Model Theory of Metric Structures
September 18 to September 22, 2006
American Institute of Mathematics,
Palo Alto, California
C. Ward Henson and Itaï Ben Yaacov
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
We will use continuous logic as a common ground for
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.
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
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
to such perturbations.
Generalising results of Zilber et al. concerning
ù-stable, ù-categorical theories:
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 Lp Banach lattices.
Ultraproducts in analysis and geometry:
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.
Material from the workshop
A list of participants.
The workshop schedule.
A report on the workshop activities.
A list of open problems.
Slides from talks
Introductory lectures by Ben-Yaacov and Henson. (plain version)
Hilbert spaces and their generic automorphisms by Berenstein.
A hastily prepared introduction to perturbations by Ben-Yaacov.
Reports from working groups
Banach spaces without stability
Non-commutative probabilities and von Neumann algebras