https://settheory.pwr.edu.pl/index.htmlTalks on the set theory seminar.enen2024-06-17T20:27:36+02:00 hourly 1 2000-01-01T12:00+00:00 Mon, 30 Jun 2014 17:35:00 +0200Talks2024-06-17T20:27:36+02:00https://settheory.pwr.edu.pl/blog/files/e03522d43bab89296eda660dbfc5668d-202.php#unique-entry-id-202https://settheory.pwr.edu.pl/blog/files/e03522d43bab89296eda660dbfc5668d-202.php#unique-entry-id-202
Location: A.4.1 C-19

Speaker: Aleksander Cieślak

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.]]>Talks2024-06-12T12:16:09+02:00https://settheory.pwr.edu.pl/blog/files/b269e6fcc5f69f53f619ebb0f0bd9dc0-201.php#unique-entry-id-201https://settheory.pwr.edu.pl/blog/files/b269e6fcc5f69f53f619ebb0f0bd9dc0-201.php#unique-entry-id-201
Location: A.4.1 C-19

Speaker: Jadwiga Świerczyńska

Title: On Q- and selective measures

Abstract: We will present some generalizations of well-known definitions of types of ultrafilters to the realm of finitely additive measures on \(\omega\). We will show a few results similar to the ones for ultrafilters: measure is selective if and only if it is a P-measure and a Q-measure, and that selective measures (Q-measures, respectively) are minimal in the Rudin-Keisler (Rudin-Blass) ordering. We will also show an example of a selective non-atomic measure. The second part will be focused on the integration: we will briefly describe Lebesgue integral with respect to finitely additive measures on \(\omega\) and prove that it is a generalization of an ultralimit. Finally, we will present an idea of further generalizations of these definitions for functionals on \(\ell^{\infty}\).]]>Talks2024-06-04T12:09:49+02:00https://settheory.pwr.edu.pl/blog/files/a1516225b4f43b27bb69b2647139ecc2-200.php#unique-entry-id-200https://settheory.pwr.edu.pl/blog/files/a1516225b4f43b27bb69b2647139ecc2-200.php#unique-entry-id-200
Location: A.4.1 C-19

Speaker: Andres Uribe-Zapata (TU Wien)

Title: Finitely additive measures on Boolean algebras: freeness and integration

Abstract: In this talk, we present an integration theory with respect to finitely additive measures on a field of sets \(\mathcal{B} \subseteq \mathcal(X)\) for some non-empty set \(X\). For this, we start by reviewing some fundamental properties of finitely additive measures on Boolean algebras. Later, we present a definition of the integral in this context and some basic properties of the integral and the integrability. We also study integration over subsets of \(X\) to introduce the Jordan algebra and compare the integration on this new algebra with the integration on \(\mathcal{B}\). Finally, we say that a finitely additive measure on \(\mathcal{B}\) is free if \(\mathcal{B}\) contains any finite subset of \(X\) and its measure is zero. We close the talk by providing some characterizations of free finitely additive measures.

This is a joint work with Miguel A. Cardona and Diego A. Mejía.

References:

[CMU] Miguel A. Cardona, Diego A. Mejía and Andrés F. Uribe-Zapata. Finitely additive measures on Boolean algebras. In Preparation.

[UZ23] Andrés Uribe-Zapata. Iterated forcing with finitely additive measures: applications of probability to forcing theory. Master’s thesis, Universidad Nacional de Colombia, sede Medellín, 2023. https://shorturl.at/sHY59.]]>Talks2024-04-22T08:20:16+02:00https://settheory.pwr.edu.pl/blog/files/d07032b4ea1e67348082bc088af55adb-199.php#unique-entry-id-199https://settheory.pwr.edu.pl/blog/files/d07032b4ea1e67348082bc088af55adb-199.php#unique-entry-id-199
Location: A.4.1 C-19

Speaker: Tomasz Żuchowski

Title: The Nikodym property and filters on \(\omega\). Part II

Abstract: For a free filter \(F\) on \(\omega\), we consider the space \(N_F=\omega\cup\{p_F\}\), where every element of \(\omega\) is isolated and open neighborhoods of \(p_F\) are of the form \(A\cup\{p_F\}\) for \(A\in F\).
In this talk we will study the family \(\mathcal{AN}\) of such ideals \(\mathcal{I}\) on \(\omega\) that the space \(N_{\mathcal{I}^*}\) carries a sequence \(\langle\mu_n\colon n\in\omega\rangle\) of finitely supported signed measures satisfying \(\|\mu_n\|\rightarrow\infty\) and \(\mu_n(A)\rightarrow 0\) for every \(A\in Clopen(N_{\mathcal{I}^*})\). If \(\mathcal{I}\in\mathcal{AN}\) and \(N_{\mathcal{I}^*}\) is embeddable into the Stone space \(St(\mathcal{A})\) of a given Boolean algebra \(\mathcal{A}\), then \(\mathcal{A}\) does not have the Nikodym property.
]]>Talks2024-04-13T07:08:33+02:00https://settheory.pwr.edu.pl/blog/files/140d47cbf98dcf66a24b97ce27bf2d3b-198.php#unique-entry-id-198https://settheory.pwr.edu.pl/blog/files/140d47cbf98dcf66a24b97ce27bf2d3b-198.php#unique-entry-id-198
Location: A.4.1 C-19

Speaker: Krzysztof Zakrzewski (MIM UW)

Title: Function spaces on Corson-like compacta

Abstract: Recall that a compact space is Eberlein compact if it is homeomorphic to a subspace of some Banach space equipped with the weak topology. A compact space is \(\omega\)-Corson compact if it embeds into a \(\sigma\)-product of real lines, that is a subspace of the product \(R^{\Gamma}\) consisting of sequences with finitely many nonzero coordinates for some set \(\Gamma\).
Every \(\omega\)-Corson compact space is Eberlein compact. For a Tichonoff space \(X\), let \(C_p(X)\) denote the space of real continuous functions on \(X\) endowed with the pointwise convergence topology.
During the talk we will show that the class \(\omega\)-Corson compact spaces \(K\) is invariant under linear homeomorphism of function spaces \(C_p(K)\) and other related results.]]>Talks2024-04-04T11:16:26+02:00https://settheory.pwr.edu.pl/blog/files/bbce80b8102df0fbcb39893e873b7cdc-197.php#unique-entry-id-197https://settheory.pwr.edu.pl/blog/files/bbce80b8102df0fbcb39893e873b7cdc-197.php#unique-entry-id-197
Location: A.4.1 C-19

