Perverse sheaves of categories and some applications

We study perverse sheaves of categories their connections to classical algebraic geometry. We show how perverse sheaves of categories encode naturally derived categories of coherent sheaves on $\mathbb{P}^1$ bundles, semiorthogonal decompositions, and relate them to a recent proof of Segal that all autoequivalences of triangulated categories are spherical twists. Furthermore, we show that perverse sheaves of categories can be used to represent certain degenerate Calabi--Yau varieties.