### About the Event

The aims of the Young Set Theory workshops are to bring together young researchers in the domain of set theory and give them the opportunity to learn from each other and from experts in a friendly environment. A long-term objective of this series of workshops is to create and maintain a network of young set theorists and senior researchers, so as to establish working contacts and help disseminate knowledge in the field.

This year's Young Set Theory Workshop will take place at the Bernoulli Center in Lausanne, Switzerland, at the end of the Descpriptive set theory and Polish groups semester. While the organization of the workshop is independent of that of the semester, we hope the proximity will promote attendance of young researchers from extra-European countries.

### Excursion and conference dinner

An excursion to the UNESCO World Heritage region of Lavaux will take place on Wednesday, 27th. It will consist either of an (easy) hike along the ancient Via Francigena, a pilgrim route connecting Canterbury and Rome, or a hike through the wine terraces. A daily train ticket will allow you to reach the region and return to Lausanne at any time, so the duration of the hike is up to you.
The conference dinner will be held at 19:30 on Thursday, 28th at the restaurant Le Vaudois, in Place de la Riponne 1, Lausanne.

### Group picture

Great pics by Chris Le Sueur, here.

### Registered Participants

##### Booklet (PDF)

Vittorio Bard, Università degli Studi di Torino.

Carolyn Barker, University of Leeds.

Thomas Baumhauer, TU Wien.

Filippo Calderoni, Università di Torino.

Raphaël Carroy, KGRC, Vienna.

Fabiana Castiblanco, Universität Münster.

Tomasz Cieśla, University of Warsaw.

Fernando Damiani, University of Rome, La Sapienza.

Ben De Bondt, Ghent University.

Carl Dean, University of Illinois at Chicago.

Ben Erlebach, University of Toronto.

Monroe Eskew, KGRC, University of Vienna.

Elliot Glazer, Harvard.

Michał Tomasz Godziszewski, Institute of Mathematics, Polish Academy of Sciences.

Jan Grebik, Institute of Mathematics of the Czech Academy of Sciences.

Fiorella Guichardaz, University Albert-Ludwigs Freiburg im Breisgau.

Andreas Hallbäck, Université Paris Diderot.

Jialiang He, Department of Mathematics, Sichuan University.

John Howe, University of Leeds.

Manuel Inselmann, Kurt Gödel Research Center for Mathematical Logic.

Sittinon Jirattikansakul, Carnegie Mellon University.

Alexander Johnson, New College of Florida.

Deborah Kant, Humboldt University Berlin.

Dominik Kirst, Saarland University.

Lukas Daniel Klausner, TU Wien.

Marlene Koelbing, KGRC, Vienna.

Michał Korch, University of Warsaw.

Chris Le Sueur, University of East Anglia.

Maxwell Levine, KGRC, Vienna.

Marc Lischka, ETH Zürich.

Richard Matthews, University of Leeds.

Ivan Ongay Valverde, University of Wisconsin-Madison.

Supakun Panasawatwong, University of Leeds.

Luis Pereira, University of Lisbon.

Alejandro Poveda, Universitat de Barcelona.

Vibeke Quorning, Copenhagen.

Milette Riis, University of Leeds.

Dan Saattrup Nielsen, University of Bristol.

Ali Sadegh Daghighi, Amirkabir University of Technology.

Jonathan Schilhan, Kurt Gödel Research Center.

David Schrittesser, Kurt Gödel Research Center, University of Vienna.

Salome Schumacher, ETH Zürich.

Roy Shalev, Bar Ilan University.

Forte Shinko, Caltech.

Nattapon Sonpanow, Chulalongkorn University, Bangkok, Thailand.

Viera Šottová, Institute of Mathematics of the Czech Academy of Sciences.

Pieter Spaas, University of California, San Diego.

Sarka Stejskalova, KGRC.

Jarosław Swaczyna, Institute of Mathematics, Łódź University of Technology.

Andrea Vaccaro, University of Pisa & York University.

