ИСТИНА

The principal subject of the talk is the electrical version of the cluster varieties of the Lusztig type. Our main result is a new vertex realization of the statistical model associated with an electrical network on circular graphs. This result has a wide set of connections with many mathematical fields: cluster algebras, Temperley-Leeb algebras, positive mathematics, and others. In this report I will focus on the vertex form of the statistical mechanical model related to the classical problem of electric networks and the phenomenon of discreet integrability associated with the electrical solution of the Zamolodchikov tetrahedron equation. The resulting integrable system is related to the Painleve equation. The talk is based on the joint work with Vassily Gorbounov.