Course overview -- Sommersemester 2022 (Archive)

A group configuration is a geometric incidence configuration consisting of 6 points which in stable theories is related to the existence of a type definable group. In the first part of the talk, we will introduce the above concepts and point out why a group gives rise to a group configuration. Moreover, by a result from Hrushovski, any group configuration in a stable theory yields the existence of a type definable group. We will discuss the basic ideas of this proof. The second part of the talk will present an application. Hrushovski and Pillay showed that any definable group in a real closed field F is locally isomorphic to the F-rational points of an algebraic group defined over F. This is achieved by considering a group configuration of the group in the algebraic closures of F.

In 1967 Lovász showed that up to isomorphism every finite relational structure A is determined by the homomorphism counts hom(B,A), i.e, by the number of homomorphisms from B to A, where B ranges over all structures (of the same vocabulary as A). Moreover, it suffices that B ranges over the structures with at most as many elements as A.

In the talk, we deal with classes C of graphs characterizable by finitely many homomorphism counts, i.e., classes for which there are finitely many graphs F_1,...,F_k such that for every graph G already hom(F_1,G),...,hom(F_k,G) determines whether G is in C. Among others, we show which prefix classes of first-order logic have the property that each class of graphs definable by a sentence of this prefix class is characterizable by finitely many homomorphism counts.

Influenced by results in real algebraic geometry, Pillay pointed out in 1988 that the space of types of an o-minimal expansion of a real closed field admits a spectral topology. With this topology, this space is quasi-compact and T_0, yet not Hausdorff. Nonetheless, the subspace of all closed points turns out to be quasi-compact and Hausdorff.

In this talk, I will relate the space of closed points with other topological spaces, such as the space of mu-types considered first by Peterzil and Starchenko. In addition, I will explain how to characterize coheir types of an o-minimal expansion of a real closed field within invariant types, using the spectral topology.

The space of blocks consists of all non-empty finite sets of natural numbers. Given any filter on the natural numbers, the sets of blocks of filter elements generate a filter on the blockspace and, vice versa, each filter on the blockspace yields a filter on the natural numbers by taking unions of filter elements. In this talk, we will make some observations about this relation and the question of whether the maximality of filters is or can possibly be preserved by it. As an application we will show how filters and coideals on the blockspace can be used as parameter sets of tree-forcings with the aim of diagonalizing a given filter on the natural numbers.

The talk will be a survey of my master's thesis. The topic of the thesis are two cardinal properties which are similar to the tree property and arise naturally from the consideration of a generalised tree. We will first introduce both properties and then outline their similarities and differences, both in the way they can be proven consistent at small cardinals and what they imply. We will show that, despite being equivalent for inaccessible cardinals, one property is strictly stronger than the other at small cardinals.

We study versions of Jensen's square principle $\square_\kappa$, a combinatorial principle that holds for all cardinals $\kappa$ in Gödel's constructible universe $L$. Cummings, Foreman, and Magidor proved that the square principle is non-compact at $\aleph_\omega$, meaning that it is consistent that $\square_{\aleph_n}$ holds for all $n<\omega$ while $\square_{\aleph_\omega}$ fails. We investigate the natural question of whether this phenomenon generalizes for singulars of uncountable cofinality. Surprisingly, we show that under some mild hypotheses, the weak square principle $\square_\kappa^*$ is in fact compact at singulars of uncountable cofinality, and that an even stronger version of these hypotheses is not enough for compactness of weak square at $\aleph_\omega$.

In 1967, Keisler introduced a partial order on first-order theories, that depends, roughly speaking, on whether certain ultrapower models of the theories in question are $\lambda^+$-saturated. In a pre-print of 2018, Douglas Urich introduced among other things a Compactness Theorem for Boolean-valued structures, generalized the machinery developed previously by Malliaris and Shelah and provided more flexible tools for proving or disproving propositions of the form ``$T_0 \trianglelefteq T_1$‘‘, making use of Boolean ultrapowers for more general complete Boolean algebras. Our thesis consists in a detailed walk-through of Ulrich‘s results and we will briefly present some of them in this talk.