The minicourse is devoted to integrable systems on cluster varieties, their deautonomization and connection with supersymmetric gauge theories. We start with the cluster Poisson varieties and describe their main properties, keeping as a basic example the Fock-Goncharov construction of cluster co-ordinates on the (affine, co-extended) Lie groups. Then we discuss how this construction leads to appearence of a completely integrable system on their Poisson subvarieties, with the most well-known example given by relativistic Toda chains, while generally these integrable systems can be alternatively defined a la Goncharov and Kenyon. The whole picture allows natural deautonomization, still keeping traces of integrability in the (discrete, non-autonomous) equations of the Painleve type, whose solutions can be constructed in terms of supersymmetric gauge theories. To do that we remind the connection between Seiberg-Witten prepotentials and algebraic integrable systems, introduce Nekrasov functions and show, that their duals (just by Fourier transform) appear in this context as isomonodromic tau-functions, solving the Hirota equations for deautonomized cluster integrable systems.

The minicourse is devoted to integrable systems on cluster varieties, their deautonomization and connection with supersymmetric gauge theories. We start with the cluster Poisson varieties and describe their main properties, keeping as a basic example the Fock-Goncharov construction of cluster co-ordinates on the (affine, co-extended) Lie groups. Then we discuss how this construction leads to appearence of a completely integrable system on their Poisson subvarieties, with the most well-known example given by relativistic Toda chains, while generally these integrable systems can be alternatively defined a la Goncharov and Kenyon. The whole picture allows natural deautonomization, still keeping traces of integrability in the (discrete, non-autonomous) equations of the Painleve type, whose solutions can be constructed in terms of supersymmetric gauge theories. To do that we remind the connection between Seiberg-Witten prepotentials and algebraic integrable systems, introduce Nekrasov functions and show, that their duals (just by Fourier transform) appear in this context as isomonodromic tau-functions, solving the Hirota equations for deautonomized cluster integrable systems.