ESI 2013 Set Theory Programme

Abstracts

Ando, Hiroshi: Ultraproducts, QWEP von Neumann algebras, and the Effros-Maréchal topology

Haagerup and Winsløw studied the space of von Neumann algebras acting on a separable Hilbert space euiqpped with so-called Effros-Maréchal topology. They proved that this topology is closely linked to the modular theory, tracial ultraproducts and Kirchberg's QWEP conjecture. They in particular showed that a separable type II_1 factor is $R^{\omega}$-embeddable if and only if it is the Effros-Maréchal limit of matrix algebras. In this talk we study further connection among ultraproducts, QWEP and Effros-Maréchal topology. The key ingredients are structural results about non-tracial ultraproducts of von Neumann algebras established last year by the speaker and Haagerup.

This is joint work with Uffe Haagerup and Carl Winsløw.

Borodulin-Nadzieja, Piotr: Geometry of analytic P-ideals

We say that an ideal on $\omega$ is represented in a Banach space $X$ if there is a sequence $(x_n)_n$ in $X$ such that $A$ belongs to the ideal if and only if $\sum_{n\in A} x_n$ is unconditionally convergent. This is a generalization of the notion of summability: an ideal is represented in $\mathbb{R}$ if and only if it is summable. We pose several questions on which ideals are represented in certain Banach spaces and we answer some of them. We show that this approach reveals some geometric properties of analytic P-ideals and has a potency of generating new examples of ideals on $\omega$.

This is a joint work with Barnabás Farkas and Grzegorz Plebanek.

Dodos, Pandelis: Some recent results in Ramsey Theory

We shall review some results in Ramsey Theory obtained, recently, by the author in collaboration with V. Kanellopoulos, N. Karagiannis and K. Tyros. Among these results are density versions of the classical pigeonhole principles of Halpern-Lauchli and Carlson-Simpson.

Elekes, Marton: Ranks on Baire class $\alpha$ functions

The well-known theory of ranks on Baire class $1$ functions was developed by Kechris and Louveau. Motivated by some problems related to paradoxic geometric decompositions, we defined various natural ranks on the Baire $\alpha$ classes. To our greatest surprise, it has turned out that all these ranks are bounded below $\omega_1$.

This is joint work with Viktor Kiss.

Gregoriades, Vassilios: Classes of Polish spaces under effective Borel isomorphism

In this talk we present results about the problem of effective Borel isomorphism between Polish spaces (otherwise $\Delta^1_1$-isomorphism when the spaces are recursively presented). As opposed to the non-effective setting, where only two Polish spaces up to Borel isomorphism exist, the picture in the effective setting is much richer. There exist strictly increasing and strictly decreasing sequences of spaces as well as infinite antichains under the natural notion of effective Borel reduction. In fact this picture occurs in two large categories of spaces, the Kleene spaces and the Spector-Gandy spaces. A key tool for our study is a mapping $T \mapsto \mathcal{N}^T$ from the space of all trees on the naturals to the class of all Polish spaces, for which every recursively presented metric space is $\Delta^1_1$-isomorphic to some $\mathcal{N}^T$ for a recursive $T$, so that the preceding spaces are representatives for the classes of $\Delta^1_1$-isomorphism. Other key tools include the Gandy Basis Theorem and Kreisel compactness. The use of the latter is inspired by a related result of Fokina-Friedman-Törnquist.

Holy, Peter: Locally $\Sigma_1$-definable Wellorders of $H(\kappa^+)$

We show that if $\kappa$ is uncountable with $\kappa^{<\kappa}=\kappa$, it is consistent to have a $\Sigma_1$-definable wellorder of $H(\kappa^+)$ and $2^\kappa$ large. This contrasts the case $\kappa=\omega$, where Mansfield showed that a $\Sigma_1$-definable wellorder of $H(\omega_1)$ implies CH.

This is joint work with Philipp Luecke.

Ikegami, Daisuke: Inner models from logics and the generic multiverse

The goal of this research is to construct a model of set theory which is "close to" HOD but easier to analyze. The motivation comes from Woodin's HOD Conjecture, which states that HOD is very "close to" V under the presence of a very strong large cardinal (extendible cardinal). HOD Conjecture is closely related to the problem of constructing a canonical extender model with a supercompact cardinal and it has striking applications to the theory of large cardinals without the Axiom of Choice.

To solve HOD Conjecture, one would expect a fine analysis of HOD. The difficulty of the analysis of HOD lies in the fact that HOD is very "non-absolute", e.g., one could force V = HOD with a proper class partial order.

Given that HOD is obtained using full second order logic in the same way as Gödel's constructible universe L via first order logic, in this talk, we use Boolean valued higher order logics and Woodin's $\Omega$-logic to construct inner models of set theory which are more "absolute" than HOD and investigate the properties of the models.

Koszmider, Piotr: The Banach space $C(N*)$ in the Cohen model.

We will present results of a joint paper with C. Brech: http://arxiv.org/abs/1211.3173 concerning the consistency of some properties of the Banach space $C(N^*)\equiv\ell_\infty/c_0$ obtained in the Cohen model. We will also mention some of many open problems concerning this Banach spaces induced by the algebra $P(N)/Fin$ which could depend on extra set-theoretic assumptions.

