A survey on the model theory of tracial von Neumann algebras

We survey the developments in the model theory of tracial von Neumann algebras that have taken place in the last 15 years. We discuss the appropriate first-order language for axiomatizing this class as well as the subclass of II1 factors and how modeltheoretic ideas were used to settle a variety of questions around isomorphism of ultrapowers of tracial von Neumann algebras with respect to different ultrafilters. We move on to more model-theoretic concerns, such as theories of II1 factors and existentially closed II1 factors, and conclude with two recent applications of model-theoretic ideas to questions around relative commutants.