7th Young Set Theory Workshop

Titles and abstracts

Mini-courses:

Piotr Koszmider

Title: Applications of generic two-cardinal combinatorics

Abstract:

We will sketch the development of a part of two-cardinal combinatorics which is often applied in constructions of various mathematical structures like Boolean algebras, topological spaces or Banach spaces. We will propose an elementary, unifying, alternative to the usual language, re-placing commonly used structures by families of sets (originally deﬁned by Velleman but unused in this form) which we call 2-cardinals. We believe that this removeslots of unnecessary information present in classical approaches to stepping-up and gives a quick practical access to many powerful combinatorial techniques relevant in applications. The main stages of the minicourse are:
• Basic deﬁnitions and facts: 2-cardinals, coherence;
• Combinatorial constructions: Kurepa trees, explicit deﬁnitions of Hausdorﬀ
gaps, ρ-like functions, property ∆, strong nonreﬂectiion;
• Forcing with side conditions in 2-cardinals, generic stepping up;
• More on constructions of mathematical structures using the above ideas:
Boolean algebras, compact spaces, Banach spaces; In this theory one mixes stepping-up tools like ρ-functions with forcing which often cannot be factored as σ-closed∗c.c.c and uses tailor-made side conditions in order to construct structures which exhibit big gaps between cardinal invariants or lack of reﬂection. Most known examples of the results are the consistency of the
• There is a dispersed compact space of countable width and Cantor-Bendixson height ω2 + 1,
• There is a Banach space of density ω2 without uncountable biorthogonal systems,
• There is a countably irredundant Boolean algebra of cardinality ω2,

ω1 versions of the above statements either can be proved in ZFC or using prin-ciples like CH or ♦. The ω2-versions require these complex strong negations of Chang’s Conjecture like 2-cardinals + forcing. And whether ω3-version of any of these statements is consistent are well-known widely open problems. We will give more examples of such results, sketch how they can be obtained and what challenges we face if we want to go further in the direction of ω3.

A survey paper: P. Koszmider, On constructions with 2-cardinals, should appear on arXiv.org before the beginning of the conference.

Lajos Soukup

Title: On properties of families of sets.

Abstract:

We define  and study some classical and contemporary properties of families of sets, such as property B, transversal property, almost disjoint, essentially disjoint, chromatic number, conflict free chromatic number, etc.We introduce and apply some basic methods to investigate the relationship between these properties:  chains of elementary submodels, Davies trees, singular cardinal compactness, Shelah's Revised GCH, and special forcing constructions. We also give some applications of these methodsin infinitary  combinatorics.

Simon Thomas

Title: A descriptive view of combinatorial group theory
Abstract:
It is well-known that descriptive set theory provides a framework for measuring the relative complexity of many naturally occurring classification problems. But it is less well-known that descriptive set theory also provides a framework for explaining the inevitable non-uniformity of many classical constructions in mathematics. In this mini-course, I will illustrate this point by considering some constructions from combinatorial group theory. For example, the Higman-Neumann-Neumann Embedding Theorem states that any countable group G can be embedded into a 2-generator group K. In the standard proof of this classical theorem, the construction of the group K involves an enumeration of a set of generators of the group G; and it is clear that the isomorphism type of K usually depends upon both the generating set and the particular enumeration that is used. One of the main results of this mini-course will be that there does not exist a more uniform construction with the property that the isomorphism type of K only depends upon the isomorphism type of G.
In the first lecture, I will explain how to formulate various uniformity problems using the notions of descriptive set theory and I will discuss the statements of the main results; and in the remaining lectures, I will present some proofs. In contrast to much of the recent work in descriptive set theory, the proofs in this mini-course will mainly involve purely set-theoretical notions such as forcing, large cardinals and Borel determinacy.

Jindrich Zapletal

Title: Borel reducibility and higher set theory

Abstract: I will introduce several methods for proving nonreducibility of analytic equivalence relations
using forcing and combinatorics of uncountable cardinals. This includes (a) the pinned cardinal
and its applications, such as a proof of nonreducibility using the failure of the singular cardinal hypothesis;
(b) generalizations of turbulence, leading to ergodicity results even when no group action is present;
(c) separation properties, resulting from the generalization of Solovay's coding to quotient spaces.

The tutorial is based on a self-titled upcoming monograph.

