Every ideal triangulation of a bordered surface with marked points has an associated quiver with potential (QP), defined in an elementary way by looking at the triangles of the triangulation and the punctures of the surface. Flips of arcs of ideal triangulations are then compatible with Derksen-Weyman-Zelevinsky's mutations of QPs. If the underlying surface has non-empty boundary, the QPs of its ideal triangulations are non-degenerate and Jacobi-finite.

Every tagged triangulation gives rise to an 'obvious' QP as well. Much less obvious is the compatibility between flips and DWZ's mutations of QPs. This compatibility, together with some non-trivial properties of the Jacobian algebras that can be established thanks to the specific form of the potentials involved, yields nice consequences for the cluster monomials in the cluster algebras associated to the surface by Fomin-Shapiro-Thurston.

The first part of the talk will be based on work done by myself a couple of years ago, the second part will be based on joint work in progress with G. Cerulli Irelli.