Aleksander Cieślak

# Aleksander Cieślak: The splitting ideal

17/06/24 20:27

Tuesday, June 18, 2024 17:15

*Location:*A.4.1 C-19*Aleksander Cieślak*

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*Title*: The splitting ideal*Abstract*: We will investigate the cardinal invariants and the Katetov position of certain ideal on \(\omega\). As a result we will obtain a new upper boundary of the covering number of the density zero ideal.# Aleksander Cieślak: Antichain numbers and other cardinal invariants of ideals

19/12/23 08:33

Tuesday, December 19, 2023 17:00

*Location:*room 601, Mathematical Institute, University of Wroclaw*Aleksander Cieślak*

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*Title*: Antichain numbers and other cardinal invariants of ideals*Abstract*: Suppose that \(J\) is an ideal on \(\omega\). The \(J\)-antichain number is the smallest cardinality of a maximal antichain in the algebra \(P(\omega)\) modulo \(J\). We will estimate the \(J\)-antichain numbers for various Borel ideals. To do so, we will focus on two features of ideals which are crucial for our construction. First one is a cardinal invariant of an ideal \(J\) which lies (strictly) in between \(\rm{add}^*(J)\) and \(\rm{cov}^*(J)\). The second one is a property which allows diagonalisation of antichains and which is similar (but not equal) to being a \(P^+\) ideal.# Aleksander Cieślak: Cofinalities of tree ideals and the shrinking property II

13/11/23 12:29

Tuesday, November 14, 2023 17:00

*Location:*room 601, Mathematical Institute, University of Wroclaw*Aleksander Cieślak*

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*Title*: Cofinalities of tree ideals and the shrinking property II*Abstract*: ILast time, given a tree type \(\mathbb{T}\), we investigated a cardinal invariant \(is(\mathbb{T})\) called "Incompatibility Shrinking Number". It was mentioned that the assumption \(is(\mathbb{T})=\mathfrak c \) implies that \( cof(t^0)>\mathfrak c\) and that \(is(\mathbb{T})\) falls in between the additivity and the covering number of the borel part \(t^0_{Bor}\). We will focus on calculating these two for various Borel ideals.# Aleksander Cieślak: Cofinalities of tree ideals and Shrinking Property

30/10/23 10:37

Tuesday, October 31, 2023 17:00

*Location:*room 601, Mathematical Institute, University of Wroclaw*Aleksander Cieślak*

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*Title*: Cofinalities of tree ideals and Shrinking Property*Abstract*: If \(\mathcal{T}\) is a collection of trees on \(\omega^\omega\), then we define the tree ideal \(t_0\) as a collection of these \(X\subset \omega^\omega\) such that each \(T\in \mathcal{T}\) has a subtree \(S\in \mathcal{T}\) which shares no branches with \(X\). We will be interested in the cofinalities of the tree ideals. In particular, we will focus on the condition, called "Incompatibility Shrinking Property", which implies that \(cof(t_0)>\mathfrak c\). We will consider under what assumptions this property is satisfied for the two types of trees, which are Laver and Miller trees which split positively according to some fixed ideal on \(\omega\).# Aleksander Cieślak: Trees and Cohen reals

01/03/23 08:02

Tuesday, March 7, 2023 17:00

*Location:*room**A.2.22 C-19***Aleksander Cieślak*

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*Title*: Trees and Cohen reals*Abstract*: We will discuss adding Cohen reals for various types of trees on Baire and Cantor space. We will distinguish that these Cohen reals can be added in a 'strong' or 'weak' way. While the former has rather pathological consequences, the latter allows certain control over the ideal related to the tree type.# Aleksander Cieślak: Marczewski ideals of product trees

24/04/22 15:37

Tuesday, April 26, 2022 17:00

*Location:*room 605, Mathematical Institute, University of Wroclaw*Aleksander Cieślak*

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*Title*: Marczewski ideals of product trees*Abstract*: We investigate Marczewski style ideals associated with the product of two tree-like forcing notions and compare these to original, one dimensional ones.# Aleksander Cieślak: Full-splitting Miller trees and Cohen reals

10/10/21 21:34

Tuesday, October 12, 2021 17:00

*Location:***Zoom.us**: if you want to participate please contact organizers*Aleksander Cieślak*

