O-minimal geometry

Syllabus

In model-theoretic terms, an expansion M of the real ordered field is o-minimal if all M-definable subsets of the reals have finitely many connected components. This can also be formulated in purely geometric terms, as a property of a collection of real sets, stable under the boolean set-operations, Cartesian products and linear projections. The sets definable in an o-minimal structure share many topological tameness properties with real algebraic and real analytic sets (good dimension theory, uniform finiteness, stratification), which makes o-minimal geometry relevant to problems in Diophantine and arithmetic geometry, non-oscillatory dynamical systems and asymptotic analysis. I will give an overview of the main results about o-minimal structures and then I will concentrate on illustrating the main methods for proving that a collection of real functions generates an o-minimal structure. There are essentially no prerequisites for this course, other than the basic undergraduate notions of algebra and analysis: the model-theoretic background needed is minimal and self-contained references will be provided to those who might need them.

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• Bâtiment Sophie Germain
  8 place Aurélie Nemours
  75013 Paris