## Matteo Viale: Well Behaving Category Forcings

Friday Set Theory Seminar (HUJI)

We shall meet next Friday (July 10th) in the Hebrew University math department building in Room 110, at 10 am.

Speaker: Matteo Viale (UNITO)

Title: Well Behaving Category Forcings

Abstract: We isolate and study certain nice properties for classes of forcings  : that of being $\kappa$-iterable (property A), that of being $\Pi^1_1$-persistent at $\kappa$ (property B), and that of having the
strong freezeability property (property C)..

We show that for classes of forcings having properties A,B,C there is a forcing axiom CFA($\Gamma$) (strentghtening the forcing axioms for $\kappa$-many dense sets for posets in $\Gamma$) with the property that no forcing in  preserving CFA($\Gamma$ ) can change the theory of the Chang model $L[Ord{}^{\le\kappa}$] with parameters in $H_{\kappa^+}$.
We also nd nine distinct classes $\Gamma$ which satisfy the above properties A,B,C for  = $\omega_1$.
For one of these classes  $\Gamma$, CFA($\Gamma$ ) implies CH (to be double checked). The other classes $\Gamma$ strengthen known forcing axioms such as MM, PFA and variations theoreof.
Moreover we can easily separate two distinct theories of the form ZFC + CFA($\Gamma_j$) using distinct $\Pi_2$-properties of the corresponding $H_{\omega_2}$.
For $\kappa>\omega_1$, we cannot give any example of a $\Gamma$ which can satisfy properties A,B,C, but any such $\Gamma$ will provide an interesting example of a forcing axiom for $\kappa$-many dense sets…..
This is a joint project with David Aspero.

## Frank Tall: PFA(S)[S] III

Place: Fields Institute (Room 210)

Date: July 3rd (13:30- 15:00)

Speaker: Franklin Tall

Title: PFA(S)[S] III

Abstract: I will use the machinery developed last time plus a little topology to finish the proof. The remaining time will be used to start discussing Alan’s proof and/or its consequences.

## P.O.I workshop in Pure and Descriptive Set Theory, Turin, September 25-26th, 2015

The logic group in Turin is pleased to announce a two-day workshop on Pure and Descriptive Set Theory. This workshop concludes the San Paolo junior PI grant “New Perspectives On the nature of Infinity”.
The workshop will take place on September 25th and 26th, 2015 in Turin University (place to be defined), and will consist in four talks on each day. The 25th will be dedicated to Descriptive Set Theory and the speakers are Andrew Marks (Los Angeles), Marton Elekes (Budapest), Raphaël Carroy (Torino) and Ben Miller (Vienna).
The 26th will be dedicated to Pure Set Theory and the speakers are Assaf Rinot (Bar-illan), David Aspero (East Anglia), Daisuke Ikegami (Kobe) and Paul Larson (Miami, Ohio).
Each day will be concluded by a brief discussion session outlining some open problems and future directions of research.
Everyone is cordially invited to attend. Please forward this announcement to anyone who could be interested in this event.

We might have some modest travel awards to graduate or recent Ph.D. students, so that they may attend the workshop. More information can be found on the webpage of the workshop:

http://logicatorino.altervista.org/POImeeting1509/index.html

We hope to see you soon in Turin!

Alessandro Andretta, Raphaël Carroy, Luca Motto Ros, and Matteo Viale

## Jindrich Zapletal: An interpreter for topologists

Wednesday, July 1, 2015, 11:00
Prague – IM AS CR, Zitna 25, seminar room, front building, third floor

Speaker: Jindrich Zapletal (University of Florida & IM CAS)

Title: An interpreter for topologists

## Appalachian Set Theory workshop: Su Gao

Appalachian set theory

# Su Gao : “Countable abelian group actions”

## Description

A number of “stubbornly open” problems about countable Borel equivalence relations concern hyperfiniteness. For instance, the increasing union problem asks if the increasing union of a sequence of hyperfinite equivalence relations is still hyperfinite. In the past decade or so, the only progress on hyperfiniteness problems has been the proof of hyperfiniteness for orbit equivalence relations of countable abelian group actions (Gao and Jackson, “Countable abelian group actions and hyperfinite equivalence relations”, Inventiones Mathematicae, 2015) and then the extension of this result to locally nilpotent groups (Schneider and Seward, “Locally nilpotent groups and hyperfinite equivalence relations”, to appear).

The hyperfiniteness proofs are based on an elaborate theory of Borel marker structures with regularity properties. Now researchers have a good understanding of which regularity properties are possible and which are beyond hope. For the proofs of negative results two new concepts and methods have been playing a key role. One of them is the introduction and construction of hyperaperiodic elements with various additional properties. The other is the introduction of new forcing notions that are special cases of the so-called orbit forcing. The workshop will be roughly divided into four lectures:

• In the first lecture we will construct some basic regular marker structures for the Bernoulli shift of ℤn. Using such marker structures, we will give an outline of the proof of hyperfiniteness for orbit equivalence relations of countable abelian group actions.
• In the second lecture we will give some advanced constructions of regular marker structures and use them to illustrate a proof that the Borel chromatic number for the free part of 2n is 3.
• In the third lecture we will construct some hyperaperiodic elements for 2n and use them to show that the continuous chromatic number for the free part of 2n is 4.
• In the fourth lecture we will consider some forcing constructions and use them to show that certain regular Borel marker structures do not exist.

