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Friday, March 13, 2015

**Abstract:** We (joint work with Hakobyan, T.) study the class $\mathcal{C}$ of two sorted structures $(F,K;\chi)$, where $F$ and $K$ are algebraically closed fields, $K$ has characteristic 0, and $\chi:F\to K$ is a generic multiplicative character which means $\chi$ is injective, multiplication preserving, and takes multiplicatively independent elements to algebraically independent elements over $\mathbb{Q}$. The examples of main interest are when $F=\mathbb{F}_{p}^{ac}$ and $K=\mathbb{Q}^{ac}$. We will discuss the following: Every strongly minimal set is non-orthogonal to $F$. Toward showing that every strongly minimal set is a Zariski Geometry, we equip $K^n$ with a natural topology and show this topology is Noetherian. We also show that the a generic character is inverse addition preserving, and induces a natural valuation ring structure on $\mathbb{Q}(\chi(F))$. This is the second of the two talks on this subject.