monopoles, problems including integer solutions to systems of quadratic forms, questions of real algebraic geometry and conjectures for elliptic functions. Several new results will be presented including the uniqueness of the tetrahedrally symmetric monopole.