Solving symplectic groupoid: quantization and integrability

We begin with the recently found solution in terms of cluster algebras of the problem of symplectic groupoid: how to describe manifolds of pairs $(B,A)$ where $B$ is an $SL_N$ matrix, $A$ is unipotent upper-triangular matrix, and $BAB^T$ is again unipotent upper-triangular matrix. Solutions obtained possess a natural Poisson and quantum algebra structures and open a wide spectrum of possibilities: from describing Teichmuller spaces of closed Riemann surfaces of arbitrary genus $g\ge 2$ to a so far hypothetical relation to Cherednik's DAHA and conformal blocks of the Liouville theory. Based on the forthcoming joint paper with Misha Shapiro.