Course overview -- Sommersemester 2023 (Archive)

A forcing order is said to be $<\kappa$-distributive iff it does not add new sequences of length $<\kappa$. A sufficient but not necessary condition for this is that the forcing is $<\kappa$-closed, i.e. any $<\kappa$-sequence of conditions has a lower bound. We introduce a strenghtening of $<\kappa$-distributivity called strong $<\kappa$-distributivity which can replace $<\kappa$-closure in many applications. A main benefit of this property is that a $<\kappa$-closed forcing remains strongly $<\kappa$-distributive in any extension by a $\kappa$-cc. order, even though it no longer necessarily is $<\kappa$-closed.

A theory is noetherian if there is a family of definable sets with the descending chain condition such that every definable set is a boolean combination of those in the family. Noetherianity captures some of the desired properties of algebraically closed fields in any characteristic or differentially closed fields in characteristic 0. Noetherian theories are in particular omega-stable and equational. In recent work with M. Ziegler, we have shown that the theory of proper pairs of algebraically closed fields in any characteristic is noetherian.

It is a well-known fact, with important applications in additive combinatorics and Ramsey theory, that the usual sum of integers may be extended to the space of ultrafilters over Z, yielding a compact right topological semigroup. The analogous construction also goes through for the product.

Recently, B. Šobot introduced two (ternary) notions of congruence on the space above. I will talk about joint work with M. Di Nasso, L. Luperi Baglini, M. Pierobon and M. Ragosta, in which the study of these congruences led us to isolate a class of ultrafilters enjoying characterisations in terms of tensor products, directed sets, profinite groups, and more.

n this talk, we present some results in the context of the generalized Baire space kappa^kappa. We prove that any suitable generalization of Laver forcing to the space κappa^κappa, for uncountable regular κappa, necessarily adds a Cohen κappa-real. We also study a dichotomy and an ideal naturally related to generalized Laver forcing. This is a joint work with Yurii Khomskii, Marlene Koelbing and Wolfgang Wohofsky.

We will present ongoing work on the model-theoretic classification of generic nilpotent groups and Lie algebras. A classical result in model theory is that all abelian groups are stable. Nilpotent groups are, in some sense, the simplest class of groups that properly contains the abelian groups. This led naturally to the question of investigating the degree of complexity of nilpotent groups.

In this talk, we will give some insight into the complexity of some generic theories of nilpotent groups. We will explain how those questions relate to more algebraic questions related to Lie algebras and we will illustrate an intriguing discontinuity of complexity when passing from generic 2-nilpotent groups to 3-nilpotent: the former are NSOP1 whereas the latter have SOP3.

The well-known Łoś-Tarski Theorem from classical model theory implies that a class of structures that is closed under induced substructures is axiomatizable in first-order logic by a sentence if and only if it has a finite set of forbidden induced finite substructures. Furthermore, by the Completeness Theorem, we can compute from the axiomatization of the class the corresponding forbidden induced substructures. This machinery fails on finite graphs as shown by our results.