Abstract：

Let $X$ be a not necessarily smooth Fano variety and denote by \Ku(X) the non-trivial semi-orthogonal component. The Categorical Torelli problem asks if \Ku(X) determines the isomorphism class of $X$. In my talk, I will briefly talk about the history of this topic including the known results and popular strategies to prove these results. Then I will survey the recent advances for (weighted) hypersurfaces, a cubic threefold with a geometric involution, del Pezzo threefold of Picard rank one, and a class of nodal prime Fano threefolds. Meanwhile, I will talk about some new approaches to solving these problems. If time permits, I will also talk about categorical Torelli problems for a class of index one prime Fano threefold as the double cover of del Pezzo threefolds. This talk is based on a series of work joint with Xun Lin, Daniele Faenzi, Zhiyu Liu, Soheyla Feyzbakhsh, Jorgen Renneomo, Xianyu Hu, Sabastian-Casalaina Martin, and Zheng Zhang.