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MODNET
Research Training Network in Model Theory



TASK III -  O-MINIMALITY AND APPLICATIONS


Description

O-minimality is a property of first-order structures with a linear ordering, simple to state (every definable subset of the model is a finite union of open intervals and points), and with very strong implications (Whitney stratification, finiteness of Vapnik-Chervonenkis dimension). Probably the most striking application of o-minimality is to Lie theory (Schmid-Vilonen), but one should also mention notable applications to asymptotics (solution of a 1911 problem of Hardy), neural networks, and database theory (O-minimality has been the main source of examples, in real mathematics, of situations where finite VC-dimension holds. This may reasonably be expected to be of use in randomized algorithms, and is of interest to some economists).


List of specific problems

    III.1. o-minimal expansions of the field of real numbers
        a) Develop homology and cohomology theories in o-minimal expansion of real closed fields
        b) Effective model-completeness of fields related to the reals with exponentiation (possibly modulo number-theoretic conjectures)
        c) Axiomatisation of the reals with exponentiation; identifying a recursively axiomatised model complete subtheory with a completion satisfying Schanuel s conjecture.
        d) Find connections between non-commutative elliptic curves (a la Soibelman) and non-standard elliptic curves arising in model theory.

    III.2. Groups in o-minimal structures
        a) Prove Pillay s conjecture that every d-dimensional group G definable in a saturated o-minimal structure has a largest Lie quotient H (with the logic topology), that if G is definably compact then H has dimension d and that if G is commutative, then H is torsion-free divisible
        b) Prove the existence of an analogue of Haar measure for definably compact groups definable over an o-minimal expansion of a field.
    III.3. Generalisations of o-minimality (e.g. weak o-minimality, C-minimality, P-minimality, quasi-o-minimality).
        a) Prove cell decomposition theorems for natural weakly ominimal classes (eg expansions of o-minimal structures by a valuation ring); classify the definable simple groups in these.
        b) Calculate VC dimension and density for P-minimal and C-minimal structures (in particular, the p-adics and ACVF).
        c) Prove cell decomposition, obtain invariants for definable sets, in  jet-o-minimal  structures.

Progress
Year 1