Cluster algebras via quivers with potentials

Cluster algebras are commutative rings of a special kind making a surprising appearance in a variety of settings, including quiver representations, Poisson geometry, Teichmuller theory, representations of semisimple groups, etc. Their structure is governed by several piecewise-polynomial and birational recurrences on a regular tree.
Although these recurrences are quite explicit and elementary, a direct proof of their conjectural properties seems to be hard. In a joint work with Harm Derksen and Jerzy Weyman an interpretation of the recurrences in question was found in terms of quivers with potentials and their representations, implying proofs of several conjectures due to Sergey Fomin and the speaker.
The lecture will provide an introduction to cluster algebras and quivers with potentials.