In this talk, I discuss a joint work with Alexey Elagin and Evgeny Shinder that lies at the intersection of birational and derived geometry. What kind of geometric information is encoded in the derived category of a variety, and how does this information behave under birational maps? We consider the case of surfaces equipped with a group action, and show that their derived categories admit a canonical (mutation-equivalence class of) semi-orthogonal decomposition that "behaves well" with respect to birational geometry. We call such decompositions "atomic". If time permits, I will discuss applications to (bi-)rationality questions. This talk is intended for non-experts of derived geometry; all notions will be explained.