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*Title*: Full-splitting Miller trees and Cohen reals*Abstract*: We will investigate tree ideal \(fm_0\) related to certain widening of Miller trees. This - so called - full Miller trees consist in taking the entire omega on split nodes instead of just its infinite subset. We will investigate cardinal invariants of \(fm_0\) and its relation to meager sets.# Aleksander Cieślak: Forcing with wider Silver

20/04/20 12:30

Tuesday, April 21, 2020 17:15

*Location:***Zoom.us**: if you want to participate please contact organizers*Aleksander Cieślak*

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*Title*: Forcing with wider Silver*Abstract*. We are going to establish basic properties of diagonal version of Silver forcing. Such forcing consists of partial functions \(p:\omega\rightarrow\omega\) with infinite codomain and \(p(n)<=n\) for each \(n\in dom(p)\). Cardinal characteristics of continuum will be calculated.# Aleksander Cieślak: Ideals of subsets of plane

10/10/17 05:57

Tuesday, October 10, 2017 17:15

*Room:*D1-215*Aleksander Cieślak*

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*Title*: Ideals of subsets of plane*Abstract*. For given two ideals I and J of subsets of Polish space X we define a Fubini product \(I \times J\) as all these subsets of plane \(X^2\) which can be covered by a Borel set B such that I-almost all its vertical sections are J-small. We will investigate how properties of factors influence properties of product.# Aleksander Cieślak: Cohen-stable families of subsets of integers

12/06/17 09:19

Tuesday, June 13, 2017 17:15

*Room:*D1-215*Aleksander Cieślak*

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*Title*: Cohen-stable families of subsets of integers*Abstract*. A mad family is Cohen-stable if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-stable. We will find condition necessary and sufficient for mad family to be Cohen-unstabe and investigate when such family exist.# Aleksander Cieślak: Indescructible tower

10/04/17 08:50

Tuesday, April 11, 2017 17:15

*Room:*D1-215*Aleksander Cieślak*

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*Title*: Indescructible tower*Abstract*. Following the Kunen's construction of m.a.d. family which is indestructible over adding \(\omega_2\) Cohen reals we provide analogous construction for indestructibe tower.# Aleksander Cieślak: Strongly meager sets and subsets of the plane

18/12/16 11:14

Tuesday, December 20, 2016 17:15

*Room:*D1-215*Aleksander Cieślak*

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*Title*: Strongly meager sets and subsets of the plane*Abstract*. We will show some results proved by J. Pawlikowski in "Strongly meager sets and subsets of the plane".# Aleksander Cieślak: Nonmeasurable images in Polish space with respect to selected sigma ideals

10/10/16 11:06

Tuesday, October 11, 2016 17:15

*Room:*D1-215*Aleksander Cieślak*

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*Title*: Nonmeasurable images in Polish space with respect to selected sigma ideals*Abstract*. We present results on nonmeasurability (with respect to a selected σ-ideal on a Polish space) of images of functions defined on Poilish spaces. In particular, we give a positive answer to the following question: Is there a subset of the unit disc in the real plane such that continuum many projections onto lines are Lebesgue measurable and continuum many projections are not? Results were obtained together with Robert Rałowski.# Aleksander Cieślak: Filters and sets of Vitali's type

19/02/16 18:43

Tuesday, February 23, 2016 17:15

*Room:*D1-215*Aleksander Cieślak*

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*Title*: Filters and sets of Vitali's type*Abstract*. In construction of classical Vitali set on \(\{0,1\}^{\omega}\) we use filter of cofinite sets to define rational numbers. We replece cofinite filter by any nonprincipal filter on \(\omega\) and ask some questions about measurability and cardinality of selectors and equevalence classes.# Aleksander Cieślak: On nonmeasurable subsets of \(\mathbb{R}\) and \(\mathbb{R}^2\)

21/10/15 21:14

Tuesday, October 27, 2015 17:15

Moreover, I will discuss the existence of large midpoint-free subsets of arbitrary subset of the real line.

*Room:*D1-215*Aleksander Cieślak*

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*Title*: On nonmeasurable subsets of \(\mathbb{R}\) and \(\mathbb{R}^2\)*Abstract*. I would like to present some results connected with the existence of a subset \(X\) of the square \([0,1]^2\) with the property that for any line \(L\) outside \([0,1]^2\) the projection \(\pi_L[X]\) is completely nonmeasurable in some interval with respect to selected \(\sigma\)-ideal with Borel base on the line \(L\).Moreover, I will discuss the existence of large midpoint-free subsets of arbitrary subset of the real line.