Deformation functors

Deformation theory is a vital tool to understand the local structure of a moduli space around a fixed object X. A systematic approach to studying (infinitesimal) deformations of X is given by defining a functor on the category of Artin rings. In this talk, I will discuss conditions for the prorepresentability of the deformation functor. I will also explain the description of the deformation functor in the case when X is smooth, where the tangent and obstruction spaces arise naturally from the cohomology of the tangent sheaf.