Beijing Institute of Mathematical Sciences and Applications

In this course I plan to present a construction of integrable systems on the Poisson cluster varieties. I plan to start with some basic facts on symplectic and Poisson manifolds, known from Hamiltonian mechanics. Then we turn to the construction of cluster varieties, where the logarithmically constant Poisson bracket in any chart is encoded by certain quiver, while different charts are related by quiver mutations. The key example of cluster varieties are the Poisson-Lie groups. The Fock-Goncharov map introduces the cluster co-ordinates in Lie groups so that the Poisson structure coincides with the Lie-Poisson structure given by classical r-matrix, whose quantization gives rise to the quantum groups. The most well-known examples of cluster integrable systems are relativistic Toda chains, and we start the discussion by understanding why they arise naturally on certain Poisson submanifolds in Lie groups. Finally I plan to come to modern issues, including the Goncharov-Kenyon construction, deautonomization of cluster integrable systems and their relation with supersymmetric gauge theories.