## Inna Pozdniakova: On monoids of monotone injective partial selfmaps of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ with co-finite domains and images

Tuesday, May 26, 2015, 18:45
Wrocław University of Technology, 215 D-1

Speaker: Inna Pozdniakova (National University of Lviv)

Title: On monoidsof monotone injective partial selfmaps of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ with co-finite domains and images

Abstract:

The speaker will discuss on the structure of the semigroup $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ of monotone injective partial selfmaps of the set of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ having co-finite domain and image, where $L_n\times_{\operatorname{lex}}\mathbb{Z}$ is the lexicographic product of an $n$-elements chain and the set of integers with the usual order.

## Taras Banakh: Separation axioms on paratopological groups and quasi-uniform spaces

Tuesday, May 26, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Taras Banakh (National University of Lviv)

Title: Separation axioms on paratopological groups and quasi-uniform spaces

Abstract:

We shall prove that each regular paratoplogical group is
completely regular thus resolving an old problem in the theory of
paratopological groups.

## Jan Grebik: Oscillations of reals and related forcings

Wednesday, May 20, 2015, 11:00
Prague – IM AS CR, Zitna 25, seminar room, front building, third floor

Speaker: Jan Grebik

Title: Oscillations of reals and related forcings

Abstract:

We will review results about oscillations of real numbers and properties of posets defined using oscillations.

## Jan van Mill: Nonhomogeneity of remainders (part 2)

Set Theory and Topology seminar (BGU)

On Tuesday Jan van Mill will continue his talk from last week.
Time: Tuesday, May 19, 12:15-13:40.

Speaker: Jan van Mill (UvA).

Title: Nonhomogeneity of remainders.

Abstract: A space $X$ is homogeneous if for any two points $x,y\in X$ there is a homeomorphism $h$ from $X$ onto itself such that $h(x) = y$.
In 1956, Walter Rudin proved that the Čech-Stone remainder $\beta\omega \setminus \omega$, where $\omega$ is the discrete space of positive integers, is not homogeneous under CH. This result was later generalized considerably by Frolik who showed in ZFC that $\beta X\setminus X$ is not homogeneous, for any nonspeudocompact space $X$. Van Douwen and Kunen proved many results that are in the same spirit.

Hence the study of (non)homogeneity of Čech-Stone remainders has a long history. In this talk we are interested in homogeneity properties of arbitrary remainders of topological spaces. We address the following general problem: when does a space have a homogeneous remainder? If $X$ is locally compact, then the Alexandroff 1-point compactification $\alpha X$ of $X$ has a homogeneous remainder. Hence for locally compact spaces, our question has an obvious answer. If $X$ is not locally compact, however, then it need not have a homogeneous remainder, as the topological sum of the space of rational numbers and the space of irrational numbers shows. Hence we consider questions of the following type: if $X$ is homogeneous, and not locally compact, does $X$ have a homogeneous remainder? We will show that if $X$ is countable and nowhere locally compact, then any remainder of $X$ has at most $\mathfrak{c}$ homeomorphisms, where $\mathfrak{c}$ denotes the cardinality of the continuum. From this we get an example of a countable topological group $G$ no remainder of which is homogeneous. We also get new and very simple proofs that familiar Čech-Stone remainders are not homogeneous.

This is joint work with A. V. Arhangel’skii.

## Jarosław Swaczyna: Generalized densities of subsets of natural numbers and associated ideals

Tuesday, May 19, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Jarosław Swaczyna (Technical University of Łódź)

Title: Generalized densities of subsets of natural numbers and associated ideals

Abstract:

Let g be a function from natural numbers to non-negative reals. We say that a subset A of naturals has g-density zero, if lim_n |{k∈A: k<n}|/g(n) = 0. It is an easy observation that family of g-density zero sets is an ideal.

I will discuss some properties of ideals obtained this way (among others, I will show that they can be generated using Solecki's submeasures). I will then examine inclusions between ideals obtained for different functions g.

I will also discuss connections between our ideals, "density-like" ideals and Erdos-Ulam ideals. I will present joint results with M. Balcerzak, P. Das and M. Filipczak.

## Jan van Mill: Nonhomogeneity of remainders

Set Theory and Topology seminar (BGU)

Time: Tuesday, May 12, 12:15-13:40.

Place: Classroom 138, Building 90.

Speaker: Jan van Mill (UvA).

Title: Nonhomogeneity of remainders.

