November 2023

# Jarosław Swaczyna: Zoo of ideal Schauder bases

24/11/23 12:02

Tuesday, November 28, 2023 17:00

*Location:*room 601, Mathematical Institute, University of Wroclaw*Jarosław Swaczyna (Łódź University of Technology)*

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*Title*: Zoo of ideal Schauder bases*Abstract*: Given a Banach space \(X\), sequence \((e_n)\) of its elements and an ideal \(I\) on natural numbers we say that \((e_n)\) is an \(I\)-Schauder base if for every \(x \in X\) there exists unique sequence of scalars \(\alpha_n\) such that series of \(\alpha_n e_n\) is \(I\)-convergent to \(X\). In such a case one may also consider coordinate functionals \(e_n^\star\). About ten years ago Kadets asked if those functionals are necessarily continuous at least for some nice ideals, e.g. the ideal of sets of density zero. During my talk I will present an answer to this question obtained jointly with Tomasz Kania and Noe de Rancourt. I will also present some examples of ideal Schauder bases which are not the classical ones. Second part will be based on ongoing work with Adam Kwela.# Diego Mejia: Ultrafilters and finitely additive measures in forcing theory

17/11/23 11:55

Tuesday, November 21, 2023 17:00

*Location:*room 601, Mathematical Institute, University of Wroclaw*Diego Mejia (Shizuoka University)*

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*Title*: Ultrafilters and finitely additive measures in forcing theory*Abstract*: We show how ultrafilters and finitely additive measures on the power set of the natural numbers can be used in forcing theory to construct models of ZFC where many classical cardinal characteristics have pairwise different values. Very recent remarkable results, like the consistency of Cichon's maximum (the constellation of Cichon's diagram where all non-dependent cardinal characteristics are pairwise different), have been proved using such techniques.# 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.# Zdenek Silber: A countably tight P(K) space admitting a nonseparable measure

06/11/23 10:44

Tuesday, November 7, 2023 17:00

It was proven by Fremlin that, under Martin's axiom and negation of continuum hypothesis, for a compact Hausdorff space \(K\) the existance of a Radon probability of uncountable type is equivalent to the exitence of a continuous surjection from \(K\) onto \([0,1]^{\omega_1}\). Hence, under such assumptions, countable tightness of \(P(K)\) implies that there is no Radon probability on \(K\) which has uncountable type. Later, Plebanek and Sobota showed that, without any additional set-theoretic assumptions, countable tightness of \(P(K\times K)\) implies that there is no Radon probability on \(K\) which has uncountable type as well. It is thus natural to ask whether the implication "\(P(K)\) has countable tightness implies every Radon probability on \(K\) has countable type" holds in ZFC.

I will present our joint result with Piotr Koszmider that under diamond principle there is a compact Hausdorff space \(K\) such that \(P(K)\) has countable tightness but there exists a Radon probability on \(K\) of uncountable type.

*Location:*room 601, Mathematical Institute, University of Wroclaw*Zdenek Silber (IM PAN)*

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*Title*: A countably tight P(K) space admitting a nonseparable measure*Abstract*: In the talk we focus on the relation of countable tightness of the space \(P(K)\) of Radon probabilty measures on a compact Hausdorff space \(K\) and of existence of measures in \(P(K)\) that have uncountable Maharam type. Recall that a topological space \(X\) has countable tightness if any element of the closure of a subset \(A\) of \(X\) lies in the closure of some countable subset of \(A\). A Maharam type of a Radon probability measure mu is the density of the Banach space \(L_1(\mu)\).It was proven by Fremlin that, under Martin's axiom and negation of continuum hypothesis, for a compact Hausdorff space \(K\) the existance of a Radon probability of uncountable type is equivalent to the exitence of a continuous surjection from \(K\) onto \([0,1]^{\omega_1}\). Hence, under such assumptions, countable tightness of \(P(K)\) implies that there is no Radon probability on \(K\) which has uncountable type. Later, Plebanek and Sobota showed that, without any additional set-theoretic assumptions, countable tightness of \(P(K\times K)\) implies that there is no Radon probability on \(K\) which has uncountable type as well. It is thus natural to ask whether the implication "\(P(K)\) has countable tightness implies every Radon probability on \(K\) has countable type" holds in ZFC.

I will present our joint result with Piotr Koszmider that under diamond principle there is a compact Hausdorff space \(K\) such that \(P(K)\) has countable tightness but there exists a Radon probability on \(K\) of uncountable type.

# Witold Marciszewski: On \(\omega\)-Corson compact spaces and related classes of Eberlein compacta

02/11/23 10:39

Friday, November 3, 2023 16:15

A compact space \(K\) is \(\omega\)-Corson compact if, for some set \(\Gamma\), \(K\) is homeomorphic to a subset of the \(\sigma\)-product of real lines \(\sigma(\mathbb{R}^\Gamma)\), i.e. the subspace of the product \(\mathbb{R}^\Gamma\) consisting of functions with finite supports. Clearly, every \(\omega\)-Corson compact space is Eberlein compact.

We will present a characterization of \(\omega\)-Corson compact spaces, and some other results concerning this class of spaces and related classes of Eberlein compacta.

This is a joint research with Grzegorz Plebanek and Krzysztof Zakrzewski, see

https://arxiv.org/abs/2107.02513

*Location:*room 601, Mathematical Institute, University of Wroclaw*Witold Marciszewski (MIM UW)*

Speaker:Speaker:

*Title*: On \(\omega\)-Corson compact spaces and related classes of Eberlein compacta*Abstract*: Recall that a compact space \(K\) is Eberlein compact if it can be embedded into some Banach space X equipped with the weak topology; equivalently, for some set \(\Gamma\), \(K\) can be embedded into the space \(c_0( \Gamma)\), endowed with the pointwise convergence topology.A compact space \(K\) is \(\omega\)-Corson compact if, for some set \(\Gamma\), \(K\) is homeomorphic to a subset of the \(\sigma\)-product of real lines \(\sigma(\mathbb{R}^\Gamma)\), i.e. the subspace of the product \(\mathbb{R}^\Gamma\) consisting of functions with finite supports. Clearly, every \(\omega\)-Corson compact space is Eberlein compact.

We will present a characterization of \(\omega\)-Corson compact spaces, and some other results concerning this class of spaces and related classes of Eberlein compacta.

This is a joint research with Grzegorz Plebanek and Krzysztof Zakrzewski, see

https://arxiv.org/abs/2107.02513