Topological structure of spaces of stability conditions and topological Fukaya type categories

This is a survey on two closely related subjects. First, we review the study of topological structure of finite type' components of spaces of Bridgeland's stability conditions on triangulated categories. The key is to understand Happel-Reiten-Smalo tilting as tiling of cells. Second, we review topological realizations of various Fukaya type categories, namely cluster/Calabi-Yau and derived categories from surfaces. The corresponding spaces of stability conditions are of tame’ nature and can be realized as moduli spaces of quadratic differentials due to Bridgeland-Smith and Haiden-Katzarkov-Kontsevich.