Plenary talks:

David Chodounsky

Title: Mathias Forcing with Filters

Abstract:

Mathias forcing M(F) is a natural forcing adding a pseudointersection for a given filter F on omega.For concrete applications, we are often interested in omega^omega bounding-like properties (almost omega^omega bounding, adding dominating real, adding eventually different real) of this forcing.These properties depend on the filter F and can be characterized by its topological combinatorial covering properties (F is considered as a subspace of 2^omega).Namely M(F) is almost bounding iff F is Hurewicz and M(F) does not add dominating reals iff F is Menger. I will present ideas behind these characterizations and discuss the connection with results of Guzman, Hrusak, Martinez, Minami and others.

The talk is based on join work with L. Zdomskyy and D. Repovs.

Aleksandra Kwiatkowska

Title: Continua from a projective Fraisse limit

Abstract:

We first review the projective Fraisse theory developed by Irwin and Solecki. Then we present several examples of projective Fraisse limits, including the one that gives rise to the pseudo-arc (this example is due to Irwin and Solecki), and the more recent one, that gives rise to the Lelek fan. Finally, we discuss several properties of the Lelek fan and of its homeomorphism group, which are proved using the projective Fraisse limit construction. This is joint work with Dana Bartosova.

Philipp Lücke

Title: Infinite fields with large free automorphism groups

Abstract:

Shelah proved that a free group of uncountable rank is not isomorphic to the automorphism group of a countable first-order structure.  In contrast, Just,  Shelah and Thomas showed that it is consistent with the axioms of set theory that there is a field of cardinality aleph1 whose automorphism group is a free group of rank 2^aleph_1. Motivated by this result,  they ask whether there always is a field of cardinality aleph_1 whose automorphism group is a free group of rank greater than aleph_1.

In my talk, I will develop general techniques that enable us to realize certain groups as the automorphism group of a field of a given cardinality. These techniques will allow us to show that the free group of rank 2^kappa is isomorphic to the automorphism group of a field of cardinality \kappa whenever kappa is a cardinal satisfying  kappa=kappa^aleph_0.  Moreover, we can use them to show that the existence of a cardinal kappa of uncountable cofinality with the property that there is no field of cardinality kappa whose automorphism group is a free group of rank greater than kappa implies the existence of large cardinals in certain inner models.

This is joint work with Saharon Shelah.

Nam Trang

Title: On a class of guessing models.

Abstract:

For an infinite cardinal kappa, we define kappa-guessing models. The notion of guessing models has been isolated by Viale and Weiss. For kappa> aleph_0, kappa-guessing models are combinatorial essence of supercompactness compatible with non-inaccessible cardinals. We prove some combinatorial consequences of guessing models as well as discuss their relationships with forcing axioms. We also discuss the consistency strength of the existence of kappa-guessing models. In particular, a theorem along this line is: the existence of aleph_2 guessing models (along with some mild cardinal arithmetic assumptions) yield models of AD_R + Theta is regular".

Konstantinos Tyros

Title: Density Theorems for words.

Abstract:

We shall present the Density Hales--Jewett theorem and the Density Carlson--Simpson theorem. The Hales--Jewett theorem is one of the most representing theorems in Ramsey theory (see [HJ]). Its density version was first  proved by H. Furstenberg and Y. Katznelson in 1991 using Ergodic Theory (see [FK2]). A combinatorial proof was discovered in 2012 and is contained in the Polymath paper (see [Pol], see also [DKT3]). However, our presentation will be based on the proof contained in [DKT3]  We will also present a density version of a theorem due to T. J. Carlson and S. G. Simpson (see [CS] and [DKT]  for its density version) concerning the space of left variable words, which consists, in particular, an extension of the Density Hales--Jewett theorem.

[CS] T. J. Carlson and S. G. Simpson, A dual form of Ramsey's theorem, Adv. Math., 53 (1984), 265-290.

[DKT3] P. Dodos, V. Kanellopoulos and K. Tyros, A simple proof of the density Hales--Jewett theorem, International Mathematical Research Notices, to appear.

[DKT] P. Dodos, V. Kanellopoulos and K. Tyros, A density version of the Carlson--Simpson theorem, Journal of the European Mathematical Society, to appear.

[FK2] H. Furstenberg and Y. Katznelson, A density version of the Hales--Jewett theorem, Journal d'Anal. Math., 57 (1991), 64-119.

[HJ] A. H. Hales and R. I. Jewett, Regularity and positional games, Trans. Amer. Math. Soc., 106 (1963), 222-229.

[Pol] D. H. J. Polymath, A new proof of the density Hales--Jewett theorem, Ann. Math., 175 (2012), 1283-1327.