## Daria Michlik: Symmetric products as cones

Tuesday, January 8, 2019, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Daria Michlik (Cardinal Stefan Wyszynski University in Warsaw)

Title: Symmetric products as cones

Abstract:
(join work with Alejandro Illanes and Veronica Martinez-de-la-Vega)
For a continuum $X$, let $F_n(X)$ be the hyperspace of all nonempty subsets of $X$ with at most $n$-points. The space $F_n(X)$ is called the n’th-symmetric product.
In
A. Illanes, V. Martinez-de-la-Vega, Symmetric products as cones, Topology Appl. 228 (2017), 36–46
it was proved that if $X$ is a cone, then its hyperspace $F_n(X)$ is also a cone.
During my talk I will discuss the converse problem. I will prove that if $X$ is a locally connected curve, then the following conditions are equivalent:
1. $X$ is a cone,
2. $F_n(X)$ is a cone for some $n\ge 2$,
3. $F_n(X)$ is a cone for each $n\ge 2$.

## Sakae Fuchino: Downward Lowenheim Skolem Theorems for stationary logics and the Continuum Problem

Tuesday, December 11, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Sakae Fuchino (Kobe University)

Title: Downward Lowenheim Skolem Theorems for stationary logics and the Continuum Problem

Abstract:

Downward Lowenheim Skolem Theorems of extended logics can be considered as reflection principles. In this talk we consider Downward Lowenheim Skolem Theorems of variations of stationary logic. Some of the strongest forms of reflection principles formulated in this way imply CH while some other imply that the continuum is very large. The results presented in this talk are further development of the results presented in the talk I gave last year in Wroclaw and will be a part of a joint paper with Hiroshi Sakai and Andre Ottenbreit Maschio Rodrigues.

## Serhii Bardyla: A topologization of graph inverse semigroups

Tuesday, November 27, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Serhii Bardyla (Ivan Franko National University of Lviv)

Title: A topologization of graph inverse semigroups

Abstract:

We characterize graph inverse semigroups which admit only discrete locally compact semigroup topology. It will be proved that if a directed graph $E$ is strongly connected and contains a finite amount of vertices then a locally compact semitopological graph inverse semigroup $G(E)$ is either compact or discrete. We describe graph inverse semigroups which admit compact semigroup topology and construct a universal object in the class of graph inverse semigroups. Embeddings of graph inverse semigroups into compact-like topological semigroups will be investigated. Also, we discuss some open problems.

## Robert Rałowski: Images of Bernstein sets via continuous functions

Tuesday, November 13, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Robert Rałowski (Wroclaw University of Science and Technology)

Title: Images of Bernstein sets via continuous functions

Abstract:

We examine images of Bernstein sets via continuous mappings. Among other results we prove that there exists a continuous function $f:\mathbb{R}\to\mathbb{R}$ that maps every Bernstein subset of $\mathbb{R}$ onto the whole real line. This gives the positive answer to a question of Osipov. This talk is based upon joint paper with Jacek Cichoń and Michał Morayne.

## Andrzej Starosolski: The Rudin-Keisler ordering of P-points under b=c

Tuesday, May 15, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Andrzej Starosolski (Silesian University of Technology)

Title: The Rudin-Keisler ordering of P-points under $\mathfrak b=\mathfrak c$

Abstract:

M. E. Rudin proved under CH that for each P-point there exists another P-point strictly RK-greater . Assuming $\mathfrak p=\mathfrak c$, A. Blass showed the same; moreover, he proved that each RK-increasing $\omega$-sequence of P-points is upper bounded by a P-point, and that there is an order embedding of the real line into the class of P-points with respect to the RK-preordering. He also asked what ordinals can be embedded in the set of P-points.
In my talk the results cited above are proved and the mentioned question is answered under a (weaker) assumption $\mathfrak b =\mathfrak c$.

## Marek Bienias: About universal structures and Fraisse theorem

Tuesday, April 24, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Marek Bienias (Łódź University of Technology)

Title: About universal structures and Fraisse theorem

Abstract:

For a given structure D of language L we can consider age of D, i.e. the family of all finitely generated L-substructures od D. It turns out that age has property (HP) and (JEP). Fraisse theorem let us revers the procedure: if K is nonempty countable family of finitely generated L-structures having properties (HP), (JEP) and (AP), then there exists exactly one (up to isomorphism) L-structure D (so called Fraisse limit) which is countable ultrahomogenous and has age K.
The aim of the talk is to define basic notions from Fraisse theory, proof the main theorem and show some alternative way of looking at the construction of Fraisse limit.

## Piotr Borodulin-Nadzieja: Tunnels through topological spaces

Tuesday, April 17, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Piotr Borodulin-Nadzieja (University of Wroclaw)

Title: Tunnels through topological spaces

Abstract:

I will show a ZFC example of a compact space (without
isolated points) through which one cannot drill a tunnel. I will discuss
when and when not $\omega^*$ has a tunnel.

## Grzegorz Plebanek: Strictly positive measures on Boolean algebras

Tuesday, March 27, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Grzegorz Plebanek (University of Wroclaw)

Title: Strictly positive measures on Boolean algebras

Abstract:

$SPM$ denotes the class of Boolean algebras possessing strictly positive measure (finitely additive and probabilistic). Together with Menachem Magidor, we consider the following problem: Assume that $B$ belongs to $SPM$ for every subalgebra $B$ of a given algebra $A$ such that $|B|\le\mathfrak c$. Does it imply that the algebra $A$ belongs to $SPM$?

It turns out that the positive answer follows from the existence of some large cardinals, while the counterexample can be found in the model of $V=L$.

## Grzegorz Plebanek: On almost disjoint families with property (R)

Tuesday, March 13, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Grzegorz Plebanek (University of Wroclaw)

Title: On almost disjoint families with property (R)

Abstract:

We consider (with A.Aviles and W. Marciszewski) almost disjoint families with some combinatorial property that has applications in functional analysis. We are looking for the minimal cardinality of m.a.d. family with property (R). It turns out that this cardinal is not greater than $non(\mathcal{N})$ the uniformity of null sets.

## Jacek Tryba: Homogeneity of ideals

Tuesday, March 6, 2018, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Jacek Tryba (University of Gdansk)

Title: Homogeneity of ideals

Abstract:

The homogeneity family of the ideal $\mathcal{I}$ is a family of subsets such that the restriction of $\mathcal{I}$ to this subset is isomorphic to $\mathcal{I}$.
We say that an ideal $\mathcal{I}$ is homogeneous if all $\mathcal{I}$-positive sets belong to the homogeneity family of $\mathcal{I}$. We investigate basic properties of this notion, give examples of homogeneous ideals and present some applications to ideal
convergence.
Moreover, we present connections between the homogeneity families and the notion of bi-$\mathcal{I}$-invariant functions introduced by Balcerzak, Głąb and Swaczyna and give answers to several questions related to this topic.