Cluster algebras: geometry of Riemann surfaces and symplectic groupoid

The first lecture is the introduction to cluster algebras (an object introduced by Fomin and Zelevinsky in around year 2000) and related Poisson and quantum algebras of geodesic functions (hyperbolic cosines of half-lengths of geodesics in Poincare uniformization of Riemann surfaces with holes) A.K.A. Goldman brackets. The cluster variables therefore provide the parameterization of Teichmuller spaces T_{g,s} of Riemann surfaces of genus g with $s>0$ holes. This is based on papers by V.V.Fock, M.Shapiro, M.Mazzocco and myself in years 1999-2010.