Speaker: Jakub Rondos (University of Vienna)

Title: Topological properties of compact spaces K that are preserved by isomorphisms of C(K)"
will be presented by

Abstract: In the talk, we present some newly discovered properties of compact Hausdorff spaces that are preserved by isomorphisms of their Banach spaces of continuous functions. ]]>Talks2024-03-25T11:14:26+01:00https://settheory.pwr.edu.pl/blog/files/66f1a622f23ac6f120c882a218946906-196.php#unique-entry-id-196https://settheory.pwr.edu.pl/blog/files/66f1a622f23ac6f120c882a218946906-196.php#unique-entry-id-196
Location: A.4.1 C-19

Speaker: Tomasz Żuchowski

Title: The Nikodym property and filters on \(\omega\). Part I

Abstract: For a free filter \(F\) on \(\omega\), we consider the space \(N_F=\omega\cup\{p_F\}\), where every element of \(\omega\) is isolated and open neighborhoods of \(p_F\) are of the form \(A\cup\{p_F\}\) for \(A\in F\).
In this talk we will study the family \(\mathcal{AN}\) of such ideals \(\mathcal{I}\) on \(\omega\) that the space \(N_{\mathcal{I}^*}\) carries a sequence \(\langle\mu_n\colon n\in\omega\rangle\) of finitely supported signed measures satisfying \(\|\mu_n\|\rightarrow\infty\) and \(\mu_n(A)\rightarrow 0\) for every \(A\in Clopen(N_{\mathcal{I}^*})\). If \(\mathcal{I}\in\mathcal{AN}\) and \(N_{\mathcal{I}^*}\) is embeddable into the Stone space \(St(\mathcal{A})\) of a given Boolean algebra \(\mathcal{A}\), then \(\mathcal{A}\) does not have the Nikodym property.]]>Talks2024-03-18T11:11:04+01:00https://settheory.pwr.edu.pl/blog/files/69d8c8d53649e8996fac62de9d43daa1-195.php#unique-entry-id-195https://settheory.pwr.edu.pl/blog/files/69d8c8d53649e8996fac62de9d43daa1-195.php#unique-entry-id-195
Location: A.4.1 C-19

Speaker: Piotr Szewczak (UKSW)

Title: Perfectly meager sets in the transitive sense and the Hurewicz property

Abstract: We work in the Cantor space with the usual group operation +. A set X is perfectly meager in the transitive sense if for any perfect set P there is an F-sigma set F containing X such that for every point t the intersection of t+F and P is meager in the relative topology of P. A set X is Hurewicz if for any sequence of increasing open covers of X one can select one set from each cover such that the chosen sets formulate a gamma-cover of X, i.e., an infinite cover such that each point from X belongs to all but finitely many sets from the cover. Nowik proved that each Hurewicz set which cannot be mapped continuously onto the Cantor set is perfectly meager in the transitive sense. We answer a question of Nowik and Tsaban, whether of the same assertion holds for each Hurewicz set with no copy of the Cantor set inside. We solve this problem, under CH, in the negative.
This is a joint work with Tomasz Weiss and Lyubomyr Zdomskyy.
The research was funded by the National Science Centre, Poland and the Austrian Science Found under the Weave-UNISONO call in the Weave programme, project: Set-theoretic aspects of topological selections 2021/03/Y/ST1/00122]]>Talks2024-03-05T07:47:50+01:00https://settheory.pwr.edu.pl/blog/files/542e7adf4bf94a6acf1a9e934236fcc7-194.php#unique-entry-id-194https://settheory.pwr.edu.pl/blog/files/542e7adf4bf94a6acf1a9e934236fcc7-194.php#unique-entry-id-194
Location: A.4.1 C-19

Speaker: Agnieszka Widz

Title: Random graph

Abstract: The Random Graph can be generated almost surely by connecting vertices with a fixed probability \(p\in(0,1)\), independently of other pairs. In my talk, I will recall the construction and explore interesting properties of the Random Graph, investigating the impact of varying probabilities for each edge. Specifically, I will characterize sequences \((p_n)_{n\in\mathbb{N}}\) for which there exists a bijection \(f\) between pairs of vertices in \(\mathbb{N}\), such that if we connect vertices \(v\) and \(w\) with probability \(p_{f(\{v,w\})}\), the Random Graph emerges almost surely.]]>Talks2024-02-26T15:32:44+01:00https://settheory.pwr.edu.pl/blog/files/5d80822e20df09a58bc917e95e6418a2-193.php#unique-entry-id-193https://settheory.pwr.edu.pl/blog/files/5d80822e20df09a58bc917e95e6418a2-193.php#unique-entry-id-193
Location: A.4.1 C-19

Speaker: Grzegorz Plebanek

Title: Aftermath of the Winter School

Abstract: We shall discuss two problem on measures on compact spaces posed by Jiri Spurny. ]]>Talks2024-01-22T07:12:18+01:00https://settheory.pwr.edu.pl/blog/files/730403defc96ed542de90672543467f0-192.php#unique-entry-id-192https://settheory.pwr.edu.pl/blog/files/730403defc96ed542de90672543467f0-192.php#unique-entry-id-192
Location: room 601, Mathematical Institute, University of Wroclaw

Speaker: Łukasz Mazurkiewicz

Title: Analytic families of trees

Abstract: Every tree can be seen as a point in a space \(P(2^<\omega)\) or \(P(\omega^<\omega)\). Therefore, families of trees are subsets of these "incarnations" of Cantor space and, as such, can be analyzed from the perspective of descriptive complexity. In this talk I would like to explore some classical families of trees with some focus put on the ones, which are analytic complete.
]]>Talks2024-01-08T05:43:01+01:00https://settheory.pwr.edu.pl/blog/files/1e5c4baf55676a3648ee42fbac1b034c-191.php#unique-entry-id-191https://settheory.pwr.edu.pl/blog/files/1e5c4baf55676a3648ee42fbac1b034c-191.php#unique-entry-id-191
Location: room 601, Mathematical Institute, University of Wroclaw

