Stability and stable groups in continuous logic

We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity

Definability of groups in ℵ0\aleph_0-stable metric structures

We prove that in a continuous $\aleph_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from \cite{BenYaacov:TopometricSpacesAndPerturbations}, allowing us to prove the theorem in case the metric is invariant under the group action; and \item Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones. \end{enumerate

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 fifteen years. We discuss the appropriate first-order language for axiomatizing this class as well as the subclass of II$_1$ factors. We discuss how model-theoretic ideas were used to settle a variety of questions around isomorphism of ultrapowers of tracial von Neumann algebras with respect to different ultrafilters before moving on to more model-theoretic concerns, such as theories of II$_1$ factors and existentially closed II$_1$ factors. We conclude with two recent applications of model-theoretic ideas to questions around relative commutants.Comment: 27 pages; first draft; comments welcome; to appear in the volume "Model theory of operator algebras" as part of DeGruyter's Logic and its Application Serie

Generically Stable Measures and Distal Regularity in Continuous Logic

We develop a theory of generically stable and smooth Keisler measures in NIP metric theories, generalizing the case of classical logic. Using smooth extensions, we verify that fundamental properties of (Borel)-definable measures and the Morley product hold in the NIP metric setting. With these results, we prove that as in discrete logic, generic stability can be defined equivalently through definability properties, statistical properties, or behavior under the Morley product. We also examine weakly orthogonal Keisler measures, characterizing weak orthogonality in terms of various analytic regularity properties. We then examine Keisler measures in distal metric theories, proving that as in discrete logic, distality is characterized by all generically stable measures being smooth, or by all pairs of generically stable measures being weakly orthogonal. We then use this, together with our results on weak orthogonality and a cutting lemma, to find analytic versions of distal regularity and the strong Erd\H{o}s-Hajnal property