8th Young Set Theory Workshop, Jerusalem, October 25-30, 2015

The 8th Young Set Theory Workshop will take place October 25-30 2015, at the Israel Institute of Advanced Studies in Jerusalem. (see posterbooklet, and Group picture)

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.

Mini Courses

Péter Komjáth ELTE

In this mini-course, we overview the various properties of the chromatic number of infinite graphs.
We will be specially interested in the coloring number, a variant, which is much easier to handle.
Menachem Magidor HUJI

Compactness and Incompactness at small Cardinals

In this mini-course, we shall study reflection and compactness cardinals for a variety of properties of mathematical structures. The properties we shall consider are typically second order. Examples: A family of sets has a transversal (one to one choice function), An Abelian group is a free Abelian group, A graph has a countable chromatic number, The structure has a definable well ordering which is a regular cardinal etc.Reflection and compactness are dual properties: $\kappa$ is a reflection cardinal for a certain property iff it is a compactness cardinal for the negation of the property.

Usually compactness properties are associated with the cardinal being a large cardinal. For example, the first supercompact cardinal is the minimal cardinal which is a strongly compact for every second order property.

In this series of talks we shall discuss the possibility of small cardinals, typically less than the first inaccessible, being compact for a given property. We shall study cases in which relatively small cardinals, like $\aleph_n$ or $\aleph_{\omega+1}$ are provably incompact for certain natural properties. On the other hand, we shall see cases in which it is consistent that relatively small cardinal is compact for interesting property.

This mini-course attempts to expose the listeners to a variety of set theoretic tools from straightforward combinatorial set theory, applications of PCF theory, forcing techniques for large cardinals, etc’. Because of time constrains, we shall not always be able to provide full proofs but hopefully the central ideas will come through. We expect familiarity with basic concepts of combinatorial set theory and basic forcing, though we try to give the definition of every concept we shall use.

W. Hugh Woodin Harvard

In these 4 lectures I will survey some of the basic issues which arise in the attempt to extend the Inner Model Program to the level of a supercompact cardinal.

1. The HOD Dichotomy, weak extender models, and universality
The starting point is the HOD Dichotomy Theorem which is an abstract version of Jensen’s Covering Theorem with L replaced by HOD.
This theorem naturally leads to the prediction that inner models with a supercompact cardinal must be close to V if that supercompact cardinal in the inner model is connected in a very weak sense to supercompactness in V.
The prediction turns out to be correct and so the fine-structural inner model for one supercompact cardinal emerges as a candidate for an ultimate version of L, if such an inner model can exist.

2. The coding obstruction
Martin-Steel extender models generalize Kunen’s inner model L[U] up to the level of superstrong cardinals. This lecture will focus on a result that shows that the natural attempt to generalize Martin-Steel extender models past the level of superstrong cardinals fails.

3. The comparison obstruction
The Mitchell-Steel extender models define the basic modern template for fine-structural extender models. These models reach the level of Woodin cardinals and assuming iteration hypotheses, the hierarchy reaches the level of superstrong cardinals. In a generalized form and assuming the same iteration hypotheses, the hierarchy can now be extended to reach the finite levels of supercompact.

The focus of this lecture is the result that the basic methodology of comparison by least disagreement must fail in the general case.

4. The amenability obstruction
The Mitchell-Steel hierarchy of models and the generalization of these models to the finite levels of supercompact, all satisfy a general amenability condition at their active stages. In this lecture a general argument is given that this must fail beyond the finite levels of supercompact.

Martin Zeman UCI

In this mini-course, I will present ideas, methods, results and current trends in inner model theory for “short” extenders; such extenders encompass all large cardinal properties below the weakest instance of supercompactness. The series of talks is intended for audience with very little or no prior exposure to inner model theory and large cardinals, so the presentation will be non-technical with emphasis on the concepts and a bit of history, and I will introduce and explain all relevant notions.

