Logic Seminar
Alex Wilkie
University of Oxford
Characterizing definable closure in certain o-minimal expansions of real closed fields, with applications
Abstract: Every closed subset of euclidean space can be represented as the zero set of a smooth function (ie an
infinitely differentiable function defined on the whole ambient space). My student, Gareth Jones, has
been investigating the question of whether this theorem (of Whitney) holds in the context of sets and
functions definable in an o-minimal expansion of the real exponential field. This turns out to require
uniform estimates on all derivatives of a given definable function. In many structures of interest these
estimates are easily seen to hold for the terms of the language, so the problem becomes one of finding
analytic operations, mapping definable functions to definable functions, which (a) preserve the
estimates and (b) eventually generate all definable functions. This is the topic of my talk.
Tuesday April 4, 2006 at 4:00 PM in SEO 427