Koszul duality and deformation theory

Koszul duality between Lie algebras and cocommutative coalgebras constructed by Hinich is the basis for formal deformation theory, at least in characteristic zero.  In this talk I explain, following Manetti, Pridham and Lurie, how Koszul duality, combined with Brown representability theorem from homotopy theory leads to representability of a formal deformation functor up to homotopy. Sometimes a formal deformation functor has a `noncommutative structure', meaning that it is defined on a suitable homotopy category of associative algebras. In this case there is a similar representability result, valid in an arbitrary characteristic. I will also discuss a generalization of this noncommutative representability theorem to the non-local case.