B02
Discrete Multidimensional Integrable Systems

The project aims at the study and classification of multidimensional discrete integrable systems. It is now an outgrowth of two projects from the first funding period of CRC: B02 “Discrete multidimensional integrable systems” and B07 “Lagrangian multiform structure and multisymplectic discrete systems”. As such, it unifies different methodological approaches of those two projects to discrete integrable systems, namely, on the one hand, the emphasis on the combinatorial issues of the underlying lattice and the geometric interpretation of integrable systems, and, on the other hand, their variational (Lagrangian) interpretation.

During the first funding period we identified the M-system as a master system unifying a great variety of different 3D systems on the cubic lattice and developed a variational approach to integrable systems based on the notion of the pluri-Lagrangian structure. Now, we intend to identify discrete integrable master equations on lattices different from Zm, playing the role similar to that of the so-called M-system (governing the integrable evolution of minors of arbitrary matrices). In addition, we want to classify discrete forms generating pluri-Lagrangian systems.This should serve as a preparation to finding the pluri-Lagrangian structure of the fundamental 3D integrable systems of cubic type, including the discrete BKP equation and its Schwarzian version, and after that to the classification of discrete 3- and 4-forms on different lattices generating pluri-Lagrangian systems. Combining both approaches, we hope to finally obtain the solution of a long-standing classification problem concerning integrable systems: why 2D integrable systems are abundant, while only a few integrable 3D systems and no integrable 4D systems are known.