The modern theory of integrability was created and developed over the last thirty years by a number of international research groups. Several approaches to integrable equations have been elaborated, which look quite different but focus on solutions of the same range of problems. One of the aims of this programme is to bring together key scientists with various background and expertise in order to elaborate a coherent view on the problem and to attempt to develop a synthetic theory which would reconcile the different approaches. This will be the first meeting on such a scale and we expect that mutual understanding of different approaches will cause a breakthrough in the whole theory of integrable equations and significantly extend its applications.
The following methods for studying integrability will be discussed
- The Painlevé approach
- The perturbation theory approach
- Asymtotic expansions methods
- The symmetry approach
In addition the "dressing method", and its many variations, will be looked at as a technique for constructing and solving integrable systems. The topic of Quantum Integrability will also be addressed. Links between the classical dressing method and various approaches to quantum systems will be studied. A fundamental issue is a classification of algebraic and differential reductions "inside" the integrable systems which have been constructed.