TASK VI -
MODEL THEORY OF GROUPS AND MODULES
Description
Groups are omnipresent in
model theory. Of particular interest here are the recent
announcements of a positive solution
to Tarski's problem (that any two
non-abelian free groups are elementarily equivalent) which has generated great
excitement in the logic community. Proofs are extremely long
and intricate, use geometrical ideas, and will require substantial
effort to be understood, and hopefully simplified. Many questions of
decidability, description of definable sets, generalisation of techniques to
some hyperbolic groups are wide open,
though stability of free groups is now known.
Model theory of modules is a well-developed branch
of model theory, with traditional connections to representation theory
of finite-dimensional algebras. One direction of research addresses
decidability questions, while the other, more ambitious, aims to extend
the model theory of modules to more general coherent categories, and to lay out
the foundations of the model theory of derived categories.
List of specific problems
VI.1. Finitely presented groups (free, polycyclic, hyperbolic, etc.).
a) Initiate fine study of definable sets in free groups and hyperbolic groups.
b) Prove that limit groups have a Krull dimension
c) Prove (or refute) stability
of certain finitely presented (not free) groups.
VI.2. Asymptotic cones
a) Find the asymptotic cone of SL(n,Z)
and other non-uniform arithmetic lattices, and of Grigorchuk groups. Extend
the Kramer-Tent proof of Margulis conjecture to affine Lambda-buildings.
VI.3. Pseudofinite groups
a) Characterise twisted simple groups
as groups definable in pseudofinite difference fields, and obtain uniformity
results for definable (maximal) subgroups
b) Show that pseudofinite stable groups are nilpotent-by-finite or construct
counterexample
c) Develop pseudofinite groups as model
for black-box recognition of structural properties of finite groups by probablilistic
methods.
VI.4: Modules
a) Show that if D is a Dedekind ring,
G a finite group, then the theory of D[G]-modules is decidable if and only
if D[G] is of finite or tame representation type.
b) Classify strongly minimal modules
over (sufficiently nice) group rings extend to quasi-minimal, finite Morley
rank; prove Vaught s conjecture for theories of modules.
c) Prove that interpretations between
categories of structures can be characterised as functors between their categories
of imaginaries
VI.5: Banach spaces
a) Prove that Hilbert spaces are exactly the Banach spaces
satisfying quantifier elimination