19 January - 3 July 2009

**Organisers:** Dr R Halburd *(University College London)*, Professor F Nijhoff *(Leeds)*, Professor M Noumi *(Kobe)*, Professor GRW Quispel *(La Trobe)* and Professor O Ragnisco *(Roma TRE)*

**Scientific Advisory Committee:** Professor MJ Ablowitz *(Boulder, Colorado)*, Professor RA Askey *(Wisconsin)*, Professor JJ Duistermaat *(Utrecht)*, Professor M Jimbo *(Tokyo)*, Professor N Joshi *(Sydney)*, Professor IM Krichever *(Columbia, New York)*, Professor SP Novikov *(Maryland)*, Professor J-P Ramis *(Toulouse)*, and Professor AP Veselov *(Loughborough)*

The theory of (ordinary and partial) differential equations is well-established and to some extent standardised. By contrast, the theory of difference equations, while more fundamental, has until recently been in its infancy, in spite of a major effort at the beginning of the 20th Century by N"orlund and the school of G.D. Birkhoff to establish the linear theory. Discrete systems can appear in two main guises: in the first case the independent variable is discrete, taking values on a lattice (e.g. finite-difference equations, such as recurrence relations and dynamical mappings), in the second case the independent variable is continuous (e.g. analytic difference equations and even functional equations).

Very recently, however, mainly through advances in the theory of exactly integrable discrete systems and the theory of (linear and nonlinear) special functions, the study of difference equations has undergone a true revolution. For the first time good and interesting examples of nonlinear difference equations admitting exact, albeit highly nontrivial, solutions were found and this has led to the formulation of novel approaches to the classification and treatment of such equations. Thus, an area has developed where several branches of mathematics and physics, that are usually distinct, come together: complex analysis, algebraic geometry, representation theory, Galois theory, spectral analysis and the theory of special functions, graph theory, and difference geometry.

Looking at recent developments in these various fields one observes that many of the areas that are clearly on the
verge of major breakthroughs have an intimate connection with discrete integrable systems.
Such fields comprise: the algebraic geometry of rational surfaces and birational maps, difference Galois theory
and isomonodromic deformation theory, Diophantine problems in number theory and *p*-adic analysis, quantum and
non-commutative discrete integrable systems, representation theory of quantum algebras and associated cluster
algebras as well as special functions, difference geometry and symmetries and conservation laws of discrete
systems. Bringing together experts from these fields for a
substantial period of time the programme aims at bringing about important cross-fertilisation
of ideas and methodologies, and thereby substantial new advances in the theory of difference equations.