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\begin{center}{\huge One-Dimensional $p$-adic Subanalytic Sets}\end{center}
\begin{center}{\large Deirdre Haskell}\end{center}
\begin{center}{\large College of the Holy Cross}\end{center}
\begin{center}{\large Worcester, MA, USA}\end{center}
\bigskip

In their 1998 paper ``$p$-adic and real subanalytic sets'', J.~Denef
and L.~van den Dries use model-theoretic techniques to develop
a notion of subanalytic sets over the $p$-adics, analogous to
the notion from algebraic geometry in the reals. They prove quantifier 
elimination for the
$p$-adics in a language containing restricted analytic functions
and a partial division function. Essentially the same proof
can be used for the reals. In particular, subanalytic subsets of the
$p$-adic line are shown to be semialgebraic. In our recent work,
L.~van den Dries, D.~Macpherson and I have extended this last
result to arbitrary models. This shows that the theory of
the $p$-adics with restricted analytic functions is P-minimal, 
analogous to the o-minimality of the reals with restricted
analytic functions.  In my talk, I will review some of the 
background material and discuss some of the methods used to
prove our theorem. 

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