For a punctured surface S and a split reductive algebraic group G such as SL_n or PGL_n, Fock and Goncharov (and Shen) consider two types of moduli spaces parametrizing G-local systems on S together with certain data at punctures. They show that these spaces have special coordinate charts, hence are birational to cluster varieties. Fock and Goncharov’s duality conjectures predict the existence of a canonical basis of the algebra of regular functions on one of these spaces, enumerated by the tropical integer points of the other space. I will give an introductory overview of this topic, briefly explain recent developments involving quantum topology and mirror symmetry of log Calabi-Yau varieties, and present some open problems if time allows.

This is a 30min slow-pace pre-seminar discussion aiming to familiarise graduate students and postdocs with basic ideas used in Thursday Analysis Seminar talk and next Wednesday Colloquium (in Analysis). The focus will be put on ideas, and technicalities are kept to a minimum. 

Topics include: heat kernel, hyperbolic surfaces, Gauss–Bonnet formula.