Differential graded Lie algebras in deformation theory

This thesis aims to provide a gentle introduction to deformation theory, and in particular to the approach based on differential graded Lie algebras. We introduce the fundamental algebraic structures at play, including differential graded Lie algebras. We establish several important facts about these objects, with particular attention to their homological and homotopical properties. We set up the general theory of abstract deformation functors, and see how it applies in case of deformations of a chain complex. Then, we define the deformation functor associated to a differential graded Lie algebra, and demonstrate with a motivating example how it can be used to describe a concrete deformation problem. In particular, we enlight the general principle asserting that in characteristic 0 every deformation problem can be described as the deformation functor associated to a DG-Lie algebra. We state the homotopy invariance theorem for the deformation functor associated to a DG-Lie algebra, we discuss its implications, and finally give an outline of its proof.