Veranstaltungsübersicht -- 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.

Guingona and Hill introduced and studied a new hierarchy of dividing lines for first-order structures, denoted by (NC_K)_K , where K ranges in the theorie of ultrahomogeneous omega-categorical Ramsey structures. In a subsequent paper, Guigonna, Hill and Scow give a characterisation in terms of (generalised) K-indiscernible sequences. In this talk, I will present a joint work with Nadav Meir and Aris Papadopoulos, in which we develop around these notions of K-indiscernibility. In particular, we will answer (negatively) a question posed by Guingona and Hill about the linearity of the NC_K hierarchy. As an application, we will also see that the ordered random graph admits a unique proper Ramsey reduct, namely the linear order.

We introduce parametrized variants of Friedman's property. $F(\lambda,\kappa)$ states that any function from $\kappa$ into $\lambda$ is constant on a closed set of order type $\omega_1$. The principle $F^+((D_i: i\in\omega_1),\kappa)$ (for $(D_i : i\in\omega_1)$ a partition of $\omega_1$) states that for any sequence $(A_i: i\in\omega_1)$ of stationary subsets of $E_{\omega}^{\kappa}$ there is a normal function $f\colon\omega_1\to\kappa$ such that $f[D_i]\subseteq A_i$. We will prove all possible implications between instances of both properties and show the optimality of our results by obtaining suitable independence results.

Motivated by structural properties of differential field extensions, Omar Leon Sanchez and I introduced the notion of a theory T being derivation-like with respect to another model complete theory $T_0$. We proved that when T admits a model companion $T^+$, then several model-theoretic properties are transferred from $T_0$ to $T^+$. These properties include completeness, quantifier elimination, stability, simplicity, and NSOP$_1$. Examples of derivation-like theories are plentiful but are typically obtained by adding extra structure to theories of fields. In this talk I will introduce the central notions, detail how the proofs work by lifting independence relations from $T_0$ to $T^+$, and give examples.