The homotopy type of associative and commutative algebras

Given  a (dg) commutative algebra, one can ask how much of its homotopy type is preserved by its associative part. More precisely one can ask if $C$ and $C'$ are commutative algebras connected by a zig-zag of quasi-isomorphisms of associative algebras  $C\stackrel{\sim}{\longleftarrow} A \stackrel{\sim}{\longrightarrow} C'$, must $C$ and $C'$ be quasi-isomorphic as commutative algebras? Despite its elementary formulation, this question turns out to be surprisingly subtle.
 
 In this talk, I will show how one can use operadic deformation theory to give an affirmative answer to this question in characteristic zero. We will also see how the Koszul duality between Lie algebras and commutative algebras allows us to use similar arguments to deduce that the homotopy type of Lie algebras is determined by the (associative) homotopy type of the corresponding universal envelopping algebras.

(Joint with Dan Petersen, Daniel Robert-Nicoud and Felix Wierstra and based on arXiv:1904.03585)