A map is said to be integrable when certain special conditions hold (existence of other commuting maps; existence of invariant functions in the context of Poisson maps; existence of Lax pairs; etc). Whilst maps may seem simpler than differential equations, their integrability is rather more complicated to analyse.
A rich source of integrable maps is through cluster algebras, which are related to sequences with the Laurent property, such as the well known Somos sequences.
The classification of these is related to the classification of quivers with certain periodicity properties. This is an exciting new area with many open questions in a diverse range of subjects.
An interesting problem is to reduce the action of a map to some invariant manifold or to split the action of the map according to symmetry properties of the first integrals. The resulting map can have surprising properties, such as commuting with other maps on this lower dimensional space.
The Integrable Systems group in Leeds, one of the leading groups worldwide working on integrable systems, has been strengthened by three recent appointments, one at a senior level. Integrable Systems are systems that, albeit highly nontrivial and nonlinear, are amenable to exact and rigorous techniques for their solvability.
They can take many shapes or forms: nonlinear evolution equations, partial and ordinary differential equations and difference equations, Hamiltonian many-body systems, quantum systems and spin models in statistical mechanics. A large number of mathematical techniques have been developed to unravel the rich structures behind these systems. The five permanent members of staff work with nine postdocs and postgraduate students.
Leeds is a major centre for the theory of discrete and quantum integrable systems.
This group represents a wide range of research activities into integrable nonlinear systems, their symmetries, solution techniques and the underlying mathematical structures, as well as more mathematical aspects of physical systems, for example quantum systems. The models comprise ordinary and partial differential and difference equations, dynamical mappings, discrete Painleve equations, Hamiltonian and many-particle systems and systems of hydrodynamic type. The theory and its specific models have wide-ranging applications, for example, in nonlinear optics, theory of water waves, integrable quantum field theory, statistical mechanics and combinatorics random matrix theory and nanotechnology.