Speaker: Piotr Borodulin-Nadzieja

Title: Fams on omega

Abstract: I will review some recent results about finitely additive measures on \(\omega\). In particular, I will talk about some new examples of such measures, motivated by the problem if there is a P-measure in the Silver model. Joint work with Jonathan Cancino and Adam Morawski.]]>Talks2023-12-19T08:33:58+01:00https://settheory.pwr.edu.pl/blog/files/7dc8d8522fda6101f23ffd3198acc987-190.php#unique-entry-id-190https://settheory.pwr.edu.pl/blog/files/7dc8d8522fda6101f23ffd3198acc987-190.php#unique-entry-id-190
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.]]>Talks2023-12-05T11:08:20+01:00https://settheory.pwr.edu.pl/blog/files/b6c39bbd95f37b35cf5e47d6f79fe2de-189.php#unique-entry-id-189https://settheory.pwr.edu.pl/blog/files/b6c39bbd95f37b35cf5e47d6f79fe2de-189.php#unique-entry-id-189
Location: room 601, Mathematical Institute, University of Wroclaw

Speaker: Daria Perkowska

Title: Non-meager filters

Abstract: In the talk I will consider filters on \(\omega\) in the measurability (and complexity) context. Also, one can distinguish some natural subclasses of non-meager filters. We say that a filter \(F\) is ccc if \(P(\omega) /F\) is ccc. Similarly, we say that a filter supports a measure if there is a probability measure \(\mu\) on \(\omega\) such that \(F = \{A: \mu(A)=1\}\). I will show that every ultrafilter supports a measure, every measure supporting filter is ccc and every ccc filter is non-meager. So, one can think about these notions as forming some hierarchy of complexity of filters. This hierarchy is strict. Next I will show that for every ultrafilter from the forcing extension (by \(\mathbb{A}\)), there is a ground model filter F such that the ultrafilter extends F and there is an injective Boolean homomorphism \(\varphi: P(\omega) /F \to \mathbb{A}\).]]>Talks2023-11-24T12:02:22+01:00https://settheory.pwr.edu.pl/blog/files/316f48e7abb0888552fd681be60a6601-188.php#unique-entry-id-188https://settheory.pwr.edu.pl/blog/files/316f48e7abb0888552fd681be60a6601-188.php#unique-entry-id-188
Location: room 601, Mathematical Institute, University of Wroclaw

Speaker: Jarosław Swaczyna (Łódź University of Technology)

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.
]]>Talks2023-11-17T11:55:58+01:00https://settheory.pwr.edu.pl/blog/files/75d2a4a2ced845e66c0aa2e3a648006a-187.php#unique-entry-id-187https://settheory.pwr.edu.pl/blog/files/75d2a4a2ced845e66c0aa2e3a648006a-187.php#unique-entry-id-187
Location: room 601, Mathematical Institute, University of Wroclaw

Speaker: Diego Mejia (Shizuoka University)

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.]]>Talks2023-11-13T12:29:52+01:00https://settheory.pwr.edu.pl/blog/files/e220624c662915a2e3e5b009f5f611e6-186.php#unique-entry-id-186https://settheory.pwr.edu.pl/blog/files/e220624c662915a2e3e5b009f5f611e6-186.php#unique-entry-id-186
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.]]>Talks2023-11-06T10:44:13+01:00https://settheory.pwr.edu.pl/blog/files/a1e135d7b90a61e585a1e204d35626df-185.php#unique-entry-id-185https://settheory.pwr.edu.pl/blog/files/a1e135d7b90a61e585a1e204d35626df-185.php#unique-entry-id-185
Location: room 601, Mathematical Institute, University of Wroclaw

Speaker: Zdenek Silber (IM PAN)

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.
]]>Talks2023-11-02T10:39:19+01:00https://settheory.pwr.edu.pl/blog/files/4096d12b550620c035f21a3814e30f3f-184.php#unique-entry-id-184https://settheory.pwr.edu.pl/blog/files/4096d12b550620c035f21a3814e30f3f-184.php#unique-entry-id-184
Location: room 601, Mathematical Institute, University of Wroclaw

Speaker: Witold Marciszewski (MIM UW)

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
]]>Talks2023-10-30T10:37:23+01:00https://settheory.pwr.edu.pl/blog/files/823a41a065b2f7cc9fc9cbec5a633565-183.php#unique-entry-id-183https://settheory.pwr.edu.pl/blog/files/823a41a065b2f7cc9fc9cbec5a633565-183.php#unique-entry-id-183
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 \(\omega\).]]>Talks2023-10-23T10:34:42+02:00https://settheory.pwr.edu.pl/blog/files/73bef591f4bf8b69ee00528022ab3f65-182.php#unique-entry-id-182https://settheory.pwr.edu.pl/blog/files/73bef591f4bf8b69ee00528022ab3f65-182.php#unique-entry-id-182
Location: room 601, Mathematical Institute, University of Wroclaw

Speaker: Maciej Korpalski

Title: Straightening almost chains into barely altenating ones

Abstract: Consider an almost chain \(\mathcal{A} = \{A_x \subset \omega: x \in X\}\) for some separable linearly ordered set \(X\). Such a chain is barely alternating if for all \(n \in \omega\) we cannot find elements \(x_1 < x_2 < x_3 < x_4\) in \(X\) satisfying \(n \in A_{x_1}, A_{x_3}\), \(n \notin A_{x_2}, A_{x_4}\). We will show that under \(MA(\kappa)\), if \(|X| \leq \kappa\), then we can straighten our almost chain \(\mathcal{A}\) into a barely alternating one by changing at most finitely many elements in each set \(A_x\).]]>Talks2023-10-16T10:33:06+02:00https://settheory.pwr.edu.pl/blog/files/2f0faad23ad395e0a8288a7779fbcc0d-181.php#unique-entry-id-181https://settheory.pwr.edu.pl/blog/files/2f0faad23ad395e0a8288a7779fbcc0d-181.php#unique-entry-id-181
Location: room 601, Mathematical Institute, University of Wroclaw

