UC Irvine

The desire for classification is ubiquitous across mathematics, and possesses two distinctgoals. First, we hope to distinguish structures up to some notion of equivalence by identifying properties that distinguish them. Secondly, we hope to understand how complex objects are, and to have tests for their simplicity. This dissertation addresses three problems from the model theory of tracial von Neumann algebras related to these themes.

The first project, presented in Chapter 2, defines a notion dubbed the uniform super McDuffproperty which captures (in a first-order way) when a tracial von Neumann algebra has II1 factorial relative commutant. This continues work on the model theory of II1 factors and their axiomatizable central sequence algebraic properties, following the work of [32]. The second project, given in Chapter 3, provides a complete characterization of which tracial von Neumann algebras admit quantifier elimination, a property from model theory which states that the complete theory of an object can be completely determined by the quantifierfree theory. Finally, the third project, presented in Chapter 4, investigates the problem of elementary equivalence of the free group factors L(Fn). In particular, we prove a trichotomy holds for the first-order analog of the fundamental group of a II1 factor, and show that a dichotomy holds for the 2-quantifier theory of the free group factors.