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 (nonseparable) models of
PAA up
to such perturbations.

Generalising results of Zilber et al. concerning
ùstable, ùcategorical theories:
We conjecture
that if T is ùstable and ùcategorical, then
the theory of beautiful pairs of models of T is
ùcategorical 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.
Material from the workshop
A list of participants.
The workshop schedule.
A report on the workshop activities.
Slides from talks
Introductory lectures by BenYaacov and Henson. (plain version)
Hilbert spaces and their generic automorphisms by Berenstein.
A hastily prepared introduction to perturbations by BenYaacov.
(plain version)
Reports from working groups
Asymptotic cones
Banach spaces without stability
Stable groups
Noncommutative probabilities and von Neumann algebras