Decorated Marked Surfaces: Calabi-Yau categories and related topics

This is a survey on the project `Decorated Marked Surfaces’, where we introduce the decoration $\mathrm{\Delta }$ on a marked surfaces $\mathbf{S}$, to study Calabi-Yau-2 (cluster) categories, Calabi-Yau-3 (Fukaya) categories, braid groups for quivers with potential, quadratic differentials and stability conditions.