Speaker: Viktoriia Brydun (Ivan Franko Lviv National University)

Title: Monad on FMS(•) (Fuzzy Metric Spaces Category)

Abstract: Monad on FMS(•) (Fuzzy Metric Spaces Category)]]>Talks2023-10-09T10:23:06+02:00https://settheory.pwr.edu.pl/blog/files/831eb14df1f94688fd076d79968a1f72-180.php#unique-entry-id-180https://settheory.pwr.edu.pl/blog/files/831eb14df1f94688fd076d79968a1f72-180.php#unique-entry-id-180
Location: room 601, Mathematical Institute, University of Wroclaw

Speaker: Arturo Martinez

Title: Cardinal invariants related to free sets

Abstract: (Joint work with T. Żuchowski) Given a function \(f\) without fixed points, an infinite set \(A\) is called free for \(f\) if \(f[A] \cap A = \emptyset\). In this talk we will discuss some cardinal invariants related to families of free sets and we will discuss their relation between some cardinal invariants related to category and measure.]]>Talks2023-06-05T10:14:43+02:00https://settheory.pwr.edu.pl/blog/files/7da78814469da3f2e96309ece1236431-179.php#unique-entry-id-179https://settheory.pwr.edu.pl/blog/files/7da78814469da3f2e96309ece1236431-179.php#unique-entry-id-179
Location: room A.4.1 C-19

Speaker: Paweł Krupski

Title: Remarks and questions on hyperspaces of knots: Borel complexity and local contractibility

Abstract: Remarks and questions on hyperspaces of knots: Borel complexity and local contractibility]]>Talks2023-05-31T10:19:08+02:00https://settheory.pwr.edu.pl/blog/files/ef5b24a7809a8c25eb4fdf790c21e61f-178.php#unique-entry-id-178https://settheory.pwr.edu.pl/blog/files/ef5b24a7809a8c25eb4fdf790c21e61f-178.php#unique-entry-id-178
Location: room A.4.1 C-19

Speaker: Piotr Szewczak (UKSW)

Title: Totally imperfect Menger sets

Abstract: A set of reals \(X\) is Menger if for any countable sequence of open covers of \(X\) one can pick finitely many elements from every cover in the sequence such that the chosen sets cover \(X\). Any set of reals of cardinality smaller than the dominating number d is Menger and there is a non-Menger set of cardinality \(d\). By the result of Bartoszyński and Tsaban, in ZFC, there is a totally imperfect (with no copy of the Cantor set inside) Menger set of cardinality \(d\). We solve a problem, whether there is such a set of cardinality continuum. Using an iterated Sacks forcing and topological games we prove that it is consistent with ZFC that \(d \lt c\) and each totally imperfect Menger set has cardinality less or equal than \(d\).

This is a joint work with Valentin Haberl and Lyubomyr Zdomskyy.

The research was funded by the National Science Centre, Poland and the Austrian Science Found under the Weave-UNISONO call in the Weave programme, project: Set-theoretic aspects of topological selections 2021/03/Y/ST1/00122.]]>Talks2023-05-24T11:41:44+02:00https://settheory.pwr.edu.pl/blog/files/0a903bd8fbf0a0f818adf2750e5d5afb-177.php#unique-entry-id-177https://settheory.pwr.edu.pl/blog/files/0a903bd8fbf0a0f818adf2750e5d5afb-177.php#unique-entry-id-177
Location: room A.4.1 C-19

Speaker: Zbigniew Lipecki (IM PAN)

Title: How noncompact is the space of Lebesgue measurable sets?

Abstract: The space in question is the space \(\mathfrak M\) of Lebesgue measurable subsets of the unit interval equipped with the usual Fréchet—Nikodym (semi)metric.

We show that there exists a sequence of elements of \(\mathfrak M\) such that their mutual distances are > 1/2. It seems to be an open problem whether "1/2" can be replaced here by a bigger constant C. We show that C must be smaller than 9/14. Moreover, we present a version of the problem in terms of binary codes.]]>Talks2023-05-17T14:02:28+02:00https://settheory.pwr.edu.pl/blog/files/18992276940aecf6613e914382cea8bd-176.php#unique-entry-id-176https://settheory.pwr.edu.pl/blog/files/18992276940aecf6613e914382cea8bd-176.php#unique-entry-id-176
Location: room A.4.1 C-19

Speaker: Barnabas Farkas (TU Wien)

Title: A tool to avoid some technical forcing arguments when working with the Hechler forcing

Abstract: I'm going to present that virtually every result saying that finite support iterations of the Hechler forcing preserve a cardinal invariant being small and its dual being large can be reduced to a single preservation theorem. In other words, this theorem eliminates the technical forcing arguments from the proofs of these results and reduces them to easy coding exercises. ]]>Talks2023-05-16T14:00:11+02:00https://settheory.pwr.edu.pl/blog/files/e8c5d5452316972a651b45957581a3d9-175.php#unique-entry-id-175https://settheory.pwr.edu.pl/blog/files/e8c5d5452316972a651b45957581a3d9-175.php#unique-entry-id-175
Location: room A.4.1 C-19

Speaker: Damian Sobota (Kurt Gödel Research Center for Mathematical Logic)

Title: On continuous operators from Banach spaces of Lipschitz functions onto \(c_0\)

Abstract: During my talk I will discuss some of our recent results concerning the existence of continuous operators from the Banach spaces \(\textrm{Lip}_0(M)\) of Lipschitz real-valued functions on metric spaces M onto the Banach space \(c_0\) of sequences converging to \(0\). I will in particular prove that there is always a continuous operator onto \(c_0\) from infinite-dimensional spaces of the form \(\textrm{Lip}_0(C(K))\) or \(\textrm{Lip}_0(\textrm{Lip}_0(M))\). (Based on an ongoing joint work with C. Bargetz and J. Kąkol).]]>Talks2023-04-19T12:50:20+02:00https://settheory.pwr.edu.pl/blog/files/71518870a65529360e46850ec5e5ecfe-174.php#unique-entry-id-174https://settheory.pwr.edu.pl/blog/files/71518870a65529360e46850ec5e5ecfe-174.php#unique-entry-id-174
Location: room A.4.1 C-19

