Course overview -- Wintersemester 2023/2024 (Archive)

The axiom of dependent choice $\mathsf{DC}$ and the axiom of countable choice $\mathsf{AC_\omega}$ are two weak forms of the axiom of choice that can be stated for a specific set: $\mathsf{DC}(X)$ assets that any total binary relation on $X$ has an infinite chain; $\mathsf{AC_\omega}(X)$ assets that any countable family of nonempty subsets of $X$ has a choice function. It is well-known that $\mathsf{DC}$ implies $\mathsf{AC_\omega}$. We show that it is consistent with $\mathsf{ZF}$ that there is a set $A\subseteq \mathbb{R}$ such that $\mathsf{DC}(A)$ holds but $\mathsf{AC_\omega}(A)$ fails.

Eine Gruppe mit trivialem Zentrum lässt sich in ihre Automorphismengruppe einbetten, die selbst auch triviales Zentrum hat. Durch Iteration bekommt man so den Automorphismenturm einer Gruppe. Wieland hat gezeigt, dass er für endliche Gruppen nach endlich vielen Schritten stationär wird. Simon Thomas hat dies auf unendliche Gruppen verallgemeinert. Der Vortrag präsentiert einige Ergebnisse aus diesem Umfeld.

We introduce the approximation property which was implicit in early work of Mitchell and later defined explicitly by Hamkins. In modern set theory, the approximation property has gotten new attention through the ineffable slender list property (ISP), introduced by Weiss in his PhD thesis. In this talk, we give a criterion for the approximation property which is very applicable to variants of Mitchell Forcing, allowing us to obtain several consistency results regarding ISP.

Präsentation der Masterarbeit.

We show that intuitionistic logic has uniform interpolation, a strong property possessed by some logics. For this we introduce the algebraic representation of intuitionistic logic (Heyting algebras) and their dual spaces (Esakia spaces). This duality is similar to that between Boolean algebras and Stone spaces for classical logic. We then show how to prove an open mapping theorem for Esakia spaces, and how uniform interpolation follows from this.

The tensor product $G\times H$ of two finite graphs $G$ and $H$ is defined by $V(G\times H) = V(G)\times V(H)$ and two vertices $(g_1,h_1)$ and $(g_2,h_2)$ being connected if $g_1 E_G g_2$ and $h_1 E_H h_2$. In 1966 Hedetniemi formulated the conjecture that $\chi(G\times H) = min(\chi(G),\chi(H))$. Only in 2019, more then 50 years later, Shitov discovered the existence of counterexamples. We will follow his proof and introduce some interesting notions along the way.