Plenary Talks by Young Researchers

Laura Fontanella HUJI

This is a joint work with Yair Hayut.

One of the most fruitful research areas in set theory is the study of the so-called reflection principles. These are statements establishing, roughly, that for a given structure (a stationary set, a tree etc.) and a given property, one can find a substructure of smaller cardinality satisfying the same property. Reflection principles are typical properties of large cardinals but can consistently hold even at small cardinals. Square principles are on the contrary anti-reflection principles as they imply the failure of several reflection principles and are false in the presence of certain large cardinals. For instance the square principle of Todorcevic [4] implies that any stationary subset of $\kappa$ may be partitioned into $\kappa$ many pairwise disjoint stationary sets such that any two such sets do not reflect simultaneously [3]. We present a technique for building models where a reflection principle and a square principle hold simultaneously at the successor of a singular cardinal. We discuss two particular principles: the so-called Delta reflection which is due to Magidor and Shelah [2], and a version of the square due to Todorcevic [4]. More precisely, we show that, starting from a suitable large cardinal assumption, one can force a model where both the Delta reflection and Todorcevic’s square hold at $\aleph_{\omega^2+1}.$

[1] L. Fontanella and Y. Hayut, Square and Delta reflection, submitted.
[2] M. Magidor and S. Shelah, When does almost free implies free, Journal of the American Mathematical Society, vol. 7 (1994), no. 4, pp. 769–830.
[3] A. Rinot, Chain conditions of products and weakly compact cardinals, Bulletin of Symbolic Logic, vol. 20 (2014) no. 03, pp. 293– 314.
[4] S. Todorcevic, Partitioning pairs of countable ordinals,
Acta Mathematica, vol. 159 (1987), no. 3-4, pp. 261–294.

Andrew Marks UCLA

Baire measurable paradoxical decompositions via matchings
The Banach-Tarski paradox states that the unit ball in $\mathbb{R}^3$ is equidecomposable with two unit balls in $\mathbb{R}^3$ by rigid
motions. In 1930, Marczewski asked whether there is such an
equidecomposition where each piece has the Baire property. In the 90s, Dougherty and Foreman gave a positive answer to this question.

We generalize Dougherty and Foreman’s result to completely
characterize which Borel group actions have Baire measurable
paradoxical decompositions. We show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. We also obtain a Baire category solution to the dynamical von Neumann-Day problem, in the spirit of Whyte and Gaboriau-Lyons: if $a$ is a nonamenable action of a group on a Polish space $X$ by Borel automorphisms, then there is a free Baire measurable action of $\mathbb{F}_2$ on $X$ which is Lipschitz with respect to $a$. The main tool we use to prove these theorems is a version of Hall’s matching theorem for Borel graphs.

This is joint work with Spencer Unger at UCLA.

Diego A. Mejia TU Wien

I present some examples of models, constructed with finite support iteration techniques, where many cardinals in Cichon’s diagram assume different values. For example, I present models where three cardinals of the right side of the diagram are separated and a model where all cardinals of the left side are separated, the latter constructed in a joint work with M. Goldstern and S. Shelah.
Diana Ojeda-Aristizabal U of T

