# Mini-Symposium Speakers

## Contributing Speakers

Charlotte Kestner
Imperial College, London
Salma Kuhlmann
University of Konstanz
Heike Mildenberger
Albert-Ludwigs University of Freiburg
Diana Carolina Montoya
Kurt Gödel Research Center, Vienna
Kaisa Kangas
University of Helsinki
Šárka Stejskalová
Kurt Gödel Research Center, Vienna

## Schedule

Monday 3rd September, 2018

Talks are 25 minutes long with 5 minutes for questions.

 14:00 Salma Kuhlmann 14:30 Diana Carolina Montoya 15:00 Kaisa Kangas 15:30 Šárka Stejskalová 16:00 Charlotte Kestner 16:30 Heike Mildenberger

## Talk Titles and Abstracts

Salma Kuhlmann -- University of Konstanz, Germany

Title: From Gödel's Incompleteness Theorem to Real Algebra

Abstract: Gödel's celebrated Theorem (which established among others the incompleteness of Peano arithmetic) has triggered great interest in the study of fragments of arithmetic in general, and in particular in finding explicit constructions of their models. In this talk, I will explain the interrelation between models of arithmetic on the one hand, and integer parts of real closed fields on the other. I will thereby highlight the interplay between the first order deductive closure of these theories versus the algebraic properties of these commutative domains.

Diana Carolina Montoya -- Kurt Gödel Research Center, Vienna, Austria

Title: Infinite combinatorics on the generalized Baire spaces

Abstract: The study of various classic set theory concepts on the generalized Baire spaces has been a subject of special interest on the last years. One specific example corresponds to the generalization of some classical cardinal invariants to this context. I will present a survey of some of the classical results and their counterparts in the extended case.

Kaisa Kangas -- University of Helsinki, Finland

Title: Categoricity and Universal Classes

Abstract: One example of a universal class is the class of models of a model complete first order theory. More generally, an abstract elementary class $(\mathcal{K}, \preccurlyeq)$ is universal if $\preccurlyeq$ is the submodel relation $\subseteq$ and for all models $\mathcal{A} \subseteq \mathcal{B} \in \mathcal{K}$, we have $\mathcal{B} \in \mathcal{K}$. In our work on universal classes, the aim was to find out whether a version of the the following statement is true: "Suppose $\mathcal{K}$ is a universal class with $LS(\mathcal{K})=\lambda$ and $\mathcal{K}$ is categorical in some cardinal $\kappa > \lambda$. Then $\mathcal{K}$ is categorical in every $\xi>\lambda$ and the models of $\mathcal{K}$ are either vector spaces or trivial (in the sense that the geometry is disintegrated)." The interesting part of the statement is that the models would be either vector spaces or trivial, which means that the reason behind the categoricity would be in the realm of classical mathematics. We proved that the statement holds after removing some "noise" from the class. In this talk, I will briefly discuss the background, present our result and discuss its implications.

Šárka Stejskalová-- Kurt Gödel Research Center, Vienna, Austria

Title: The tree property

Abstract: We will discuss the tree property and its effect on the continuum function. For a regular cardinal $\kappa$, we say that $\kappa$ has the tree property if there are no $\kappa$-Aronszajn trees. It is known that the tree property has the following non-trivial effect on the continuum function:

(*) If the tree property holds at $\kappa^{++}$, then $2^{\kappa} > \kappa^{+}$.

We will present original results regarding the tree property which suggest that (*) is the only restriction which the tree property puts on the continuum function in addition to the usual restrictions provable in ZFC.

Charlotte Kestner -- Imperial College, London, UK

Title: Some results in distal theories Non-forking formulas in distal NIP theories

Abstract: One of the aims of model theory is to classify mathematical structures according to properties of their definable sets. I will give an introduction to distal theories, a relatively new area of the classification that has received a lot of attention recently. I will then go on to discuss some results in distal theories. In particular, time allowing, I will discuss the definable $(p,q)$-theorem for distal theories, and the more recent result that $T$ is distal provided it has a model $\mathcal{M}$ such that the theory of the Shelah expansion of $\mathcal{M}$ is distal. This is joint work with G. Boxall.

Heike Mildenberger -- Albert-Ludwigs University of Freiburg, Germany

Title: Preserving a $P$-Point and Diagonalising an Ultrafilter

Abstract: We show that under CH there is a forcing that preserves a $P$-point and diagonalises another ultrafilter. We report on our work towards iterating such a procedure.

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