
MODNET
Research Training Network in Model Theory
TASK III  OMINIMALITY AND APPLICATIONS
Ominimality is a property of firstorder 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 VapnikChervonenkis dimension). Probably the most striking application of ominimality is to Lie theory (SchmidVilonen), but one should also mention notable applications to asymptotics (solution of a 1911 problem of Hardy), neural networks, and database theory (Ominimality has been the main source of examples, in real mathematics, of situations where finite VCdimension holds. This may reasonably be expected to be of use in randomized algorithms, and is of interest to some economists).
III.1. ominimal expansions of the field of real numbers III.2. Groups in ominimal structures
a) Prove Pillay s conjecture that every ddimensional group G definable in a saturated ominimal 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 torsionfree divisible b) Prove the existence of an analogue of Haar measure for definably compact groups definable over an ominimal expansion of a field. III.3. Generalisations of ominimality (e.g. weak ominimality, Cminimality, Pminimality, quasiominimality). a) Prove cell decomposition theorems for natural weakly ominimal classes (eg expansions of ominimal structures by a valuation ring); classify the definable simple groups in these. b) Calculate VC dimension and density for Pminimal and Cminimal structures (in particular, the padics and ACVF). c) Prove cell decomposition, obtain invariants for definable sets, in jetominimal structures. Progress Year 1 