Model Theory of Metric Structures

September 18 to September 22, 2006

at the

American Institute of Mathematics, San Jose, California

organized by

Itaï Ben Yaacov and C. Ward Henson

Original Announcement

This workshop 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.