An Absence of Quantifier Reduction for $\textrm{II}_1$ Factors, using Quantum Expanders -SC 1432
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 $\textrm{II}_1$ factor with a theory which is model-complete. In this talk, we discuss work on the question of quantifier reduction for general tracial von Neumann algebras. In particular, we prove a complete classification for which tracial von Neumann algebras admit complete elimination of quantifiers. Furthermore, we show that no II$_1$ factor (satisfying a weaker assumption than CEP) has a theory that is model complete by using Hastings’ quantum expanders. This is joint work with Ilijas Farah and David Jekel.