Speaker: Sławomir Solecki (Cornell University)

Title: Incomparable Borel linear subspaces of \(\mathbb{R}\) (over \(\mathbb{Q}\))

Abstract: We present a construction of a large family of Borel linear subspaces of \(\mathbb{R}\) (over \(\mathbb{Q}\)), which are incomparable with respect to Borel linear embeddings (over \(\mathbb{Q}\)). A version of this construction answers a question by Frisch and Shinko.]]>Talks2023-04-18T14:28:28+02:00https://settheory.pwr.edu.pl/blog/files/1eef2f15dc5427fa38b1470da7bf31e3-173.php#unique-entry-id-173https://settheory.pwr.edu.pl/blog/files/1eef2f15dc5427fa38b1470da7bf31e3-173.php#unique-entry-id-173
Location: room A.4.1 C-19

Speaker: Jonathan Cancino (Czech Academy of Sciences)

Title: On nwd-MAD families

Abstract: The cardinal invariant a(nwd) is defined as the minimal cardinality of an uncountable maximal antichain of the power set of the rational modulo the nowhere dense ideal. This cardinal invariant was introduced by J. Steprans, and he proved that in the Laver's model it is \(\omega_1\), and the pseudointersection number p is a lower bound for it. In this talk we will prove some related results, for example, the additivity of the meager ideal is a lower bound for a(nwd), thus improving Steprans theorem, as well as some facts about the structure of nwd-MAD families.]]>Talks2023-04-04T11:47:16+02:00https://settheory.pwr.edu.pl/blog/files/f19cc48177b85e189262ce2a7b89bbe4-172.php#unique-entry-id-172https://settheory.pwr.edu.pl/blog/files/f19cc48177b85e189262ce2a7b89bbe4-172.php#unique-entry-id-172
Location: room A.4.1 C-19

Speaker: Arkady Leiderman (Ben-Gurion University of the Negev, Beer Sheva, Israel)

Title: On \(\Delta\)-spaces

Abstract: \(\Delta\)-spaces have been defined by a natural generalization of a classical notion of \(\Delta\)-sets of reals to Tychonoff topological spaces; moreover, the class \(\Delta\) of all \(\Delta\)-spaces consists precisely of those \(X\) for which the locally convex space \(C_p(X)\) is distinguished. A systematic study of the class \(\Delta\) was originated in my joint papers [1], [2].

The talk will be devoted to some results obtained in a recent joint work with Paul Szeptycki (Canada). The aim of this work is to better understand the boundaries of the class $\Delta$, by presenting new examples and counter-examples.

1) We examine when trees considered as topological spaces equipped with the interval topology belong to \(\Delta\).

In particular, we prove that no Souslin tree is a \(\Delta\)-space. Other main results are connected with the study of

2) \(\Psi\)-spaces built on maximal almost disjoint families of countable sets; and

3) Ladder system spaces.

There exists an Isbell-Mrówka \(\Psi\)-space \(X\) (which is in \(\Delta\)) such that one-point extension \(X_p = X \cup \{p\}\) of \(X\) has uncountable tightness at the point \(p\), for some \(p \in \beta(X) \setminus X\).

It is consistent with CH that all ladder system spaces on \(\omega_1\) are \(\Delta\)-spaces.

We show that in forcing extension of ZFC obtained by adding one Cohen real, there is a ladder system space on \(\omega_1\) which is not in \(\Delta\).



[1] Jerzy Kąkol and Arkady Leiderman, A characterization of \(X\) for which spaces \(C_p(X)\) are distinguished and its applications, Proc. Amer. Math. Soc., series B, 8 (2021), 86-99.

[2] Jerzy Kąkol and Arkady Leiderman, Basic properties of \(X\) for which the space \(C_p(X)\) is distinguished, Proc. Amer. Math. Soc., series B, (8) (2021), 267-280.

]]>Talks2023-03-28T12:22:45+02:00https://settheory.pwr.edu.pl/blog/files/5fa6937a019eaede97efc063800ab543-171.php#unique-entry-id-171https://settheory.pwr.edu.pl/blog/files/5fa6937a019eaede97efc063800ab543-171.php#unique-entry-id-171
Location: room A.4.1 C-19

Speaker: Tomasz Zuchowski

Title: Katětov order on Borel ideals

Abstract: An ideal \(\mathcal{I}\) on \(\omega\) is Katětov reducible to ideal \(\mathcal{J}\) if there is a function \(f:\omega\to \omega\) such that if \(I\in\mathcal{I}\) then \(f^{-1}[I]\in\mathcal{J}\). The existence of such reduction is related to some cardinal invariants and other properties of considered ideals. We will present some examples of Borel ideals with or without Katětov reductions between them. Furthermore we will prove a structural dichotomy about Katětov order for all Borel ideals.

The presented results are from the paper “Katětov order on Borel ideals” by Michael Hrusak.]]>Talks2023-03-16T17:29:20+01:00https://settheory.pwr.edu.pl/blog/files/ae7047866e37e301383a50ae732649ea-170.php#unique-entry-id-170https://settheory.pwr.edu.pl/blog/files/ae7047866e37e301383a50ae732649ea-170.php#unique-entry-id-170
Location: room A.4.1 C-19

Speaker: Sebastian Jachimek

Title: Combinatorial Banach spaces

Abstract: Combinatorial space is a type of Banach space induced by (some) family of sets in a certain way. During the talk I will present examples of families (of subsets of natural numbers) and spaces related with them. Furthermore, I will consider properties of these spaces, in particular in the context of containing isomorphic copy of \(c_0\) and \(\ell_1\).]]>Talks2023-03-09T20:47:57+01:00https://settheory.pwr.edu.pl/blog/files/76350cc2f6104393c92c12173532ffbf-169.php#unique-entry-id-169https://settheory.pwr.edu.pl/blog/files/76350cc2f6104393c92c12173532ffbf-169.php#unique-entry-id-169
Location: room A.2.22 C-19