Charles Valentin, Université Paris Diderot - IMJ-PRG.

Alessandro Vignati, Institut de Mathematiques de Jussieu - Paris Rive Gauch.

Wolfgang Wohofsky, Universität Kiel.

### Practical Information

The closest airport is Geneva International Airport. Lausanne is also fairly well connected by train with some major cities in Europe, check the Swiss Federal Railways website for more information.

The Bernoulli Center is on the EPFL campus, which can be reached via a 10-15 minutes metro ride from the city center. Here is a video showing the path from the EPFL metro station to the Bernoulli Center.

### Program

Monday Tuesday Wednesday Thursday Friday
09:00-10:00 Registration Paolini Halbeisen Brickhill Pequignot
10:00-11:00 Rosendal Rinot Sinapova Sinapova Farah
11:00-11:30 Coffee Break Coffee Break Coffee Break Coffee Break Coffee Break
10:30-12:30 Rosendal Discussion Session Sinapova Discussion Session Farah
12:30-14:00 Lunch Lunch Lunch Lunch
14:00-15:00 Montoya Rinot Excursion Sinapova
15:00-16:00 Rinot Rosendal Farah
16:00-16:30 Coffee Break Coffee Break Coffee Break
16:30-17:30 Rinot Rosendal Farah

### Abstracts

#### Generalised closed unbounded and stationary sets (Slides)

##### Hazel Brickhill

The notions of closed unbounded and stationary set are central to set theory. I will introduce a new generalisation of these notions, and describe some of their basic theory. Surprisingly for a new concept is set theory, generalised closed unbounded and stationary sets are very simple to deﬁne and accessible. They are closely related to the phenomena of stationary reﬂection and indescribability and can be characterised in terms of derived topologies. These notions are being used to answer questions about provability logic, and promise a range of further applications.

#### Quotients and universality

##### Ilijas Farah

Quotients are always smaller and simpler than the original structures. In this tutorial we will see how naive and wrong the first sentence of this abstract is. Passing to a quotient frequently brings an increase in the `definable cardinality' and results in objects such as the Boolean algebra $$P(\mathbb N)/\text{Fin}$$ or the Calkin algebra. These objects were a notable source of inspiration (and sometimes frustration) for mathematicians, and set theorists in particular, for a long time. Among other properties of these objects, we will explore their set-theoretic malleability - the fact that almost any nontrivial statement about them is independent from ZFC.

#### Introduction to permutation models (Slides)

##### Lorenz Halbeisen

First, it will be shown how permutation models are constructed. As a prototype of a permutation model we consider Mostowski's model. Then, as an application, a modified version of Mostowski's model will be constructed in which every integral domain has a 5th-root function, but not every family of 5-element sets has a choice function.

#### Cardinal invariants and the generalized Baire spaces (Slides)

##### Diana Montoya

The study of the set-theoretical properties of the generalized Baire spaces, i.e. the spaces $$\kappa^\kappa$$, for $$\kappa$$ an uncountable cardinal has received special attention in the last years. Particularly, my research has been focused on the study of the generalization of some classical cardinal invariants to this context. Since 1995, with the paper Cardinal invariants above the continuum from James Cummings and Saharon Shelah [CumSh], investigations of this generalization, as well as the interactions between these cardinal notions in several models of ZFC, have been developing and by now, there is an extensive literature on the topic.

The first part of this talk aims to show the state of the art and some of the main results on this subject, and most importantly, it is devoted to present the differences and similarities with the classical case (the one where $$\kappa$$ is countable) and to motivate and present some interesting open questions.

The second part will be more specific. Its purpose is to exemplify through a particular case – the study of the independence number – in both, the countable and uncountable cases the first part of the talk. The concept of independence was first introduced by Fichtenholz and Kantorovic to study the space of linear functionals on the unit interval and since then, independent families have been an important object of study in the combinatorics of the real line. Particular interest has been given, for instance, to the study of their definability properties and to their possible sizes. In this part, I will present some recent results obtained in joint work with Vera Fischer.

