Research Training Network in Model Theory
Research > Task I: Pure model theory

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 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.1: Theoretical stability and simplicity

a) Prove a group configuration theorem for 2-simple 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 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 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 omega-stable examples showing n-ampleness gives a proper hierarchy ; build strongly minimal such examples; find connections with pseudo-analytic structures
b) Build: nilpotent groups of finite Morley rank and exponent, and class greater than 2; omega-stable 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 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 constructions.


Year 1, Year 2, Year 3, Year 4

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