  In this paper we develop a bridge between model theory, geometric topology,
and geometric group theory. In particular, we investigate the Ivanov
Metaconjecture from the point of view of model theory, and more broadly we seek
to answer the general question: why does the curve graph of a surface play such
a central role in the study of surfaces and mapping class groups?
  More specifically, we consider a surface $\Sigma$ of finite type and its
curve graph $\mathcal C(\Sigma)$, and we investigate its first-order theory in
the language of graph theory. Crucially, $\mathcal C(\Sigma)$ is
bi-interpretable with a certain object called the augmented Cayley graph of the
mapping class group of the surface. We use this bi-interpretation to prove that
the theory of the curve graph is $\omega$--stable, to compute its Morley rank,
and to show that it has quantifier elimination with respect to the class of
$\forall\exists$--formulae. We also show that many of the complexes which are
naturally associated to a surface are interpretable in $\mathcal C(\Sigma)$.
This shows that these complexes are all $\omega$--stable and admit certain a
priori bounds on their Morley ranks. We are able to use Morley ranks to prove
that various complexes are not bi--interpretable with the curve graph. As a
consequence of quantifier elimination, we show that algebraic intersection
number is not definable in the first order theory of the curve graph. Finally,
we prove that the curve graph of a surface enjoys a novel phenomenon that we
call interpretation rigidity. That is, if surfaces $\Sigma_1$ and $\Sigma_2$
admits curve graphs that are mutually interpretable, then $\Sigma_1$ and
$\Sigma_2$ are homeomorphic to each other. Along the way, numerous technical
results are obtained.
