Discrete Integrable Systems

Description and aim

In the field of Discrete Integrable Systems several branches of mathematics and physics, that are usually distinct, come together: complex analysis, algebraic/symplectic geometry, representation theory, difference/arithmetic geometry, graph theory, and the theory of special functions. This workshop intends to exploit state of the art expertise in the mentioned areas to attack the most prominent and challenging open problems related to Discrete Integrable Systems. The workshop will target the following five themes.

1.    Multi-dimensional Consistency, Lagrangian Forms Can we extend the classification with respect to multi-dimensional consistency, to include non-scalar cases. What is the significance of the new variational principle connected to Lagrangian forms, e.g. in physics applications.

2.    Symmetries, Integrals, Conservation laws, and Symplectic structures What are the relations between (the number of) symmetries, integrals and conservation laws? Can one decide on symplectivity without deriving a symplectic structure? How to systematically construct symplectic structures.

3.    Reductions and Solutions How to construct (explicit) solutions for periodic reductions, Painlevé reductions, or solutions of other type, i.e. finite gap, rational solutions.

4.    Detection and testing of Discrete Integrable Systems What are currently the most effective methods? Can we identify relations between different notions of integrability, e.g. growth versus consistency?

5.    Geometric approaches to Discrete Integrable Systems Algebraic geometry has been successfully applied to the Painlevé equations, and to the QRT maps. Can this be extended to higher dimensional mappings, and to lattices?

The workshop aims to initiate new research and research collaborations, in the field of Discrete Integrable Systems, and in closely related areas.