Course overview -- Sommersemester 2024 (Archive)

A type in a stable theory $T$ is stationary if it has a unique non-forking extension. By adding imaginary elements to a model of $T$, types over algebraically closed sets in the expanded structure become stationary. If $T$ does not eliminate imaginaries, the question arises whether types over algebraically closed subsets in the original model (real subsets) are also stationary.

After reviewing all the above notions, we will discuss this problem for the theory $T_P$ of beautiful pairs of models of a stable theory $T$ introduced by Poizat. By a result of Pillay and Vassiliev, this theory does not have (geometric) elimination of imaginaries if an infinite group is definable in $T$. We will prove that types over real algebraically closed sets in $T_P$ are stationary.

In 1975, Friedman introduced the property $F(\kappa)$, stating that every subset of $\kappa$ either contains or is disjoint from a closed set of ordertype $\omega_1$. Famously, this property follows from the power forcing axiom ``Martin's Maximum''. In this talk, we introduce posets which force the negation of this property and other related notions and investigate the patterns in which these properties can fail in connection to large cardinals.

One way to study the properties of the infinite cardinals is to examine the extent to which they can be changed by forcing. In 1969 and 1970, Bukovsk{\'y} and Namba independently showed that $\aleph_2$ can be forced to be an ordinal of cofinality $\aleph_0$ without collapsing $\aleph_1$. The forcings they used and their variants are now known as Namba forcing. Shelah proved that Namba forcing collapses $\aleph_3$ to an ordinal of cardinality $\aleph_1$. In a 1990 paper, Bukovsky and Coplakova asked whether there can be an extension that collapses $\aleph_2$ to an ordinal of cardinality $\aleph_1$ without collapsing $\aleph_3$. We will show that a slight strengthening of local precipitousness on $\aleph_2$ due to Laver allows us to construct such an extension.

I will present a general way of building some examples of NSOP1 theories as limit of some Fraïssé class satisfying strong conditions. We take interest in the properties of independence relations in these theories. In particular these limits will satisfy existence and we can compute Kim-forking and forking inside of them. These theories also come with a stationary independence relation. This study is based on results of Baudisch, Ramsey, Chernikov and Kruckman.

Homogeneous structures and their reducts can be used to model many computational problems from finite model theory as constraint satisfaction problems (CSPs). In this talk I will give a survey on open model-theoretic problems for such structures that are relevant for obtaining complexity classification results for the corresponding CSPs. In particular, I will discuss finite homogeneous Ramsey expansions, reconstruction of structures up to bi-interpretability from the abstract automorphism group, and Thomas's conjecture about closed supergroups.