[CumSh] J. Cummings, S. Shelah, Cardinal invariants above the continuum, Ann. Pure Appl. Logic 75 (1995), 251--268.

#### On Groups Admitting no Polish Group Topology and Other Things (Slides)

##### Gianluca Paolini

We make an overview of some recent results (joint with S.Shelah) on the algebraic restrictions put on an uncountable group by the admissibility of a Polish group topology. We then present some new results (also joint with S.Shelah) in the theory of reconstruction of model theoretic properties of a countable structure from the topological properties of its automorphism group. Finally, we give some applications to homogeneous model theory and Fraïssé limits.

#### Colorful well-foundedness (Slides)

##### Yann Pequignot

Mathematicians imagine a myriad of objects, most of them infinite, and inevitably followed by an infinite suite. One way to understand them consists of arranging them, ordering them. This act of organising objects amounts to considering an instance of the very general mathematical notion of a quasi-order. Well-founded quasi-orders play a crucial role in many areas of mathematics, and so do the stronger notions of well-quasi-orders and better-quasi-orders.

In the first part of the talk, I will give an introduction to better-quasi-orders. This should eventually lead us to their definition that hinges on a particularly interesting Borel graph: the shift graph. Its chromatic number is 2, but its Borel chromatic number is $$\aleph_0$$. While the $$\mathbb{G}_0$$-dichotomy states that there exists a minimal analytic graph with uncountable Borel chromatic number, it was open until recently whether the shift graph is in some sense minimal among graphs with infinite Borel chromatic number. Owing to complexity considerations, this turns out not to be the case and in fact there is no analogue of the $$\mathbb{G}_0$$-dichotomoy at $$\aleph_0$$.

#### In praise of C-sequences (Slides)

##### Assaf Rinot

Ulam and Solovay showed that any stationary set may be split into two. Is it also the case that any fat set may be split into two? Shelah and Ben-David proved that, assuming GCH, if the successor of a singular cardinal carries a special Aronszajn tree, then it also carries a distributive Aronszajn tree. What happens if we relax "special Aronszajn" to just "Aronszajn"? Shelah proved that the product of two $$\omega_2$$-cc posets need not be $$\omega_2$$-cc. How about the product of countably many $$\omega_2$$-Knaster posets?

It turns out that a common strategy for answering all of the above questions is the study of C-sequences. In this series of lectures, we shall provide a toolbox for constructing C-sequences, and unveil a spectrum of applications.

#### Geometry of Polish groups

##### Christian Rosendal

We will present the basics of the geometric theory of Polish groups, which generalises geometric group theory for finitely generated groups and the non-linear geometry of Banach spaces.

#### Strengthening the tree property

##### Dima Sinapova

A major theme in set theory is how much compactness we can consistently obtain in the universe. Compactness is the phenomenon when a given property holding for every smaller substructure of some object implies that property holds for the object itself. Compactness is usually a consequence of large cardinals, but often can hold at successors. A related topic is obtaining large cardinal properties are small cardinals.

A key instance of compactness is the tree property, and certain strengthenings. These principles capture the combinatorial essence of large cardinals in the following sense: for an inaccessible $$\kappa$$, $$\kappa$$ is weakly compact iff it has the tree property; $$\kappa$$ is strongly compact if it has the strong tree property, and $$\kappa$$ is supercompact iff ITP holds at $$\kappa$$. (Sometimes the latter is referred to as the super tree property). An old project in set theory is to consistently obtain these principles at every regular cardinal greater than $$\omega_1$$. That would require large cardinal hypotheses and many violations of SCH.

In this tutorial we will present what is known about these principles, and the state of the art of getting the tree property and its strengthenings at many cardinals simultaneously. We will also discuss their interaction with the continuum function. Finally, we will go over another strengthening, the so called ISP, and the notion of guessing models.

##### Contacts

You can contact the organizers at youngsettheory2018@gmail.com.
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