## H. Jerome Keisler: Randomizations of scattered theories

Place: Bahen Centre (Room BA6183)

Date: 30-June-2015 (14:00-15:30)

Speaker: H. Jerome Keisler.
Title: Randomizations of scattered theories

Abstract: Consider a sentence $\phi$ of the infinitary logic $L_{\omega_1, \omega}$. In 1970, Morley introduced the notion of a scattered sentence, and showed that if $\phi$ is scattered then the class $I(\phi)$ of isomorphism types of countable models of $\phi$ has cardinality at most $\aleph_1$, and if $\phi$ is not scattered then $I(\phi)$ has cardinality continuum. The absolute form of Vaught’s conjecture for $\phi$ says that if $\phi$ is scattered then $I(\phi)$ is at most countable. Generalizing previous work of Ben Yaacov and the author, we introduce here the notion of a separable model of $\phi^R$, which is a separable continuous structure whose elements are random elements of a model of $\phi$. We say that $\phi^R$ has few separable models if every separable model of $\phi^R$ is uniquely characterized up to isomorphism by a function that assigns probabilities summing to one to countably many elements of $I(\phi)$. In a previous paper, Andrews and the author showed that if $\phi$ is a complete first order theory and $I(\phi)$ is at most countable then $\phi^R$ has few separable models. We show here that this result holds for all $\phi$, and that if $\phi^R$ has few separable models then $\phi$ is scattered. Hence if the absolute Vaught conjecture holds for $\phi$, then $\phi^R$ has few separable models if and only if $I(\phi)$ is countable, and also if and only if $\phi$ is scattered. Moreover, assuming Martin’s axiom for $\aleph_1$, we show that if $\phi$ is scattered then $\phi^R$ has few separable models.

## Franklin Tall: PFA(S)[S] II

Place: Fields Institute (Room 210)
Date:  26-June-2015 (13:30-14:45)
Speaker: Franklin Tall
Title: PFA(S)[S] II
Abstract: This is a continuation of last week’s lecture. Last week’s lecture was largely motivation; this lecture will be mainly technical, developing the method. If you really want to attend and missed last week, contact me and I will give you something to read.

## Shimon Garti: Galvin’s property

Students Set Theory Seminar (HUJI)

We shall meet this Wednesday, as usual, in Ross 70 at 11:00 for Magidor’s lecture, and again at 14:00 for the students seminar (details below).

Shelah’s lecture will also take place as usual and will continue from last week.

This week in the students seminar.

Speaker: Shimon Garti

Title: Galvin’s property

Abstract: We shall see that Galvin’s property (for large intersection of club sets) follows from the Devlin-Shelah weak diamond, but independent of it. Moreover, it is independent also over Martin’s axiom. However, we suspect that the PFA implies the negation of Galvin’s property, and we shall prove a weak form of this result.

See you there!

## Asaf Karagila: Restrictions on forcings that change cofinalities

Friday Set Theory Seminar (HUJI)

We shall meet this Friday (June 26th) in the Hebrew University math department building in Room 110, at 10 am.

Speaker: Asaf Karagila (HUJI)

Title: Restrictions on forcings that change cofinalities

Abstract: Given a regular cardinal kappa, we want to know what sort of “nice” properties a forcing can have while making kappa singular. For target cofinality omega we have the Prikry forcing which is homogeneous and does not add bounded subsets to kappa. But if we want the cofinality to be uncountable we run into problems. For example, sigma-closed forcings cannot change cofinalities without collapsing cardinals.

We will investigate a couple of nice properties a forcing might have, weaken them, and show that under reasonable conditions a forcing with these conditions cannot change the cofinality of a cardinal to be uncountable without collapsing it. Joint work with Yair Hayut.

See you there!

## BEST 2015 slides

The 22nd BEST conference was held June 14–17 in San Francisco, CA.

Mirna Džamonja – FAC and WQO orders and their ordinal invariants
Luke Serafin – Cardinal invariants of generalized continua
Ari Brodsky – A microscopic approach to higher Suslin tree constructions
Ola Kwiatkowska – Dynamics of the homeomorphism group of the Lelek fan
Paul Ellis – The conjugacy problem for countable homogeneous structures
Trevor Wilson – Covering properties of derived models
Miha Habic – Joint Laver diamonds
Bill Chen – Scales in Prikry extensions
Rodrigo Dias – Productively countably tight spaces and selective games
Steven Clontz – Limited information strategies in infinite games
Liljana Babinkostova – The selective strong screen ability game
Spencer Unger – Baire measurable paradoxical decompositions via matchings
Diana Ojeda-Aristizábal – Finite forms of Gowers’ FINk theorem