“The Skein Algebra and Hall algebras”

In the early 90's, Turaev showed that the q=1 limit of skein algebra of a surface gives a quantization of its character variety, and he asked for an explicit description for general q. We discuss these results and an answer to his question for the torus (joint with H. Morton). We then discuss applications and related results involving the Heisenberg category, the Hall algebra of an elliptic curve, and the Hall algebras of the Fukaya categories of surfaces. (All objects in this abstract will be defined, and the second half involves joint works with Cooper, Cautis, Lauda, Licata, Morton, and Sussan.)