Generic bases for surface cluster algebras with arbitrary coefficients

In the last 15 years there has been a lot of research aimed at finding and understanding various types of bases for cluster algebras. One of the proposed bases is the “generic basis”, shown recently by Fan Qin to be indeed a basis for injective reachable upper cluster algebras with full-rank extended exchange matrices. In this talk, based on joint work with Christof Geiss and Jan Schröer, I will sketch an elementary proof of the linear independence of the generic basis for cluster algebras arising, with arbitrary geometric coefficients (not necessarily of full rank), from punctured surfaces with non-empty boundary. I shall also sketch our strategy to show that the generic basis generates the (upper) cluster algebra (more accurately, the Caldero-Chapoton algebra, which in general sits between the cluster algebra and the upper cluster algebra).