For $X,Y$ topological spaces and natural numbers $l,m,n$ we write $X\longrightarrow(Y)^n_{l,m}$ if for every $l$-coloring of $[X]^n$, the unordered $n$-tuples of elements of $X$, there exists $Z\subseteq X$ homeomorphic to $Y$ such that $c\upharpoonright[Z]^n$ takes at most $m$ colors. An example of such an arrow relation is $\mathbb{Q}\longrightarrow(\omega^k+1)^2_{l,2k}$ where $l,k$ are natural numbers and both $\mathbb{Q}$ and $\omega^k+1$ are endowed with the order topology. This relation follows after identifying $\mathbb{Q}$ with $\mathrm{FIN}$, the collection of finite non empty subsets of $\mathbb{N}$ with the topology of pointwise convergence, and applying Hindman’s Theorem and its higher dimensional versions. Baumgartner found that in fact for every natural number $k$ there exists a countable ordinal $\gamma_k$ such that $\gamma_k\longrightarrow(\omega^k+1)^2_{l,2k}$. Recently C. Pina obtained an optimal value for $\gamma_2$, namely she proved that $\omega^{\omega^{\omega}}$ is the least ordinal $\gamma$ that satisfies $\gamma\longrightarrow(\omega^2+1)^2_{l,4}$ for every $l$. When working in a countable ordinal instead of $\mathbb{Q}$, a finer analysis is needed and Hindman’s Theorem is no longer useful. The key of Pina’s result is the use of certain families of finite sets to represent countable ordinals.Using families of finite sets to represent countable ordinals, we begin our study with ordinals of the form $\omega\cdot k+1$ with $k\geq 2$, and find that for every countable ordinal $\gamma$ there exists a 3-coloring of $[\gamma]^2$ that can’t be reduced in a copy of $\omega\cdot 2+1$. We set out to find for each $m\geq 3$ the least ordinal $\gamma$ such that for every $l$ we have that $\gamma\longrightarrow(\omega\cdot 2+1)^2_{l,m}$. It turns out that if $\gamma\longrightarrow(\omega\cdot 2+1)^2_{l,3}$ for every $l$, then already $\gamma\geq \omega^{\omega^{\omega}}$. We carry out a similar analysis for ordinals of the form $\omega\cdot k+1$ with $k>2$.

This is joint work with William Weiss from the University of Toronto.

Yizheng Zhu Munster

The higher sharp
We establish the descriptive set theoretic representation of the mouse $M_n^\#$, which is called $0^{(n+1)\#}$. At even levels, $0^{(2n)\#}$ is the higher level analog of Kleene’s O; at odd levels, $0^{(2n+1)\#}$ is the unique iterable remarkable level-$(2n+1)$ blueprint.

Plenary Talks By Local Researchers

Ari Brodsky BIU

Reduced powers of Souslin trees
We study the relationship between a $\kappa$-Souslin tree $T$ and its reduced powers $T^\theta/\mathcal U$.
Previous works addressed this problem from the viewpoint of a single power $\theta$, whereas here, tools are developed for controlling different powers simultaneously.
As a sample corollary, we obtain the consistency of an $\aleph_6$-Souslin tree $T$ and a sequence of uniform ultrafilters $\langle \mathcal U_n\mid n<6\rangle$ such that $T^{\aleph_n}/\mathcal U_n$ is $\aleph_6$-Aronszajn iff $n<6$ is not a prime number.
This is joint work with Assaf Rinot at Bar-Ilan University.
Menachem Kojman BGU

The density function on infinite cardinals behaves well asymptotically. A few of its properties and uses will be sketched.
Saharon Shelah HUJI

We deal with the theory of iterated forcing for inaccessible $\lambda$, not adding $\lambda$-Cohens. Our starting point is the paper [Sh:1004].
Boaz Tsaban BIU

This lecture will be an elementary exposition to the paper [1].
For simplicitly, we consider only Abelian groups. An abelian topological group is an abelian group equipped with a topology, such that addition and negation are continuous. Since toplogical groups are homogenous, their topology is to a large extent determined by the topology around a single point, say, the neutral element of the group. Consequently, the most important cardinal invariant of a topological group, its “character”, is the minimal cardinality of a local base at the neutral element. In particular, a topological group is metrizable if and only if it has countable character.

Let X be a topological space. The space X generates the Abelian group A(X) of all linear combinations of elements of X with integer coefficients, and with no additional relations. The topology on A(X) is determined by the request that every continuous map on X (into an abelian topological group) extends to a homomorphism on A(X). It turns out that computing the character of this simply defined group requires most sophisticated set theoretic methods. We will show why this is the case (by introducing the so-called Pontryagin van-Kampen duality and modifying it to our needs), demonstrate the usefulness of set theory in this context, and propose purely set theoretic open problems that must be addressed in order to make progress in some natural directions concerning this mostly open problem.

