Program

The conference took place from Monday, April 30th to Friday, May 4th. Here is the detailed program (an underlined name means that slides or lecture notes are available, click on the name to view them):

Time	Monday	Tuesday	Wednesday	Thursday	Friday
9:15-10:05	I. Farah 1	I. Farah 2	A. Louveau 3	I. Farah 3	F. Schlutzenberg
10:15-11:05	S. Todorcevic 1	S. Todorcevic 2	S. Todorcevic 3	S. Todorcevic 4	I. Farah 4
11:05-11:25	Pause	Pause	Pause	Pause	Pause
11:25-12:15	S. Cox	C. Brech	I. Neeman 2	C. Martinez-Ranero	D. Milovich
12:30-14:00	Lunch	Lunch	Lunch	Lunch	Lunch
14:00-16:00	Discussions	Discussions		Discussions
16:00-16:50	A. Louveau 1	I. Neeman 1		A. Louveau 4
17:00-17:50	V. Fischer	A. Louveau 2		I. Neeman 3
18:10-19:00		D. Ikegami		I. Neeman 4
19:30	Dinner	Dinner	Dinner	Dinner

There were 4 tutorials, each with 4 sessions of 50 minutes, as well as invited research talks (click on the button at the end of each line below to make the abstract unfold).

Tutorials:

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 (lecture notes )

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 ω₁ where every operator is of the form λI + S, where S has separable range. I will present the underlying ideas behind those constructions.

Research talks:

Christina Brech , Forcing nonuniversal Banach spaces (slides)

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 (slides)

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 \emph{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 (slides)

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 (slides)

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. (slides)

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 (slides)

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, Comparison and Measures in Inner Models (slides)

We'll give an overview of the analysis of measures in inner models. This will include a description of the comparison process, which is essential to this analysis and many other proofs in inner model theory. We will focus first on \(L[U]\), then state some generalizations. A knowledge of measurable cardinals, ultrapowers and constructibility should be enough to follow most of what's covered.

Last updated May 5, 2012.