Deformations of strictly unital A-infinity algebras

I'll start by giving a gentle introduction to $A$-infinity algebras and their Hochschild cochains, using a rudimentary graphical calculus. The goal of the talk is to show how to construct a $dg-Lie$ algebra extension of the Hochschild cochains, that controls the unital deformations of a strictly unital $A$-infinity algebra (as opposed to the non-unital deformations given by the Hochschild cochains). This is motivated by some concrete questions in homological commutative algebra; in particular, the first non-trivial case of the construction gives a conceptual reason why matrix factorizations describe representations of a hypersurface singularity.