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.