Course overview -- Wintersemester 2021/2022 (Archive)

In recent years a strict connection between set theory and social welfare relations has been studied in theoretical economics. In this talk I present some generalised versions of redistributional equity principles for infinite populations. More specifically we focus on the representation and construction of social welfare relations satisfying these generalised principles and we also combine them with other known efficiency and intergenerational equity principles in economic theory; in particular, important roles from set theory are played by ultrafilters and non-Ramsey sets.

In 1973, Shelah showed that the theory of existentially closed differential fields of characteristic 0 although complete and totally transcendental admits the maximal number of models in any uncountable cardinality. This was extended by Hrushovski-Sokolovic and independently by Pillay to countable models in the 90’s.

In my talk, I will discuss how to recover (local and global versions of) Shelah’s result from the study of a specific family of differential equations: the differential equations of the form y’’/y’ = f(y) where f(y) is an arbitrary rational function introduced by Poizat in the 90's.

This is joint work with J. Freitag, D. Marker and R. Nagloo.

Given an abelian group G, a corner is a a subset of pairs of the form ${(x,y), (x+g, y), (x, y+g)}$ with $g$ non trivial. Ajtai and Szemerédi proved that, asymptotically for finite abelian groups, every dense subset $S$ of $G\times G$ contains an corner. Shkredov gave a quantitative lower bound on the density of the subset $S$. In this talk, we will explain how model-theoretic conditions on the subset $S$, such as local stability, will imply the existence of corners and of cubes for (pseudo-)finite abelian groups. This is joint work with D. Palacin (Madrid/Freiburg) and J. Wolf (Cambridge).

Forcing with perfect trees is a major topic of research in set theory. One example is Namba forcing, which was originally developed as an example of a forcing that is $(\aleph_0,\aleph_1)$-distributive but not $(\aleph_0,\aleph_2)$-distributive. A recent paper of Dobrinen, Hathaway, and Prikry shows that a classical singular Namba forcing $P_\kappa$ is $(\omega,\nu)$-distributive for $\nu<\kappa$ if $\kappa$ is a singular strong limit cardinal of countable cofinality. The authors then ask whether this result generalizes, i.e. if $P_\kappa$ is (cf$(\kappa),\nu)$-distributive for $\nu<\kappa$ if $\kappa$ has uncountable cofinality. In joint work with Heike Mildenberger, we answer this question in the negative by showing that in this case $P_\kappa$ is not (cf$(\kappa),2)$-distributive.

Baldwin and Lachlan proved that an uncountably categorical structure is largely controlled by a strongly minimal set D and the Canonical Base Property (CBP) states that over a realization of a stationary type, its canonical base is always almost internal to the strongly minimal set D. Chatzidakis, Moosa and Pillay showed under the assumption of the CBP that every almost D-internal type transfers internality on intersections and more generally on quotients. Both properties do not hold in the uncountably categorical structure without the CBP, produced by Hrushovski, Palacín and Pillay.

In this talk, I will show that transfer of internality of quotients already implies the CBP and present the counter-example to the CBP as an additive cover of the complex numbers. In order to show that this structure does not transfer internality, we must consider imaginary elements (definable equivalence classes) and obtain a connection between elimination of finite imaginaries and the failure of the CBP.

Compressibility is a certain isolation notion suited to NIP theories. One definition of distality of a theory (a crucial notion with useful combinatorial consequences) is that every type is compressible. I will discuss some good properties of compressibility and their consequences, which include the existence of "compressibly atomic" models over arbitrary sets in countable NIP theories, and uniform honest definitions for an NIP formula.

Joint work with Itay Kaplan and Pierre Simon.

For 1≤k<ω we introduce k-distality and strong k-distality as properties of first-order theories, which both coincide with distality for k=1. With these properties we define the (strong) distality rank of a theory, and we give examples of theories with (strong) distality rank m for all ordinal numbers m between 1 and ω. We prove that the two ranks coincide for strongly minimal theories by providing a characterization in terms of the algebraic closure.

Der Satz von Lindström charakterisiert die Prädikatenlogik erster Stufe als stärkste Logik, in welcher der Kompaktheitssatz und der Satz von Löwenheim-Skolem abwärts gelten. Um den Satz sauber formulieren zu können, muss zunächst geklärt werden, was unter einer Logik zu verstehen ist. Hierzu wird in diesem Vortrag der von Chang und Keisler eingeführte Begriff einer abstrakten Logik diskutiert. Anschließend kann der Satz von Lindström präzise formuliert und bewiesen werden. Im Beweis, der sich ebenfalls am Vorgehen von Chang und Keisler orientiert, wird die Charakterisierung von elementarer Äquivalenz mithilfe von Back-and-Forth Systemen eine wichtige Rolle spielen.

The famous Cherlin-Zilber Algebraicity conjecture proposes that any infinite simple $\aleph_1$-categorical group is isomorphic to a simple algebraic group over an algebraically closed field.

In my talk, I will first explain the current state of the Cherlin-Zilber conjecture. I will then introduce a recent approach towards this conjecture, which is based on the notion of a supertight automorphism. I will discuss a result, proven jointly with P. Ugurlu, on “small” infinite simple groups of finite Morley rank with a supertight automorphism whose fixed-point subgroup is pseudofinite.