The purpose of this research, largely undertaken in collaboration with logicians, is to study operator algebras using a recent variant of classical model theory that interacts well with the structures of functional analysis. A basic goal is to determine how well this "continuous" model theory distinguishes analytic structures, and which properties it captures. Among the applications is new information about isomorphisms and embeddings between structures and their ultrapowers, as these can be readily reinterpreted using continuous model theory. The project also develops the continuous version of many model theoretic concepts, such as stability, primeness, quantifier elimination, model companions, and existential closedness, both for general logical theorems and new paradigms in functional analysis. Up to this point the emphasis has been on tracial von Neumann algebras, but the approach is expected to be fruitful for C*-algebras, subfactors, algebras equipped with an automorphism, etc.
The past few years have seen several surprising applications of logical techniques in functional analysis, mostly coming from combinatorial and descriptive set theory. This project aims to import concepts from model theory, the study of the relation between mathematical objects and their logical properties. Given an object, suppose one knows the answer to all the questions that can be formulated in a specific way (first-order syntax) -- how well does one understand the object? In general one cannot recover the object itself, but this "logical information" is relevant to many lines of research, and only recently has an appropriate version of model theory been applied to operator algebras. This project will continue to develop the ties between logic and functional analysis, in particular supporting travel for researchers and training for graduate students. The results should offer insights into active topics with applications to other branches of mathematics, for instance the Connes Embedding Problem and logical stability.
