Appears in collection : Combinatorics and Arithmetic for Physics : Special Days

While cluster algebras generally are not finitely generated, reddening sequences offer a more relaxed notion of finiteness. The existence of a redden- ing sequence has far reaching consequences for a cluster algebra (generic finite dimensionality of the Jacobian, numeric Donaldson-Thomas invariants, canonical bases). While it is not clear how to determine if a cluster algebra admits a redden- ing sequence, in this talk we discuss some cases in which reddening sequences have been established.