In [1], Fock and Goncharov introduced a pair of moduli spaces (X, A) dual to each other which generalized the enhanced Teichmuller space and the decorated Teichmuller space. In the first lecture, we will introduce some basic tools like their parameterization, monodromy computation, cluster ensemble structure, induced Poisson structure, Goncharov–Shen potential [2] and some related topics. In the second lecture, we will introduce their duality conjecture.

参考文献：

[1] Moduli spaces of local systems and higher Teichmuller theory, Publications Mathématiques de l’IHES 103 (2006), p. 1-211.

[2] Geometry of canonical bases and mirror symmetry, Inventiones mathematicae 202 (2015), 487-633.

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