In this dissertation we generalize the supra-SIM property, a partition regular property of subsets of the integers, to a property we call the supra-SBM property of subsets of finitely generated virtually nilpotent groups. We achieve this by defining a measure on the unit ball of asymptotic cones which we show satisfies the Lebesgue density theorem, allowing us to use this measure in place of the Lebesgue measure in the original definition for the integers. These proofs make use of various concepts from nonstandard analysis, measure theory, and combinatorial number theory. We prove that, under an additional assumption, the supra-SBM property is a partition regular property for subsets of finitely generated groups of nilpotency class 2, and conjecture that this holds for all finitely generated virtually nilpotent groups.

In the second chapter, we initiate the study of computable presentations of real and complex C-star-algebras under the program of effective metric structure theory. With the group situation as a model, we develop corresponding notions of recursive presentations and word problems for C-star-algebras, and show some analogous results hold in this setting. Famously, every finitely generated group with a computable presentation is computably categorical, but we provide a counterexample in the case of C-star-algebras. On the other hand, we show every finite-dimensional C-star-algebra is computably categorical.

In the third chapter, building on previous work by Franklin et al., we show if X is a compact subset of R which consists of the Cantor set and only finitely many intervals and points, then C(X; R) is arithmetically categorical as a real Banach space and C(X; C) is arithmetically categorical as a complex C-star-algebra. For any compact subset X ⊆ R, we show there is an arithmetical embedding of X into R in computable presentations of C(X; R). As a consequence, if C(X; R) or C(X; C) admit a computable presentation, then X is homeomorphic to an arithmetically closed subset of R. Also, multiplication is arithmetical in any computable presentation of C(X; R).

In this thesis, we study expansions of the additive group of integers (Z, +). In particular, weare interested in expansions of the group structure on Z which preserve some of the model- theoretic properties of the group. For example, it is well known that (Z, +) is superstable of Lascar rank 1. Much work has been done to determine which expansions of (Z, +) by a predicate yield a structure which is still (super)stable. Relatedly, one can ask about which predicates yield expansions that, while unstable, are still (super)simple. Using notions from algorithmic randomness, we show that for almost all subsets A of the natural numbers, the expansion of (Z, +) by the set ±A = {a ∈ Z : |a| ∈ A} is unstable, supersimple, and has Lascar rank 1, providing one answer to a question posed by Bhardwaj and Tran. Additionally, we show how the expansions of (Z, +) by the semiprimes (products of two primes) and separately the quasiprimes (numbers with no small prime factors) also satisfy some similar classification results by proving that these sets behave generically with respect to the additive structure on Z. In the case of the semiprimes, this comes from assuming an unproven number-theoretic result similar to Dickson’s conjecture for the prime numbers. In the case of the quasiprimes, we use results from sieve theory to derive the needed genericity.

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.

We adapt a partial notion of EF games to a continuous logic for unbounded pointed metric structures, and use this to investigate the elementary equivalence of certain metric structures. Particular focus is placed upon EF games between asymptotic cones of symmetric spaces $X$ arising from semisimple Lie groups. Kramer, et al. showed that, depending on the truth of CH, there is, up to homeomorphism, either $1$ or $2^{2^{\aleph_0}}$-many asymptotic cones of $X$ as one varies the choice of ultrafilter. This leaves open the possibility that all such asymptotic cones are elementarily equivalent. Towards a proof of the elementary equivalence, we utilize the fact that these asymptotic cones are known to be isometric to the point spaces of certain nondiscrete affine $\R$-buildings. We investigate the building structure and demonstrate the elementary equivalence of parallel classes of walls, which are fundamental to the classification of affine $\R$-buildings.