Course overview -- Wintersemester 2024/2025

Imaginaries are a way of introducing canonical representatives of equivalence classes. Given a theory, an important question is whether equivalence classes are already coded in the models of the theory so that one can avoid working with imaginaries (i.e. we say that the theory then eliminates imaginaries).

In this talk, we want to present a criterion that yields the failure of elimination of imaginaries due to equivalence classes that arise as cosets of subgroups. We will illustrate the main ideas by considering the example of the theory of beautiful pairs of algebraically closed fields. Pillay and Vassiliev proved that this theory does not have elimination of imaginaries. In this talk, we will present a different proof that generalizes to more theories of fields.

Motivated by work of Hrushovski on pseudofinite fields with an additive character we investigate the theory ACFA+ which is the model companion of the theory of difference fields with an additive character on the fixed field working in (a mild version of) continuous logic. Building on results by Hrushovski we can recover it as the characteristic 0 asymptotic theory of the algebraic closure of finite fields with the Frobenius-automorphism and the standard character on the fixed field. We characterise 3-amalgamation in ACFA+ and obtain that ACFA+ is simple as well as a description of the connected component of the Kim-Pillay group. If time permits we present some results on higher amalgamation.