Calabi—Yau dg algebras are the derived-noncommutative analogue of smooth Calabi—Yau varieties. I’ll talk about a generalised notion of Calabi-Yau structure for dg (co)algebras, before giving a brief review of algebra-coalgebra Koszul duality, and explaining why this `nonsmooth Calabi—Yau’ condition is dual to a symmetric Frobenius condition. There is also an analogous one-sided version: Gorenstein (co)algebras are Koszul dual to Frobenius (co)algebras. I’ll talk about a surprising example: commutative complete local Gorenstein k-algebras, when equipped with their natural topology, are pseudocompact Frobenius algebras. This is in some sense a reflection of Matlis duality. As an application of the above theory, we obtain a new characterisation of Poincaré duality spaces, which for simply connected spaces recovers Félix-Halperin-Thomas’s notion of Gorenstein space. This is joint work with Joe Chuang and Andrey Lazarev.