Abstract: A space $X$ is homogeneous if for any two points $x,y\in X$ there is a homeomorphism $h$ from $X$ onto itself such that $h(x) = y$.
In 1956, Walter Rudin proved that the Čech-Stone remainder $\beta\omega \setminus \omega$, where $\omega$ is the discrete space of positive integers, is not homogeneous under CH. This result was later generalized considerably by Frolik who showed in ZFC that $\beta X\setminus X$ is not homogeneous, for any nonspeudocompact space $X$. Van Douwen and Kunen proved many results that are in the same spirit.

Hence the study of (non)homogeneity of Čech-Stone remainders has a long history. In this talk we are interested in homogeneity properties of arbitrary remainders of topological spaces. We address the following general problem: when does a space have a homogeneous remainder? If $X$ is locally compact, then the Alexandroff 1-point compactification $\alpha X$ of $X$ has a homogeneous remainder. Hence for locally compact spaces, our question has an obvious answer. If $X$ is not locally compact, however, then it need not have a homogeneous remainder, as the topological sum of the space of rational numbers and the space of irrational numbers shows. Hence we consider questions of the following type: if $X$ is homogeneous, and not locally compact, does $X$ have a homogeneous remainder? We will show that if $X$ is countable and nowhere locally compact, then any remainder of $X$ has at most $\mathfrak{c}$ homeomorphisms, where $\mathfrak{c}$ denotes the cardinality of the continuum. From this we get an example of a countable topological group $G$ no remainder of which is homogeneous. We also get new and very simple proofs that familiar Čech-Stone remainders are not homogeneous.

This is joint work with A. V. Arhangel’skii.

## Ilijas Farah: Recent progress on Naimark’s problem

Friday Set Theory Seminar (HUJI)

We shall meet this Friday (May 15th) in the Hebrew University
math department building in Room 110, at 10 am.

Speaker: Ilijas Farah (York)

Title: Recent progress on Naimark’s problem

Abstract: In 2004 Akemann and Weaver used diamond on omega_1 to solve a long-standing Naimark’s problem about representations of C*-algebras (all definitions and intuition will be provided). It is still not known whether their result can be proved in ZFC, CH, or even from diamond on some other cardinal. I will say a few words on the recent
progress.

## Tomasz Żuchowski: Tukey types of orthogonal ideals

Tuesday, May 12, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Tomasz Żuchowski (University of Wroclaw)

Title: Tukey types of orthogonal ideals

Abstract:

A partial order P is Tukey reducible to partial order Q when there exists a function f:P->Q such that if A is a bounded subset of Q then f^{-1}[A] is a bounded subset of P. The existence of such reduction is related to some cardinal invariants of considered orders. We will show Tukey reductions between some special ideals of subsets of N with the inclusion order and other partial orders.

## ​Nili Vilkin: ​Intuitionistic logic

Set Theory and Topology seminar (BGU)

Time: Tuesday,
​May 5th, 12:15-13:40.

Place: Seminar room -101, Math building 58.

Speaker: ​Nili Vilkin (BGU).

Title:  ​Intuitionistic logic.

Abstract:
​Intuitionistic logic, also referred as constructive logic, differs from classical logic in that it has no notion of truth. Truth is replaced by the notion of proofs, thus instead of using truth tables, the meaning of the logical connectives is given in terms of proofs. I will present a formal system in which the proofs are constructed, and also an algebraic semantics for intuitionistic logic (Heyting algebra), demonstrating that some classical tautologies such as the principle of excluded middle “P or not P”, are not constructively provable.

IMPORTANT NOTICE: This week’s talk will be given in Hebrew.

## Diego A. Mejía: Separating the left side of Cichon’s diagram

Friday Set Theory Seminar (HUJI)

We shall meet this Friday (May 8th) in the Hebrew University
math department building in Room 110, at 10 am.

Speaker: Diego A. Mejía (TU WIEN)

Title: Separating the left side of Cichon’s diagram

Abstract: It is well known that, with finite support iterations of ccc posets, we can obtain models where 3 or more cardinals of Cichon’s diagram can be separated. For example, concerning the left side of Cichon’s diagarm, it is consistent that \aleph_1 < add(N) < cov(N) < b < non(M)=cov(M)=c. Nevertheless, getting the additional strict inequality non(M) < cov(M) is a challenge because subposets of E, the standard ccc poset that adds an eventually different real, may add dominating reals (by Pawlikowski, 1992).

We construct a model of \aleph_1 < add(N) < cov(N) < b < non(M) < cov(M)=c with the help of chains of ultrafilters that allows to preserve certain unbounded families. This is a joint work with M. Goldstern and S. Shelah.

See you there!