Task I: Pure model theory
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 first-order context.
This includes not only questions about simple theories and
non-elementary classes, but also more recent developments such as
stable domination, where a `controlling part' of a structure is
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
I.1: Theoretical stability and simplicity
List of specific problems
a) Prove a group configuration theorem for 2-simple theories
I.2: Amalgamation constructions a la Hrushovski
b) Isolate conditions under which
independence in simple theories is governed by stable formulas, or use (aii)
below to obtain counterexamples.
c) Find an omega-categorical, simple, non-low 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 o-minimal and simple analogues, and find connections to thorn forking
f) Study the hierarchy: stable ->
omega-simple ->(n+1)-simple -> n-simple -> simple.
g) Prove the conjecture of Hrushovski
that the existence of a finitely axiomatizable, non-trivial strongly minimal
theory is equivalent to the existence of an infinite, finitely presented
division ring. Investigate the corresponding conjecture of Ivanov for the
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.
a) Construct omega-stable examples
showing n-ampleness gives a proper hierarchy ; build strongly minimal such
examples; find connections with pseudo-analytic structures
I.3: Topological methods in model theory
b) Build: nilpotent groups of finite
Morley rank and exponent, and class greater than 2; omega-stable Lie algebras
over finite field.
a) Develop model theory of the recent notion of a profinite structure, find examples.
I.4: Automorphism groups
a) Examine G-compactness for automorphism
groups of countable structures.
b) Prove that countable saturated, omega-stable structures are reconstructable
from their automorphism groups (e.g. via the small index property); prove
the small index property for omega-categorical structures from Hrushovski
, Year 3
, Year 4