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VIII

MODNET
Research Training Network in Model Theory



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   


Progress
Year 1