[1] Chis, Cristina; Ferrer, M. Vincenta; Hernandez, Salvador; Tsaban, Boaz, The character of topological groups, via bounded systems, Pontryagin-van Kampen duality and pcf theory, J. Algebra, vol. 420 (2014), pp. 86–119.

Program Committee:

List of participants:

1 Uri Abraham
2 Giorgio Audrito
3 Gianluca Basso
4 Ur Ben-Ari-Tishler
5 Tom Benhamou
6 Mariam Beriashvili
7 Jeffrey Bergfalk
8 Eilon Bilinsky
9 Douglas W. Blue
10 Hazel Brickhill
11 Ari M. Brodsky
12 Marta Burczyk
13 Filippo Calderoni
14 Miguel Antonio Cardona Montoya
15 Fabiana Castiblanco
16 David Chodounsky
17 Shani Cohen
18 Tal Cohen
19 Klaudiusz Czudek
20 Vincenzo Dimonte
21 Stamatis Dimopoulos
22 Ohad Drucker
23 Piotr Drygier
24 Jin Du
25 Laura Fontanella
26 Kevin Fournier
27 Shimon Garti
28 Michel F. Gaspar
29 Moti Gitik
30 Fiorella Guichardaz
31 Elliot Glazer
32 Ishay Golinsky
33 Jan Grebik
34 Miha E. Habic
35 Nir Hakeyni
36 Yair Hayut
37 Jacob Hilton
38 Stefan Hoffelner
39 Haim Horowitz
40 Manuel Inselmann
41 Joanna Jureczko
42 Itay Kaplan
43 Asaf Karagila
44 Burak Kaya
45 Yechiel Kimchi
46 Martin Koberl
47 Marlene Koelbing
48 Menachem Kojman
49 Peter Komjath
50 Micha l Korch
51 Regula Krapf
52 Adam Kwela
53 Chris Lambie-Hanson
54 Chris Le Sueur
55 Maxwell Levine
56 Azriel Levy
57 Dani Livne
58 Menachem Magidor
59 Andrew Marks
60 Paul McKenney
61 Nadav Meir
62 Diego A. Mejıa
63 Omer Mermelstein
64 Marcin Michalski
65 Kaethe Minden
66 Diana Carolina Montoya
67 Nikodem Mrozek
68 Dan S. Nielsen
69 Diana Ojeda-Aristizabal
70 Yann Pequignot
71 Luıs Pereira
72 Assaf Rinot
73 Gil Sagi
74 Damian Sobota
75 Jan Stary
76 Sarka Stejskalova
77 Silvia Steila
78 Jaroslaw Swaczyna
79 Dorottya Sziraki
80 Anda-Ramona Tanasie
81 Fabio E. Tonti
82 Jacek Tryba
83 Boaz Tsaban
84 Milette Tseelon-Riis
85 Spencer Unger
86 Andrea Vaccaro
87 Jonathan L. Verner
88 Alessandro Vignati
89 Thilo V. Weinert
90 Gabriel Zanetti Nunes Fernandes
91 Wolfgang Wohofsky
92 W. Hugh Woodin
93 Martin Zeman
94 Yizheng Zhu
95 Tomasz Zuchowski

Links to previous meetings:

Attachments

5 responses to “8th Young Set Theory Workshop, Jerusalem, October 25-30, 2015

  1. Pingback: Young Set Theory 2015 | Asaf Karagila

  2. Video recordings of all talks is available here. Enjoy!

  3. And finally the people can see a video lecture by Prof. Shelah. Thanks to the organizers for providing it.

  4. Pingback: 9th Young Set Theory Workshop, Copenhagen, June 13-17, 2016 | Set Theory Talks

  5. Pingback: 10th Young Set Theory Workshop, Edinburgh, July 10-14, 2017 | Set Theory Talks

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