Research > Task VI: Model theory of groups and modules
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
nonabelian 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.
a) Initiate fine study of definable sets in free groups and hyperbolic groups.
a) Find the asymptotic cone of SL(n,Z)
and other nonuniform arithmetic lattices, and of Grigorchuk groups. Extend
the KramerTent proof of Margulis conjecture to affine Lambdabuildings.
a) Characterise twisted simple groups
as groups definable in pseudofinite difference fields, and obtain uniformity
results for definable (maximal) subgroups.
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. a) Prove that Hilbert spaces are exactly the Banach spaces satisfying quantifier elimination. Progress Year 1, Year 2, Year 3, Year 4.

Last updated: September 9 2009 15:53  Please send your corrections to: 