Quantum mirrors of log Calabi-Yau surfaces and higher genus curve counting

We present three results, at the intersection of tropical geometry, enumerative geometry, mirror symmetry and non-commutative algebra. 1. A correspondence between Block-Göttsche q-refined tropical curve counting and higher genus log Gromov-Witten theory of toric surfaces. 2. A correspondence between q-refined two-dimensional Kontsevich-Soibelman scattering diagrams and higher genus log Gromov-Witten theory of log Calabi-Yau surfaces. 3. A q-deformation of the Gross-Hacking-Keel mirror construction, producing a deformation quantization with canonical basis for the Gross-Hacking-Keel families of log Calabi-Yau surfaces. These results are logically dependent: the proof of the third result relies on the second, whose proof itself relies on the first. Nevertheless, each of them is of independent interest.