Speaker: Grzegorz Plebanek

Title: Countable extensions of compact lines

Abstract: For a compact space \(K\), we say that \(L\) is a countable discrete extension of \(K\) if \(L\) is compact and consists of \(K\) and a countable set of isolated points. We investigate some properties of such extenions for separable compact lines \(K\). This is directly related to properties of almost chains of subsets of \( \mathbb{N}\). ]]>Talks2023-03-01T08:02:19+01:00https://settheory.pwr.edu.pl/blog/files/777c2f08ba9df72eb708c17495203b18-168.php#unique-entry-id-168https://settheory.pwr.edu.pl/blog/files/777c2f08ba9df72eb708c17495203b18-168.php#unique-entry-id-168
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.]]>Talks2023-01-23T14:38:27+01:00https://settheory.pwr.edu.pl/blog/files/15519a6840be0048230764ed29f5e104-167.php#unique-entry-id-167https://settheory.pwr.edu.pl/blog/files/15519a6840be0048230764ed29f5e104-167.php#unique-entry-id-167
Location: room C11-3.11

Speaker: Bill Mance (Adam Mickiewicz University in Poznan)

Title: Descriptive complexity in number theory and dynamics

Abstract: Informally, a real number is normal in base \(b\) if in its \(b\)-ary expansion, all digits and blocks of digits occur as often as one would expect them to, uniformly at random. Kechris asked several questions involving descriptive complexity of sets of normal numbers. The first of these was resolved in 1994 when Ki and Linton proved that the set of numbers normal in base \(b\) is \(\Pi_3^0\)-complete. Further questions were resolved by Becher and Slaman. Many of the techniques used in these proofs can be used elsewhere. We will discuss recent results where similar techniques were applied to solve a problem of Sharkovsky and Sivak and a question of Kolyada, Misiurewicz, and Snoha. Furthermore, we will discuss a recent result where the set of numbers that are continued fraction normal, but not normal in any base \(b\), was shown to be complete at the expected level of \(D_2(\Pi_3^0)\). An immediate corollary is that this set is uncountable, a result (due to Vandehey) only known previously assuming the generalized Riemann hypothesis.

]]>Talks2023-01-17T09:14:49+01:00https://settheory.pwr.edu.pl/blog/files/42d4c611dac1ac6591c991f265772ddc-166.php#unique-entry-id-166https://settheory.pwr.edu.pl/blog/files/42d4c611dac1ac6591c991f265772ddc-166.php#unique-entry-id-166
Location: room C11-3.11

Speaker: Łukasz Mazurkiewicz

Title: Ideal analytic sets

Abstract: We will consider examples of analytic sets which are not Borel. We will focus on, so called, complete analytic sets. Firstly, we will consider ideals on naturals (naturally treated as subsets of the Cantor space). Secondly, we will consider the family of Silver trees. We will compare the later example with theorem of Kechris-Louveau-Woodin.]]>Talks2023-01-09T15:18:42+01:00https://settheory.pwr.edu.pl/blog/files/cd1e3e712bda8cee3740f09131693193-165.php#unique-entry-id-165https://settheory.pwr.edu.pl/blog/files/cd1e3e712bda8cee3740f09131693193-165.php#unique-entry-id-165
Location: room C11-3.11

Speaker: Artsiom Ranchynski

Title: Ultrafilters avoiding measures

Abstract: A point \(x\) avoids measures if whenever \(\mu\) is a measure such that \(\mu({x})=0\), then \(x\) does not belong to the support of \(\mu\). In this talk I will construct a point avoiding non-atomic measures in the Stone-Cech compactification of naturals. I will discuss the relation of such points to other special points in \(\beta N\).]]>Talks2022-12-08T09:57:06+01:00https://settheory.pwr.edu.pl/blog/files/acfd23cdf1856a12c36d598aa9c6db58-164.php#unique-entry-id-164https://settheory.pwr.edu.pl/blog/files/acfd23cdf1856a12c36d598aa9c6db58-164.php#unique-entry-id-164
Location: room C11-3.11

Speaker: Jacek Jachymski (Łódź University of Technology)

Title: Between Cantor and Smulian: the intersection theorem and its applications

Abstract: I will present an intersection theorem for a descending sequence of closed sets for which a convexity type condition is satisfied. However, the condition applies to the whole sequence, not separately to individual sets. I will show that the following theorems follows easily from this result: Smulian's theorem about characterization of reflexive spaces, theorem about the convex set in Hilbert space, and Browder-Gohde-Kirk's theorem about fixed points of nonexpansive mappings. ]]>Talks2022-12-06T08:10:05+01:00https://settheory.pwr.edu.pl/blog/files/5e79c3765adeb4d64e401f94dae4baa8-163.php#unique-entry-id-163https://settheory.pwr.edu.pl/blog/files/5e79c3765adeb4d64e401f94dae4baa8-163.php#unique-entry-id-163
Location: room C11-3.11

Speaker: Mikołaj Krupski (University of Warsaw)

Title: \(\kappa\)-pseudocompactness and uniform homeomorphisms of function spaces

Abstract: A Tychonoff space \(X\) is called \(\kappa\)-pseudocompact if for every continuous mapping \(f\) of \(X\) into \( {\mathbb{R}}^\kappa\) the image \(f(X)\) is compact. This notion generalizes pseudocompactness and gives a stratification of spaces lying between pseudocompact and compact spaces. It is well known that pseudocompactness of \(X\) is determined by the uniform structure of the function space \(C_p(X)\) of continuous real-valued functions on \(X\) endowed with the pointwise topology. In respect of that A.V. Arhangel'skii asked in [Topology Appl., 89 (1998)] if analogous assertion is true for \(\kappa\)-pseudocompactness. We provide an affirmative answer to this question.]]>Talks2022-11-29T10:12:39+01:00https://settheory.pwr.edu.pl/blog/files/69e5a29e664d4c443ab7a6002d18fd9c-162.php#unique-entry-id-162https://settheory.pwr.edu.pl/blog/files/69e5a29e664d4c443ab7a6002d18fd9c-162.php#unique-entry-id-162
Location: room C11-3.11

