Aleksander Cieślak
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
Speaker: Aleksander Cieślak
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.
Location: room 601, Mathematical Institute, University of Wroclaw
Speaker: Aleksander Cieślak
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
Speaker: Aleksander Cieślak
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.
Location: room 601, Mathematical Institute, University of Wroclaw
Speaker: Aleksander Cieślak
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
Speaker: Aleksander Cieślak
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 \(\omega\).
Location: room 601, Mathematical Institute, University of Wroclaw
Speaker: Aleksander Cieślak
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
Speaker: Aleksander Cieślak
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.
Location: room A.2.22 C-19
Speaker: Aleksander Cieślak
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
Speaker: Aleksander Cieślak
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.
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Aleksander Cieślak
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
Speaker: Aleksander Cieślak
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.
Location: Zoom.us: if you want to participate please contact organizers
Speaker: Aleksander Cieślak
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
Speaker: Aleksander Cieślak
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.
Location: Zoom.us: if you want to participate please contact organizers
Speaker: Aleksander Cieślak
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
Speaker: Aleksander Cieślak
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.
Room: D1-215
Speaker: Aleksander Cieślak
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
Speaker: Aleksander Cieślak
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.
Room: D1-215
Speaker: Aleksander Cieślak
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
Speaker: Aleksander Cieślak
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.
Room: D1-215
Speaker: Aleksander Cieślak
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
Speaker: Aleksander Cieślak
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".
Room: D1-215
Speaker: Aleksander Cieślak
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
Speaker: Aleksander Cieślak
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.
Room: D1-215
Speaker: Aleksander Cieślak
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
Speaker: Aleksander Cieślak
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.
Room: D1-215
Speaker: Aleksander Cieślak
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
Room: D1-215
Speaker: Aleksander Cieślak
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.
Room: D1-215
Speaker: Aleksander Cieślak
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.