Research > Task III: O-minimality and applications
Task III: O-minimality and applications
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).
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
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-freedivisible
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.
Year 1, Year 2, Year 3, Year 4.
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