Harry Braden, School of Mathematics, University of Edinburgh Wednesday 28th April 14:05-14:55pm, New Law School Seminar 030 (Building F10). Abstract: 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.