Title: "On mutation classes of quivers with constant number of arrows and derived equivalences." Abstract: The Bernstein-Gelfand-Ponomarev reflection is a combinatorial operation on quivers (directed graphs) defined at vertices which are sinks or sources and preserves the number of arrows. From a representation-theoretic perspective, it induces derived equivalence between the path algebras. The Fomin-Zelevinsky mutation extends this operation to arbitrary vertices on a combinatorial level, whereas the mutation of quivers with potentials (QP) introduced by Derksen, Weyman and Zelevinsky does this algebraically. However, in general the number of arrows is no longer preserved and the corresponding Jacobian algebras are not necessarily derived equivalent. In this talk we will characterize all the quivers with the property that performing arbitrary sequences of mutations does not change their number of arrows. It turns out that these quivers arise from ideal triangulations of certain marked bordered surfaces in the sense of Fomin, Shapiro and Thurston. This combinatorial property has also a representation-theoretic counterpart: to each such quiver there is naturally associated potential such that performing arbitrary sequences of QP mutations does not change the derived equivalence class of the corresponding Jacobian algebra. Most of these algebras are finite-dimensional, but some of them are infinite-dimensional. The latter resemble the 3-Calabi-Yau algebras despite not being so. All the notions will be explained during the talk.