## Assaf Shani: Borel equivalence relations and symmetric models

HUJI Logic Seminar

We will have a meeting of the Logic Seminar this Wednesday 11/7, 11-13 Ross 70A.

Title:  Borel equivalence relations and symmetric models
Speaker: Assaf Shani
Abstract. We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of $S_\infty$, and the study of symmetric models of set theory without choice, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998).
For example, we show that the equivalence relation $\cong^\ast_{\omega+1,0}$ is strictly below $\cong^\ast_{\omega+1}$ in Borel reducibility. By results of Hjorth-Kechris-Louveau, $\cong^\ast_{\omega+1}$ corresponds to $\Sigma^0_{\omega+1}$ actions of $S_\infty$, while $\cong^\ast_{\omega+1,0}$ corresponds to $\Sigma^0_{\omega+1}$ actions of “well behaved” closed subgroups of $S_\infty$, e.g., abelian groups.
We further apply these techniques to study the Friedman-Stanley jumps. For example, we find a topology on the domain of $=^{++}$ so that $=^{++}\restriction C$ is Borel bireducible with $=^{++}$, for any comeager set $C$. This answers a question of Zapletal, based on the results of Kanovei-Sabok-Zapletal (2013).
For these proofs we analyze the models $M_n$, $n<\omega$, developed by Monro (1973), and extend his construction past $\omega$, through all countable ordinals. This answers a question of Karagila (2016), e.g., establishing separation between the $\omega$ and $\omega+1$’th Kinna-Wagner principle.

## Martin Goldstern: Higher Random Reals

HUJI Logic Seminar
Tuesday 29/5 13:30-15:00 Ross 63
Title: Higher Random Reals
Speaker: Martin Goldstern
Abstract:
The set of real numbers is often identified with
Cantor Space 2^omega, with which it shares many important
properties: not only the cardinality, but also other
“cardinal characteristics” such as cov(null), the smallest
number of measure zero sets needed to cover the whole space,
and similarly cov(meager), where meager=”first category”;
or their “dual” versions non(meager) (the smallest
cardinality of a nonmeager set) and non(null).

Many ZFC results and consistency results (such as
“cov(meager) lessequal non(null), but no inequality between
cov(null) and cov(meager) is provable in ZFC”)
are known.

Recent years have seen a renewed interest in “higher reals”,
i.e., elements of 2^kappa, where kappa is usually an inaccessible
cardinal. Meager sets have a natural generalisation to this
context, namely “kappa-meager” sets (using the <kappa-box product
topology), but what is the natural generalisation of the
ideal of null sets?

In my talk I will present an ideal null_kappa recently introduced
by Saharon Shelah, and some ZFC and consistency results from a
forthcoming joint paper with Thomas Baumhauer and Saharon Shelah,
such as “cov(null_kappa) lessequal non(null_kappa)”, and
“consistently, cov(meager_kappa) > cov(null_kappa)”.

## Asaf Karagila: What can you say about critical cardinals?

HUJI Logic Seminar

Tuesday 1/5 at Math 209 13:30-15:30
Speaker: Asaf Karagila
Title: What can you say about critical cardinals?
Abstract. We isolate the property of being a critical point, and prove some basic positive properties of them. We will also prove a lifting property that allows lifting elementary embedding to symmetric extensions, and outline a construction that shows that it is consistent that a successor of a critical cardinal is singular. This is a recent work with Yair Hayut.

## Yair Hayut: Filter compactness and squares

BIU seminar in Set Theory

April 8, 2018

Speaker: Yair Hayut (TAU)

Title: Filter compactness and squares

Abstract. Strongly compact cardinals are characterized by the property that any $\kappa$-complete filter can be extended to a $\kappa$-complete ultrafilter. When restricting the cardinality of the underlying set, we obtain a nontrivial hierarchy. For example, when requiring the extension property to hold only for filters on $\kappa$, we obtain Gitik’s $\kappa$-compact cardinals, which are known to be consistently weaker than $\kappa$ being $\kappa^+$-strongly compact.

In this talk I will focus on the level by level connection between the filter extension property and the compactness for $L_{\kappa,\kappa}$. Using the compactness, I will show that if $\kappa$ is $\kappa$-compact then $\square(\kappa^{+})$-fails.

## Omer Mermelstein: ​Searching for template structures in the class of Hrushovski ab initio geometries

BGU seminar in Logic, Set Theory and Topology.
Tuesday, ​December ​​​26th, 12:15-13:3, Seminar room -101, Math building 58.
Speaker:​ ​ Omer Mermelstein (BGU)

Title: ​​Searching for template structures in the class of Hrushovski ab initio geometries

Abstract. Zilber’s trichotomy conjecture, in modern formulation, distinguishes three flavours of geometries of strongly minimal sets — disintegrated/trivial, modular, and the geometry of an ACF. Each of these three flavours has a classic “template” — a set with no structure, a projective space over a prime field, and an algebraically closed field, respectively. The class of ab initio constructions with which Hrushovski refuted the conjecture features a new flavour of geometries — non-modular, yet prohibiting any algebraic structure.

