Course overview -- Sommersemester 2018

Wednesday,
2018-04-25
16:30-18:00
Raum 404, Ernst-Zermelo-Str. 1
Amin Shuaib: A Model Without Aleph-2 Suslin Trees
Wednesday,
2018-05-09
16:30-18:00
Raum 404, Ernst-Zermelo-Str. 1
Jörg Flum: Verallgemeinerte Quantorenelemination und was nun?

Wir charakterisieren Klassen von (endlichen und unendlichen) Strukturen, die eine verallgemeinerte Quantorenelimination erlauben. Dabei erlaubt die Klasse K verallgemeinerte Quantorenelimination, wenn jede in der Logik der ersten Stufe definierbare Eigenschaft in K bereits durch eine Anzahl q von Quantoren ausgedrückt werden kann, die nur von K abhängt. Falls q = 0 gewählt werden kann, erhalten wir somit den klassischen Begriff der Quantorenelimination.

Wednesday,
2018-05-16
16:30
Raum 404, Ernst-Zermelo-Str. 1
Adrian Mathias: The Paris-Harrington theorem

Let IRT, the infinite form of Ramsey's theorem, be the statement that for $c$ a finite non-empty set and $\pi:[\omega]^k \longrightarrow c$ there is an infinite $Y \subset \omega$ with $\pi$ constant on $[Y]^k$. Such an $Y$ is called homogeneous for $\pi$.

Let FRT, the finite form of Ramsey's theorem, be the statement that for $c$ a finite non-empty set and every positive $m$ and $k$ in $\omega$ there is an $n \in \omega$ such that whenever $X$ is a set of size $n$ and $\pi: [X]^k \longrightarrow c$ there is a set $Y \in [X]^m$ which is homogeneous for $\pi$.

A (finite) subset $Z$ of $\omega$ is called large if the size of $Z$ is at least $\min Z$.

The Paris-Harrington statement, PH, is FRT with the strengthened conclusion that the homogenous set $Y$ may be taken to be large.

FRT may be deduced from IRT, as can PH; FRT may be proved in Peano arithmetic PA, thus without using the axiom of infinity; the remarkable result (1972, published 1977 in the Handbook of Mathematical Logic) of Paris and Harrington is that PH is too strong to be provable in PA.

This talk will expound the work of many authors to show that PH fails in many non-standard models of PA.

Wednesday,
2018-05-30
16:05
Raum 404, Ernst-Zermelo-Str. 1
Martin Ruzicka: Iterated Ultrapowers in Set Theory

Given a measurable cardinal in an inner model of set theory we can construct its ultrapower, which is smaller than any of its factors. Following Kunen's work, we explain this process and its iteration.

Wednesday,
2018-06-20
16:30-18:00
Raum 404, Ernst-Zermelo-Str. 1
Philip Dittmann: Anwendungen von Typen in der Theorie reeller Körper

Ein angeordneter Körper heißt archimedisch, falls die Menge der ganzen Zahlen in ihm unbeschränkt ist. Es ist leicht zu sehen, dass diese Eigenschaft nicht erststufig axiomatisierbar ist, weshalb es sich anbietet, schwächere Bedingungen zu studieren. Dies führt auf sehr natürliche Weise zur Sprache von Typen im Sinne der Modelltheorie, und einigen interessanten (und recht subtilen) Fragen zu deren Realisierbarkeit.

Wednesday,
2018-07-04
16:30-18:00
Raum 404, Ernst-Zermelo-Str. 1
Elias Baro: Commutators of simply-connected o-minimal groups

Groups definable in an o-minimal expansion of a real closed field can be seen as a non-standard version of a Lie group (if the real closed field is the real field then such a group is actually a Lie group). For example, algebraic groups over a real closed field are o-minimal groups. In fact, the behaviour of o-minimal groups rests in between algebraic groups and Lie groups. The definability of the derived subgroup is a good example of this dichotomy.

The commutator subgroup of an algebraic group is again algebraic. However, the commutator of a Lie group may not be a Lie subgroup (there are even solvable counterexamples). The commutator of an o-minimal group may not be definable; in previous work with Jaligot and Otero, we proved that it is so if the group is solvable. Commutators play an important role in any category of groups; they played a crucial role in Conversano-Onshuus-Starchenko's characterisation of which solvable Lie groups are definable in an o-minimal expansion of the real field.

In this talk I will present an overview of these results and provide new insights concerning commutators of simply-connected o-minimal groups.

Wednesday,
2018-07-11
16:30-18:00
Raum 404, Ernst-Zermelo-Str. 1
Brendan Stuber-Rousselle: Baumforcings und Ideale
Wednesday,
2018-07-18
16:30-18:00
Raum 404, Ernst-Zermelo-Str. 1
Isabel Müller: TBA

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