We study expansions of a vector space V over a field F, possibly with extra structure, with a generic submodule over a subring of F. We construct a natural expansion by existentially defined functions so that the expansion in the extended language satisfies quantifier elimination. We show that this expansion preserves tame model theoretic properties such as stability, NIP, NTP1, NTP2 and NSOP1. We also study induced independence relations in the expansion.

Abstract We give a model-theoretic account for several results regarding sequences of random variables appearing in Berkes and Rosenthal [12]. In order to do this, • We study and compare three notions of convergence of types in a stable theory: logic convergence, i.e., formula by formula, metric convergence (both already well studied) and convergence of canonical bases. In particular, we characterise א0-categorical stable theories in which the last two agree. • We characterise sequences that admit almost indiscernible sub-sequences. • We apply these tools to the theory of atomless random variables (ARV). We characterise types and notions of convergence of types as conditional distributions and weak/strong convergence thereof, and obtain, among other things, the Main Theorem of Berkes and Rosenthal.

Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-basedω-categorical geometric theory interprets an infinite vector space over a finite field.

Abstract We provide a general theorem implying that for a (strongly) dependent theoryTthe theory of sufficiently well-behaved pairs of models ofTis again (strongly) dependent. We apply the theorem to the case of lovely pairs of thorn-rank one theories as well as to a setting of dense pairs of first-order topological theories.

Abstract. We interpret the algebra of finite rank operators as imaginaries inside a Hilbert space. We prove that the Hilbert space enlarged with these imaginaries has built-in canonical bases.

Abstract We generalize tools and results from first order stable theories to groups inside a simple stable strongly homogeneous model.