Cluster algebras: geometry of Riemann surfaces and symplectic groupoid

The second lecture will be about the novel advances in cluster algebra applications to geometry:
We consider the symplectic groupoid: pairs (B,A) with A unipotent upper-triangular matrices and $B\in G{L}_n$ being such that $\tilde{A}=BA{B}^T$ are also unipotent upper-triangular matrices. We explicitly solve this groupoid condition using Fock–Goncharov–Shen cluster variables and show that for B satisfying the standard semiclassical Lie–Poisson algebra, the matrices B, A, and $\tilde{A}$ satisfy the closed Poisson algebra relations. Identifying entries of A and $\tilde{A}$ with geodesic functions on a closed Riemann surface of genus $g=n-1$, we are able construct the geodesic function $G_B$ for geodesic joining two halves of the Riemann surface. We thus obtain the complete cluster algebra description of Teichmüller space $T_{2,0}$ of closed Riemann surfaces of genus two . We discuss also the generalization of our construction for higher genera. (based on joint paper ArXiv:2304.05580 with Misha Shapiro).