Graph connections, (wild) character varieties, and generating functions in symplectic geometry

We will discuss a natural (pre)-symplectic structure associated to an arbitrary flat graph connection on a Riemann surface and its invariance properties. This allows to efciently parametrize (wild) character varieties using Fock-Goncharov coordinates and provide explicit log-canonical coordinates for several types of Poisson structures; Goldman on the standard character variety, Flaschka-Newell-Boalch on Stokes' manifolds and Ugaglia-Bondal Poisson structures.
In the case of (wild) character varieties, this construction allows to define the generating functions of symplectic polarizations and identify them with the classical notion of isomonodromic tau functions of the Japanese school.
Based on works with Dmitry Korotkin, Fabrizio Del Monte, Sofia Tarricone.