Final Activity Report Summary - INTEGRABILITY (Analysis of a novel class of integrable Partial Differential Equations) Many different type phenomena in physics are described by nonlinear partial differential equations. Some of these equations have a very special structure tied into them which is called integrability. Several powerful methods are available for analysing and constructing solutions for integrable equations. In this project we studied numerous problems, using such equations, which described: 1. the propagation of waves, either in water or optic fibres, and 2. rotating disks in Einstein relativity theory. In particular, we considered problems with a boundary, for which the solution data was known on the boundary of some domain. The problem consisted of constructing the solution in the interior. By applying a novel method based on spectral theory, we showed how several boundary value problems for physically relevant equations could be solved. The solutions could be used to understand how electromagnetic waves evolved in optical fibres and how an astrophysical disk of dust rotated in space.