Research > Task VI: Model theory of groups and modules
Task VI: Model theory of groups and modules
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.
VI.1: Finitely presented groups (free, polycyclic, hyperbolic, etc.)
a) Initiate fine study of definable sets in free groups and hyperbolic groups.
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.
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.
Year 1, Year 2, Year 3, Year 4.
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