## 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.

## Itai Ben Yaacov: Reconstruction for non-aleph0-categorical theories?

HUJI Logic Seminar

This Wednesday, 8 November, we will have a meeting of the Logic Seminar. The meeting will be in Math 209, 8 November (Wednesday), 11:00 – 13:00.

Speaker: Itai Ben Yaacov
Title: Reconstruction for non-aleph0-categorical theories?

Abstract. It is a familiar fact (sometimes attributed to Ahlbrandt-Ziegler, though it is possibly older) that two aleph0-categorical theories are bi-interpretable if and only if their countable models have isomorphic topological isomorphism groups. Conversely, groups arising in this manner can be given an abstract characterization, and a countable model of the theory (up to bi-interpretation, of course) can be reconstructed.
More recently, it was observed that various (interpretation-invariant) properties of an aleph0-categorical theory (stability, NIP, …) correspond to familiar dynamical properties of the automorphism group, with a few nice applications.
Since stability and NIP have nothing to do with aleph0-categoricity, one may ask whether the correspondence between theories (up to bi-interpretation) and topological groups can be extended to theories which are not aleph0-categorical? There are some indications that a positive answer is possible, if one is willing to replace groups with more general objects.
See you there,
Yatir

## Chris Lambie-Hanson: Squares, ascent paths, and chain conditions

BIU seminar in Set Theory

On 06/11/2017, 13-15, Building 505, Room 65

Speaker: Chris Lambie-Hanson

Title: Squares, ascent paths, and chain conditions

Abstract. Two topics of interest in modern set theory are the productivity of chain conditions and the existence of higher Aronszajn trees. In this talk, we discuss generalizations of both of these topics and their connections with various square principles. In particular, we will prove that, if $\kappa$ is a regular uncountable cardinal and $\square(\kappa)$ holds, then:
1) for all regular $\lambda < \kappa$, there is a $\kappa$-Aronszajn tree with a $\lambda$-ascent path;
2) there is a $\kappa$-Knaster poset $\mathbb{P}$ such that $\mathbb{P}^{\aleph_0}$ is not $\kappa$-c.c.
Time permitting, we will also present a complete picture of the relationship between the existence of special trees and the existence of Aronszajn trees with ascent paths at the successor of a regular cardinal. This is joint work with Philipp Lücke.