The 2012 Young Set Theory Workshop will take place between April 30th and May 4th 2012 in Luminy, France.

The aim of this conference is to bring together PhD students, postdocs and young researchers in Set Theory in order to learn from leading researchers in the field, hear about the latest research and to discuss research issues in a co-operative environment.

**Tutorial leaders:**

- Ilijas Farah
- Alain Louveau
- Itay Neeman
- Stevo Todorcevic

**Post-doc talks:**

- Christina Brech
- Sean Cox
- Vera Fischer
- Daisuke Ikegami
- Carlos Martinez-Ranero
- David Milovich
- Farmer Schlutzenberg

**Abstracts**

- Ilijas Farah, Some applications of set theory to operator algebras

The use of set-theoretic methods in theory of operator algebras has led to remarkable advances over the last decade. I will present some of the results and directions for future research. - Alain Louveau, Introduction to Effective Descriptive Set Theory .

In this series of lectures, I will provide a short overview of the basics of effective DST, concentrating on the tools this theory provides for proving various dichotomy results. I will also try to compare the effective techniques with the new techniques developped recently by Ben Miller to get dichotomies. There are no precise pre-requisites, although some knowledge of the theory of computable functions, and/or of classical DST, would help. And some reading of parts of Chapters 3 and 4 of Moschovakis’s book Descriptive Set Theory would be even better. - Itay Neeman, Forcing iterations with finite side conditions.

We describe a proof of the proper forcing axioms that uses a finite support iteration with models as side conditions to ensure properness. We will describe the poset of side conditions in detail, and include some additional applications. If time permit we will discuss generalizations of the construction that yield a higher analog for the proper forcing axiom that involves meeting \(\aleph_2\) dense sets in a specific class of posets that preserve both \(\aleph_1\) and \(\aleph_2\). - Stevo Todorcevic,Walks on ordinals and their characteristics

Since its introduction in 1987, the method of minimal walks on ordinals has been useed in multiple settings. For example: in combinatorial set theory, it reduces Chang’s conjecture to a purely combinatorial statement of Ramsey type; in general topology, it was used by Moore in order to construct, in ZFC, a regular, hereditarily Lindelöf, non separable topological space (2006); in Banach space theory, it allows the construction of a reflexive Banach space with a Schauder basis of length ω_{1}where every operator is of the form λI + S, where S has separable range. I will present the underlying ideas behind those constructions. - Christina Brech , Forcing nonuniversal Banach spaces

A classical result due to Banach and Mazur states that the Banach space \(C[0,1]\) is universal for separable Banach spaces, that is, every separable Banach space can be embedded into \(C[0,1]\). For \(\kappa = \omega_1\) or \(\kappa = \mathfrak{c}\), we address the question on the existence of a universal object for some classes of Banach spaces of density \(\kappa\). The existence of a universal compact space of weight \(\kappa\) for some classes (which holds under CH) guarantees the existence of corresponding universal Banach spaces. For the nonexistence of these objects, the situation changes and we have to improve consistency results concerning compact spaces. We will present and sketch the proof of some results obtained by forcing. - Sean Cox, Some topics related to bounding by canonical functions

Let \(\kappa\) be an uncountable regular cardinal. Define a well-founded partial order \(\le^*\) on \({}^\kappa \kappa\) by: \(f \le^* g\) iff \(\{ \alpha < \kappa \ | \ f(\alpha) \le g(\alpha) \}\) contains a closed unbounded set. There are many interesting (and usually independent) questions about this partial order; for example, are the so-called*canonical functions*cofinal in this partial order? This topic is closely related to saturated ideals on \(\kappa\) and Chang’s Conjecture, and although these topics are very well understood for \(\kappa = \omega_1\), much less is known about the case \(\kappa = \omega_2\). I will discuss some joint work with Zeman which might shed some light on this and related “bounding properties” at \(\omega_2\). I will also discuss some related work with Viale, which deals with conflicts between bounding properties at \(\omega_2\) and strong forcing axioms. - Vera Fischer, Mad families, splitting families and large continuum

Let \(\kappa<\lambda\) be regular uncountable cardinals. Using finite support iteration of ccc posets we obtain the consistency of \(\mathfrak{b}=\mathfrak{a}=\kappa<\mathfrak{s}=\lambda\). If \(\mu\) is a measurable cardinal and \(\mu<\kappa<\lambda\), then using similar techniques we obtain the consistency of \(\mathfrak{b}=\kappa<\mathfrak{a}=\mathfrak{s}=\lambda\). - Daisuke Ikegami , Gale-Stewart games and Blackwell games

