Abstract:

I will present a manifestly positive formula computing any wall-function in a rank-2 scattering diagram generated by two initial lines. The coefficients of these wall-functions enumerate partial tilings of the region above certain Dyck paths. By lifting the rank-2 positivity to higher ranks, we obtain a proof of the positivity conjecture for Chekhov-Shapiro's generalized cluster algebras. This is joint work with Amanda Burcroff and Kyungyong Lee.