Project B2

Geometry of discrete integrability

The main goal of this project is an analytical and structural study as well as a classification of discrete integrable systems, with an emphasis on the geometrical structures behind integrability, such as multidimensional consistency. This notion is crystallized during the development of discrete differential geometry in recent years, which was intimately related with the development of discrete integrable systems. On the one hand, most (if not all) interesting special classes of surfaces and coordinate systems, smooth and discrete, turn out to be integrable, i.e., to be described via integrable systems. On the other hand, a geometric interpretation provides us with new insights into the nature of integrability.
The main open problems we are going to investigate in this project are:

• Are there integrable systems in dimensions greater than three?
• How many different 3-dimensional integrable systems exist?
• Which incidence theorems in geometry can be made discrete evolution equations and, on the other hand, can all discrete integrable systems be interpreted as particular incidence theorems?
• What are discrete integrable elliptic systems?
• What are new features of discrete integrable systems on non-cubical lattices?