Research > Task III: Ominimality and applications
Task III: Ominimality and applications
Description
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).
a) Develop homology and cohomology theories in ominimal expansion of real closed fields
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 torsionfreedivisible
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. Progress Year 1, Year 2, Year 3, Year 4.

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