Starting from the determinacy of Chess by Zermelo, the theory of determinacy of games with perfect information has been developed exclusively. Among those games, Gale-Stewart games are general infinite games which have applications to set theory, model theory and some theoretical computer science. Apart from that, the research in games with imperfect information has started in game theory since von Neumann’s minimax theorem and Blackwell games are one of the few infinite games with imperfect information which are tractable to discuss their determinacy. In this talk, we discuss the connection between the determinacy of Gale-Stewart games and that of Blackwell games. Our main result is that assuming the Axiom of Dependent Choice, the axiom of determinacy of Blackwell games with reals is equivalent to that of Gale-Stewart games with reals. This is joint work with W. Hugh Woodin. - Carlos Martinez-Ranero, Contributions towards a fine structure theory of Aronszajn orderings.

We shall present some results concerning the structure of the class of Aronszajn lines as well as the class of Aronszajn trees ordered by the embeddability relation. We shall show that, assuming PFA, the class of A-lines is well quasi-ordered by embeddability. Moreover, we prove that the structure of the class of A-lines is analogous to that of countable linear orderings. We shall also consider the class of A-trees and in this case we prove that, assuming PFA, the class of A-trees is universal for linear orderings of cardinality at most \(\aleph_2\). - David Milovich, Davies trees and their applications

A Davies tree is a method of performing transfinite recursive constructions longer than \(\omega_1\) yet proceeding one countable piece at a time. At any given stage, the tree organizes all previous stages into finitely many nice pieces. I will discuss how a simpler structure induces a canonical Davies tree, extending the applicability of the Davies tree and simplifying its use. I will also survey some proofs that use Davies trees. The original such proof (Davies, 1963) shows (in ZFC) that the plane is a countable union of rotations of graphs of functions. Most recently, I have used a Davies tree in a proof that every openly generated compactum is a continuous image of a homogeneous openly generated compactum. - Farmer Schlutzenberg, +0 iteration trees on countable substructures of V

In their paper Iteration Trees, Martin and Steel proved some theorems regarding the iterability of countable substructures of rank initial segments of V . Two key theorems in that paper, 3.12 and 4.3, make the hypothesis that the iteration tree in question is +1, or +2, respectively. We will give an introduction to iterability and iteration trees, then outline a recent result showing that the theorems mentioned above can be proved without the “+n” hypotheses. Familiarity with ultrapowers and measurable cardinals should hopefully be enough to follow the main ideas.

**Further information:**

0) Webpages:

The webpage of the conference is:

http://math.univ-lyon1.fr/~melleray/yst2012-info.html

Official registration will soon take place on the CIRM website (before

registering, an account has to be created):

http://www.cirm.univ-mrs.fr/index.html/spip.php?rubrique2

1) Registration fee:

The amount to register will be at most 205 Euros. This will cover food and

lodging from Sunday April 29th evening to Friday May 4th (checkout 6pm).

Those who are willing to stay at the CIRM until Saturday morning will have

the possibility to do so, but we will not be able to cover the

corresponding expenses (at most 90 Euros, depending on the kind of bedroom).

Payment of the fee will have to be done online. Details in a soon coming

email, and on the website of the conference.

2) Grants:

The ASL will bring support via the possibility of obtaining travel awards

for graduate students who are members of the ASL:

http://www.aslonline.org/studenttravelawards.html

Please note that the deadline for asking for such a grant is three months

before the meeting. In our case, that means that applications should be

sent very shortly, before the end of January 2012.

We will also be able to bring partial financial support for some

participants who are not supported by their institutions. To apply for

such a grant, participants should contact Lionel Nguyen Van Thé

(lionel@latp.univ-mrs.fr), and ask their supervisor to send directly a

short reference letter. The deadline for applying is February 29th.

2) Research statement:

We would like all the participants to send us a research statement (pdf)

file, which will be made available on the web page.

And last, again, please transfer this message to anybody who might be

interested in attending.

Looking forward to meeting you in Luminy,

Julien Melleray

Lionel Nguyen Van Thé

Todor Tsankov

Matteo Viale

Slides are now available.