Speaker: Arturo Martinez Celis

Title: Some combinatorics related to the Michael space problem III

Abstract: In this talk, we will continue the construction of a Michael space from an ultrafilter. We will show the consistency of ZFC + There are no Michael Ultrafilters and we will discuss some open questions.]]>Talks2022-11-21T09:09:24+01:00https://settheory.pwr.edu.pl/blog/files/31522d2abd53e61e7429e79fe5e6ea62-161.php#unique-entry-id-161https://settheory.pwr.edu.pl/blog/files/31522d2abd53e61e7429e79fe5e6ea62-161.php#unique-entry-id-161
Location: room C11-3.11

Speaker: Arturo Antonio Martínez Celis Rodríguez

Title: Some combinatorics related to the Michael space problem II

Abstract: In this talk we will continue the construction of a Michael space from an ultrafilter. The main goal is to show that the existence of a selective ultrafilter (plus \(\varepsilon\ge 0\)) is enough to construct a Michael space. If the time allows it, we will show a model of ZFC without Michael ultrafilters.]]>Talks2022-11-10T18:54:28+01:00https://settheory.pwr.edu.pl/blog/files/77812eac8738c4398e2419042e000e2e-160.php#unique-entry-id-160https://settheory.pwr.edu.pl/blog/files/77812eac8738c4398e2419042e000e2e-160.php#unique-entry-id-160
Location: room C11-3.11

Speaker: Daria Michalik (Jan Kochanowski University in Kielce)

Title: Blocking properties of the diagonal in Cartesian product

Abstract: In [1], the authors present six kinds of blocking properties for points in continua. We can consider the same properties for subcontinua. During my talk I will present some results concerning the blocking properties of the diagonal in Cartesian product. Among others, I will show a new characterisation of the interval.

[1] J. Bobok, P. Pyrih and B. Vejnar, Non-cut, shore and non-block points in continua, Glas. Mat. Ser. III 51 (71) (2016), 237–253.]]>Talks2022-11-08T05:58:36+01:00https://settheory.pwr.edu.pl/blog/files/df9b64cc9bf1cba33c7325264dd79c69-159.php#unique-entry-id-159https://settheory.pwr.edu.pl/blog/files/df9b64cc9bf1cba33c7325264dd79c69-159.php#unique-entry-id-159
Location: room C11-3.11

Speaker: Damian Głodkowski (University of Warsaw)

Title: A Banach space C(K) reading the dimension of K

Abstract: For every natural number \(n\) I construct (assuming Jensen's diamond principle) a compact space \(K_n\) such that whenever \(L\) is a compact space and the Banach spaces of continuous functions \(C(K_n)\) and \(C(L)\) are isomorphic, the covering dimension of \(L\) is equal to \(n\). The constructed space is a modification of Koszmider's example of a compact space \(K\) with the property that every bounded linear operator \(T\) on \(C(K)\) is a weak multiplication (i.e. it is of the form \(T(f)=gf+S(f)\), where \(g\) is an element of \(C(K_n)\) and \(S\) is weakly compact). In the talk I will give a sketch of the construction and focus on the differences between my and the original space. The talk will be based on https://arxiv.org/abs/2207.00149.
]]>Talks2022-10-20T14:47:44+02:00https://settheory.pwr.edu.pl/blog/files/f6e777a6ec2fef1f696e55bcf437dbb5-158.php#unique-entry-id-158https://settheory.pwr.edu.pl/blog/files/f6e777a6ec2fef1f696e55bcf437dbb5-158.php#unique-entry-id-158
Location: room C11-3.11

Speaker: Adam Morawski

Title: P-measures in models without P-points

Abstract: P-points are ultrafilters in which every decreasing sequence of sets from the filter has a pseudointersection (in a sense an intersection modulo finite sets) in the filter. Quite similarly P-measures (known in the literature as measures with additive property*) are finitely additive probability measures in which every decreasing sequence of sets has a pseudointersection with measure equal to the limit of measures of sets from the sequence.

It is not hard to see that (a characteristic function of) a P-point is a P-measure. However, a question whether the existence of P-measures implies the existence of P-points remains open.

I will talk about current knowledge of the problem including my and Piotr Borodulin-Nadzieja's efforts and results - based on the Silver forcing and its variations.]]>Talks2022-10-17T19:07:47+02:00https://settheory.pwr.edu.pl/blog/files/084009b1cd188c01a483e3207e82603f-157.php#unique-entry-id-157https://settheory.pwr.edu.pl/blog/files/084009b1cd188c01a483e3207e82603f-157.php#unique-entry-id-157
Location: room C11-3.11

Speaker: Antonio Martínez Celis Rodríguez

Title: Some combinatorics related to the Michael space problem

Abstract: A Lindelöf space is Michael if it has non-Lindelöf product with the Baire space. In this talk (series of talks?) we will review some of the combinatorics required to construct one of these spaces. The main goal is to show that the existence of a selective ultrafilter (plus \(\varepsilon\ge 0\)) is enough to construct a Michael space.]]>Talks2022-10-13T14:46:12+02:00https://settheory.pwr.edu.pl/blog/files/fb1d8822495d86096bda55590620a84f-156.php#unique-entry-id-156https://settheory.pwr.edu.pl/blog/files/fb1d8822495d86096bda55590620a84f-156.php#unique-entry-id-156
Location: room C11-3.11

Speaker: Paweł Krupski

Title: On the hyperspace of simple closed curves in the plane

Abstract: The Vietoris hyperspace of simple closed curves in the plane will be discussed toward its desirable characterization. In particular, the local contractibility will be shown.