In this talk we take a step towards defining “template” structures for the class of (CM-trivial) ab initio Hrushovski constructions. After presenting intuitively the standard ab initio Hrushovski construction, we generalize Hrushovski’s predimension function, showing that the geometries associated to certain Hrushovski constructions are, essentially, ab initio constructions themselves. If time permits, we elaborate on how these geometric structures may generate the class of geometries of ab initio constructions under the Hrushovski fusion operation.

## Omer Ben-Neria: Singular Stationarity and Set Theoretic Generalizations of Algebras

HUJI Logic Seminar

This Wednesday, 27 December, we will have a meeting of the Logic Seminar. The meeting will be in Ross 63 (notice room change) , 27 December (Wednesday), 11:00 – 13:00.

Speaker: Omer Ben-Neria
Title: Singular Stationarity and Set Theoretic Generalizations of Algebras
Abstract. The set theoretic generalizations of algebras have been introduced in the 1960s to give a set theoretic interpretation of usual algebraic structures. The shift in perspective from algebra to set theory is that in set theory the focus is on the collection of possible algebras and sub-algebras on specific cardinals rather than on particular algebraic structures. The study of collections of algebras and sub-algebras has generated many well-known problems in combinatorial set theory (e.g., Chang’s conjecture and the existence of small singular Jonsson cardinals).
In the 1990s Foreman and Magidor used algebras to initiate a study of Singular Stationarity, i.e., a study of alternative notions of stationarity for subsets of singular cardinals. They introduced and developed two notions of singular stationarity called Mutual Stationarity and Tight Stationarity, and used their findings to prove a fundamental result concerning generalized nonstationary ideals.
The two notions of singular stationarity have been studied in the last two decades, and the main purpose of the talk to describe the related known and recent results.
I will start by giving some background material on algebras and stationary sets, and describe the history of a well-known problem by Jonsson. We will then proceed to describe the work done on the notion of Mutually Stationary Sequences and sketch a recent proof which is based on the existence of special types of strong filters.
In the second part of the talk, we will connect the notions of Singular Stationarity to results from PCF theory and Extender-based forcing methods.

## Yair Hayut: Subcompact cardinals

TAU forcing seminar

On 04/Dec/17, 9-11, Yair Hayut will be speaking on Subcompact cardinals.

Abstract. During the investigation of the existence of squares in core models, Jensen isolated the concept of subcompact cardinals. Subcompactness is weaker than supercompactness and stronger than superstrength. It is a natural candidate for the consistency strength of simultaneous failure of $\square(\kappa)$ and $\square_\kappa$. A work of Schimmerling and Zeman indicates that under some mild assumptions, subcompactness is the only possibility for failure of squares at core models of the form L[E].

In this talk I will define the relevant large cardinals notions, and talk about Zeman’s theorem for the consistency of failure $\square_{\aleph_{\omega}}$ from a measurable subcompact cardinal.

## Bill Chen: A small Dowker space

BIU seminar in Set Theory

Speaker: Bill Chen (BGU)

There will be two talks: November 27 and December 4.

First talk: A small Dowker space, part 1

Abstract. A topological space is said to be “Dowker” if it is normal but its product with the unit interval is not normal. In this lecture, we shall present a construction, due to Balogh, of a Dowker space of size continuum.

Second talk: A small Dowker space, part 2

Abstract. Last week, we presented an approach for constructing a Dowker space of size continuum, ending up with a statement of a lemma that would yield such a space. In this talk, we shall prove this lemma.

## Yair Hayut: Chang’s Conjecture at many cardinals simultaneously

HUJI Logic Seminar
This Wednesday, 22 November, we will have a meeting of the Logic Seminar. The meeting will be in Math 209, 22 November (Wednesday), 11:00 – 13:00.

Speaker: Yair Hayut
Title: Chang’s Conjecture at many cardinals simultaneously

Abstract. Chang’s Conjecture is a strengthening of Lowenheim-Skolem-Tarski theorem. While Lowenheim-Skolem-Tarski theorem is provable in ZFC, any instance of Chang’s Conjecture is independent with ZFC and has nontrivial consistency strength. Thus, the question of how many instances of Chang’s Conjecture can consistently hold simultaneously is natural.
I will talk about some classical results on the impossibility of some instances of Chang’s Conjecture and present some results from a joint work with Monroe Eskew.

## Eilon Bilinsky: Uncountable set of reals with a single condensation point

BIU seminar in Set Theory On 20/11/2017, 13-15, Building 505, Room 65

Speaker: Eilon Bilinsky (TAU)

Title: Uncountable set of reals with a single condensation point

Abstract. We construct a model of ZF with an uncountable set of reals having a unique condensation point. This answers a question of Sierpinski from 1918.