
MODNET
Research Training Network in Model Theory
TASK I  PURE MODEL THEORY
Description
Pure model theory is the core of model theory,
and its development is essential to new applications. The main
focus is centered around taking ideas from (geometric) stability
theory and applying them 'beyond' the stable firstorder context.
This includes not only questions about simple theories and
nonelementary classes, but also more recent developments such as
stable domination, where a `controlling part' of a structure is
stable.
Even within the classical stable context there are still important
questions to be answered, such as understanding the possible
geometries on strongly minimal sets, and Vaught's conjecture for
superstable theories.
List of specific problems
I.1Theoretical stability and simplicity
a) Prove a group configuration theorem for 2simple theories
b) Isolate conditions under which
independence in simple theories is governed by stable formulas, or use (aii)
below to obtain counterexamples.
c) Find an omegacategorical, simple, nonlow theory.
d) Develop forking in unstable contexts
(e.g. thorn forking, where one needs to find connections between thorn ranks
and ranks in stability/simplicity theory)
e) Find new unstable structures with
many stable, stably embedded sets and develop model theory of stable domination;
develop ominimal and simple analogues, and find connections to thorn forking
f) Study the hierarchy: stable >
omegasimple >(n+1)simple > nsimple > simple.
g) Prove the conjecture of Hrushovski
that the existence of a finitely axiomatizable, nontrivial strongly minimal
theory is equivalent to the existence of an infinite, finitely presented
division ring. Investigate the corresponding conjecture of Ivanov for the
trivial case.
h) Develop stability theory for almost
elementary classes, and examine groups in this setting.
i) Investigate compact abstract theories,
with applications to the model theory of Banach spaces.
I.2. Amalgamation constructions a la Hrushovski
a) Construct omegastable examples
showing nampleness gives a proper hierarchy ; build strongly minimal such
examples; find connections with pseudoanalytic structures
b) Build: nilpotent groups of finite
Morley rank and exponent, and class greater than 2; omegastable Lie algebras
over finite field.
I.3. Topological methods in model theory
a) Develop model theory of the recent notion of a profinite structure, find examples.
I.4. Automorphism groups
a) Examine Gcompactness for automorphism
groups of countable structures. b)
Prove that countable saturated, omegastable structures are reconstructable
from their automorphism groups (e.g. via the small index property); prove
the small index property for omegacategorical structures from Hrushovski
constructions.
Progress
Year 1
