Model Theory of Metric Structures

This workshop, sponsored by AIM and the NSF, will focus on the use of model theoretic ideas in analysis and metric geometry, bringing together model theorists and specialists from a few key application areas for a period of intense discussions. A diverse combination of backgrounds will allow the participants to explore from new angles certain examples, applications, and theoretical problems that define the frontier of research on the model theory of metric structures.

A major goal of this workshop is to overcome communication barriers between model theorists and analysts. We will use continuous logic as a common ground for collaboration. This recently developed logic combines familiar semantic constructs from analysis with the syntactic framework of first order logic.

A new phenomenon, which does not exist in ordinary model theory, is that metric structures can be naturally perturbed. Experience shows that restating questions "up to perturbation" may be essential for a smooth general theory to be developed.

Principal topics on which the workshop will focus are:

• The theory of probability algebras with a generic automorphism (PAA): The theory PAA is stable, but not superstable, even though the theory of pure probability algebras is superstable. We hope to remedy this anomaly by considering models and types only up to small perturbations of the automorphism (this is also of interest for other superstable theories). We would also like to find an elegant classification of (non-separable) models of PAA up to such perturbations.
• Generalising results of Zilber et al. concerning ù-stable, ù-categorical theories: We conjecture that if T is ù-stable and ù-categorical, then the theory of beautiful pairs of models of T is ù-categorical up to perturbations of the predicate defining the submodel. This is known for Hilbert spaces, probability algebras and L^p Banach lattices.
• Ultraproducts in analysis and geometry: Ultraproduct constructions have been effectively used in functional analysis (geometry of Banach spaces, von Neumann algebras, operator spaces) and metric space geometry (Gromov convergence, asymptotic cones of finitely generated groups). This provides an interface between continuous logic and these areas, and yields many instructive examples and phenomena.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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