A basic question in model theory is whether a theory admits any kind of quantifier reduction. One form of quantifier reduction is called model completeness, and broadly refers to when arbitrary formulas can be "replaced" by existential formulas.
Prior to the negative resolution of the Connes Embedding Problem (CEP), a result of Goldbring, Hart, and Sinclair showed that a positive solution to CEP would imply that there is no II$_1$ factor with a theory which is model-complete. In this talk, we discuss work on the question of quantifier reduction for tracial von Neumann algebras. In particular, we prove that no II$_1$ factor has a theory that is model complete by using Hastings' quantum expanders and a weaker assumption than CEP. This is joint work with Ilijas Farah and David Jekel.