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{\huge The model theory of analytic and smooth functions}
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{\large A course of 3 lectures by A.J. Wilkie (Oxford)}\end{center}
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In his excellent course at Logic Colloquium 94 van 
den Dries discussed the notion of $o$-minimality.   
He showed that
collections that can be realised as the definable 
sets in an $o$-minimal structure (based, usually, on the ordered
domain of real numbers) have many pleasant geometrical 
and topological properties.   Indeed, it seems that almost all
such properties previously thought special to, say, the 
collection of semi-algebraic sets or of sub-analytic sets
(which, due to the work of Tarski and Gabrielov/van den 
Dries-Denef respctively, can be so realised) can be established
in the general setting of $o$-minimality.

In this course I shall briefly review van den Dries' 
article in the proceedings of LC'94 (which has just 
appeared) as
motivation for my main theme, which is to present a 
practical method for deciding whether a given expansion of the
ordered domain of real numbers by analytic or, more generally, 
smooth functions is $o$-minimal.   In particular this
method may be applied to the Pfaffian functions, which 
are analytic functions satisfying certain simple systems of
partial differential equations.

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