Integrable models are systems in quantum physics which can be solved exactly, because they have a large number of conserved quantities and thus a high degree of symmetry. They appear in many guises - field theories, spin chains, models of statistical mechanics, models of a fixed number of interacting particles - and can also be subsumed into models with even more symmetry, such as conformal field theories and 'superintegrable' models (such as the harmonic oscillator, the Kepler system in Newtonian gravity and the hydrogen atom!).
The last few decades have seen a resurgence of exactly-solvable physical systems, which appear ubiquitously in modern fundamental physics. For example, they have been at the heart of recent advances in our understanding of the relationship between gauge field theory and string theory (the so-called 'gauge-string correspondence'). The mathematical beauty of such models is often seen in new algebraic structures (usually 'quantum groups' of some kind), and we have been involved in the discovery and development of many of these.
For information about possible PhD projects, see here.