Le Maître, François: Topological generators for full groups

A theorem of Dye asserts that two full groups of ergodic pmp equivalence relations are isomorphic iff the equivalence relations are orbit equivalent, so that the study of full groups might provide new invariants for pmp equivalence relations. Here we will focus on the topological rank of the full group, that is, the minimal number of elements needed to generate a dense subgroup. We will also discuss "genericity phenomenons" for topological generators, motivated by the Schreier-Ulam theorem which states that the generic pair in a compact metrisable connected group does generate a dense subgroup. If time permits, some results on non ergodic equivalence relations will be mentioned.

Marcone, Alberto: The complexity of isometric embeddability between ultrametric Polish spaces with fixed set of distances

(abstract TBA)

Mejía, Diego: Rothberger gaps in $F_\sigma$ ideals

We investigate a cardinal invariant associated with gaps of a given $F_\sigma$-ideal of $\omega$, which we call the Rothberger number of the ideal. We concentrate on the class of fragmented ideals and prove that it is consistent to obtain infinitely many different Rothberger numbers associated to such ideals. Also, we present some examples of fragmented ideals with Rothberger number equal to $\aleph_1$.

Mildenberger, Heike: Finitely Many Near-Coherence Classes of Ultrafilters

I will describe a recent forcing construction that establishes for any finite number $n\geq 2$ that it is consistent relative to ZFC that there are eactly $n$ near coherence classes of ultrafilters. This settles the old question on the possible numbers of near coherence classes: any finite non-zero number or $2^c$ are the only possibilities, and for each of them we have a model. The main new technique is to destroy selective ultrafilters by a forcing in a way that they can be complemented to selective ultrafilters in the forcing extension.

Rinot, Assaf: Hedetniemi’s conjecture for uncountable graphs

It is proved that if $V=L$, then for every successor cardinal κ, there exist graphs $G$ and $H$ of size and chromatic number κ, for which the tensor product graph $G×H$ is countably chromatic. This solves a longstanding open problem of Hajnal, and establishes the consistency of the failure of the infinite weak Hedetniemi conjecture.

Rosendal, Christian: Large scale geometry of metrisable groups

Large scale geometry of finitely generated or locally compact groups has long been one of the cornerstones of geometric group theory and its connections with harmonic and functional analysis. However, many of the groups of interest in logic, topology and analysis fail to be locally compact, such as automorphism groups of countable structures, diffeomorphism and isometry groups. For these there has been no canonical way of defining their large scale structure, as it is possible, e.g., with the word metric on a finitely generated group. Moreover, recently many groups have turned out to have no non-trivial large scale structure at all, despite being non-compact. We present a theory of large scale structure of metrisable groups and among other things determine the necessary and sufficient conditions for this structure to be unique up to coarse or quasi-isometric equivalences. Applications to model theory will be presented.

Sargsyan, Grigor: On the strength of the unique branch hypothesis (UBH)

We show that certain failures of UBH imply the existence of a non-tame mouse. This significantly improves the previous lower bound due to Steel.

This is a joint work with Nam Trang.

Schindler, Ralf: Does $\Pi^1_1$ determinacy yield $0^\#$?

It is unknown whether $\Pi^1_1$ determinacy yields $0^\#$ in 3rd order arithmetic. We show that 3rd order arithemtic plus Harrington's principle "there is a real $x$ such that every $x$-admissible is an $L$-cardinal" is equiconsistent with ZFC plus a remarkable cardinal, and we also discuss strengthenings of Harrington's principle.

This is joint work with Cheng Yong.

Seward, Brandon: Locally nilpotent groups and hyperfinite equivalence relations

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this talk I will discuss a positive solution to this question when the acting group is locally nilpotent. This extends previous results obtained by Gao–Jackson for abelian groups and by Jackson–Kechris–Louveau for finitely generated nilpotent-by-finite groups.

This is joint work with Scott Schneider.

Slutskyy, Kostyantyn: Automatic continuity for homomorphisms into free products

The main concept of automatic continuity is to establish conditions on topological groups $G$ and $H$ under which any homomorphism from $G$ into $H$ is necessarily continuous. Typically one of the groups is assumed to be very special, while the conditions on the other group are relatively mild.

We shall start with an overview of the automatic continuity results in the situation when the range group is a free group, a free product, or a free product with amalgamation. Of particular interest for us will be homomorphisms $f : G\to A*B$ from a completely metrizable group $G$ into a free product $A*B$ endowed with the discrete topology. The new result to be discussed is that any such homomorphism is necessarily continuous, unless its image is contained in one of the factors.

Sokic, Miodrag: Semilattices

We consider the class of finite semilattices with respect to the Ramsey property.

Tsankov, Todor: Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups

It is a well known fact that there is a narrow correspondence between properties of $\omega$-categorical structures and those of their automorphism groups. We investigate properties on the group side that correspond to stability and it turns out that an $\omega$-categorical structure is stable iff every continuous function on the group that is both left and right uniformly continuous is weakly almost periodic. As an application, we show that every such group is minimal, i.e., every continuous surjective homomorphism to another topological group is open, generalizing previous results for particular groups of Stoyanov, Uspenski, Glasner, and others. Our methods can be applied in the setting of both classical and continuous logic.

This is joint work with Itaï Ben Yaacov.