My thesis is a proof of a conjecture of Kontsevich stating that the Hochschild cohomology (in the case of dgVect, this is really Hochschild cochains!) of an E_n algebra is the universal E_{n+1} algebra acting on it. The notion on an E_{n+1} algebra acting on an E_n algebra is defined using the Swiss Cheese operad of Voronov. A rewritten and improved version is available here.

A simple example of this theorem is an algebra A acting on a vector space V. In this case n=0 and the role of the E_0 algebra is played by the vector space V, and the role of the E_1 algebra is played by A. The swiss cheese operad tells us, roughly, that the notion of an action of A on V is a map AxV---> V satisfying certain conditions, chief among which is compatibility with the algebra structure on A. The "universal" example of such an algebra A is the vector space End(V) with algebra structure given by composition of maps. Furthermore, to give an action of A on V is equivalent to giving a map of associative algebras A--> End(V).

The theorem proven in my thesis is that this still works (in a homotopy theoretic sense) if you replace A by an E_{n+1} algebra, V by an E_n algebra, and End(V) by Hoch(V), the E_n-Hochschild cohomology of V.