May 2015

Szymon Żeberski: Applications of Shoenfield Absoluteness Lemma

Tuesday, June 2, 2015 17:15

Room: D1-215

Speaker:
Szymon Żeberski

Title: Applications of Shoenfield Absoluteness Lemma

Abstract. We will recall Shoenfield Absoluteness Lemma about \(\Sigma^1_2\) sentences. We will show applications of this theorem connected to topological and algebraic structure of Polish spaces in publications co-authored by the speaker.

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

Room: D1-215

Speaker:
Inna Pozdniakova

Title: On monoids of 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

Room: D1-215

Speaker:
Taras Banakh

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.

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

Tuesday, May 19, 2015 17:15

Room: D1-215

Speaker:
Jarosław Swaczyna

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

Abstract. Let \(g: \omega \rightarrow [0, \infty)\). We say that \(A \subset \omega\) has \(g\)-density zero, if \(\lim_{n \rightarrow \infty} \frac{A \cap n}{g(n)} = 0\). 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.

Tomasz Żuchowski: Tukey types of orthogonal ideals

Tuesday, May 12, 2015 17:15

Room: D1-215

Speaker:
Tomasz Żuchowski

Title: Tukey types of orthogonal ideals

Abstract. A partial order \(P\) is Tukey reducible to partial order \(Q\) when there exists a function \(f:P\to Q\) 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 \(\mathbb{N}\) with the inclusion order and other partial orders.