Abstract: Fixed point Floer cohomology of a symplectic automorphism categorifies the Lefschetz trace formula. On the categorical side, twisted Hochschild homology of an automorphism on an A infinity category can be understood as a categorical trace. I will explain how the twisted open-closed maps are used to relate these two invariants in the setting of Fukaya categories of Landau-Ginzburg models. Applications to symplectic topology and mirror symmetry will be discussed, especially for Lefschetz fibrations. This is joint work with Paul Seidel.