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 \(B\in GL_n\) being such that \(\tilde A=BAB^T\) 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).