Minimal definable sets in difference fields.

I will speak about the Zilber trichotomy for weakly minimal difference varieties, and the definable structure on them.

A difference field is a field with a distinguished automorphism $\sigma$. Solution sets of systems of polynomial difference equations like

$3 x \sigma(x) +4x +\sigma^2(x) +17 =0$ are the quantifier-free definable subsets of difference fields. These \emph{difference varieties} are similar to varieties in algebraic geometry, except uglier, both from an algebraic and from a model-theoretic point of view.

ACFA, the model-companion of the theory of difference fields, is a supersimple theory whose minimal (i.e. U-rank $1$) types satisfy the Zilber's Trichotomy Conjecture that any non-trivial definable structure on the set of realizations of a minimal type $p$ must come from a definable one-based group or from a definable field. Every minimal type $p$ in ACFA contains a (weakly) minimal quantifier-free formula $\phi_p$, and often the difference variety defined by $\phi_p$ determines which case of the Zilber Trichotomy $p$ belongs to.