Abstract: In joint work with Patrick Speissegger we have recently constructed
a Hardy field which contains all transition maps of plane polynomial vector fields
at hyperbolic singularities and all unary functions definable in the o-minimal
expansion of the real field by restricted analytic functions and exponentiation.
In this talk I explain these notions and their connections and discuss how
o-minimality can make some contributions to questions around Hilbert 16.