Joint work with Krzysztof Omiljanowski.]]>Talks2022-06-10T08:16:08+02:00https://settheory.pwr.edu.pl/blog/files/d06386f9d0b7675b82cbea8b5e7b525c-155.php#unique-entry-id-155https://settheory.pwr.edu.pl/blog/files/d06386f9d0b7675b82cbea8b5e7b525c-155.php#unique-entry-id-155
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Eliza Jabłońska (AGH University of Science and Technology)

Title: From the Steinhaus property to the Laczkovich one

Abstract: Let \(X\) be a locally compact Abelian Polish group, \(\mathcal{B}(X)\) be the family of all Borel subsets of \(X\) and \(\mathcal{F}\subset 2^{X}\). We consider the following Steinhaus' type properties:

• \((S^+)\): \((A+B)\neq\emptyset\) for every \(A,B\in\mathcal{B}(X)\setminus \mathcal{F}\),
• \((S^-)\): \(0\in (A-A)\) for every \(A\in\mathcal{B}(X)\setminus \mathcal{F}\),
• \((D^+)\): \(A+B\) is non-meager for every \(A,B\in\mathcal{B}(X)\setminus \mathcal{F}\),
• \((D^-)\): \(A-A\) is non-meager for every \(A\in\mathcal{B}(X)\setminus \mathcal{F}\).

It is known that the family \(\mathcal{M}\) of all meager sets as well as the family \(\mathcal{N}\) of all sets of Haar measure zero satisfy each of these conditions. We prove that the family \(\mathcal{M}\cap\mathcal{N}\) satisfies \((S^-)\), \((D^+)\), \((D^-)\) although it does not satisfy \((S^+)\). We also show that the \(\sigma\)-ideal \(\sigma\overline{\mathcal{N}}\subset \mathcal{M}\cap\mathcal{N}\) generated by closed sets of Haar measure zero satisfies only \((D^+)\) and \((D^-)\) which leads us to the Laczkovich property. This is joint work with T. Banakh, I. Banakh, Sz. Głąb and J. Swaczyna.
]]>Talks2022-06-02T09:43:13+02:00https://settheory.pwr.edu.pl/blog/files/77f052fa29849e3779b8c0cef25764b5-154.php#unique-entry-id-154https://settheory.pwr.edu.pl/blog/files/77f052fa29849e3779b8c0cef25764b5-154.php#unique-entry-id-15417:15

Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Jonathan Cancino (Czech Academy of Sciences)

Title: Ideal independent families and ultrafilters

Abstract: A family \(\mathscr{I}\subseteq[\omega]^\omega\) is called ideal independent if no element \(A\in\mathscr{I}\) is almost contained in the union of finitely many other elements in \(\mathscr{I}\). The ideal independence number, denoted by \(\mathfrak{s}{mm}\), is defined as the minimal cardinality of a maximal ideal independent family. We will review some results about ideal independent families and the cardinal invariant \(\mathfrak{s}{mm}\). In particular we will prove that the ultrafilter number is a lower bound for \(\mathfrak{s}{mm}\). Also, we will see that the spectrum of ideal independent families, defined as the family of all cardinalities of maximal ideal independent families, can be quite rich. If time allows, we will sketch a proof that consistently \(\mathfrak{s}{mm}<\mathfrak{a}_T\), where \(\mathfrak{a}_T\) is the minimal cardinality of a family of disjoint compact sets covering the Baire space. This is joint work with V. Fischer and C. B. Switzer.]]>Talks2022-05-30T09:19:14+02:00https://settheory.pwr.edu.pl/blog/files/8807315f440ba5903098cb27c1d2421e-153.php#unique-entry-id-153https://settheory.pwr.edu.pl/blog/files/8807315f440ba5903098cb27c1d2421e-153.php#unique-entry-id-153
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Rafał Filipów (University of Gdańsk)

Title: Does there exist a Hindman space which is not a van der Waerden space?

Abstract: Both Hindman spaces and van der Waerden spaces were defined by M. Kojman (Proc. AMS 130(2002), no. 3 and no. 6) with the aid of Hindman's finite sum theorem and van der Waerden's theorem on arithmetic progressions, respectively. Then M. Kojman and S. Shelah (Proc. AMS 131(2003), no. 5) proved that there exists a van der Waerden space which is not a Hindman space. The question whether there exists a Hindman space which is not a van der Waerden space is still open. In my talk I will show how this question about topological spaces can be reduced to a question only about Katetov order of two ideals of subsets of \(\mathbb{N}\). This result is from our joint paper with K. Kowitz, A. Kwela nad J. Tryba (Proc. AMS 150(2022), no. 2).]]>Talks2022-05-23T08:00:48+02:00https://settheory.pwr.edu.pl/blog/files/83f104d9cfa0c27e0df23b22fae6cde9-152.php#unique-entry-id-152https://settheory.pwr.edu.pl/blog/files/83f104d9cfa0c27e0df23b22fae6cde9-152.php#unique-entry-id-152Thursday, May 26, 2022 17:15

Location: room P.01 C-11

Speaker: Krzysztof Leśniak (Nicolaus Copernicus University in Toruń)

Title: Enriching IFS fractals with symmetry

Abstract: Let \(\mathcal{F}=(X; f_i:i\in I)\) be an iterated function system (IFS) consisting of a finite number of Banach contractions \(f_i\) acting on a complete metric space \(X\). According to the seminal result of Hutchinson (1981), \(F\) admits an attractor, denoted by \(A_{\mathcal{F}}\). Let \(g:X\to X\) be a \(p\)-periodic isometry, \(p>1\), which admits a (not necessarily unique) fixed point.

Proposition: The IFS \(\widetilde{\mathcal{F}} =F \cup \{g\}\) admits a semiattractor \(A^{\flat}\) (in the Lasota—Myjak sense) which is compact and \(g\)-symmetric.

The question arises, whether \(A^{\flat}\) is an ordinary attractor. The answer is `yes'.

Theorem (L & Snigireva): \(A^{\flat}\) is an attractor of any of the following contractive IFSs
\begin{eqnarray*}
\mathcal{G}
= (X; \;\; g^{-j}\circ f_i\circ g^j \;\;: i\in I, j\in\mathbb{Z}_p),
%\label{eq:IFS-Gconj}
\\
GF
= (X; \;\; g^k\circ f_i\circ g^j \;\;: i\in I, j,k\in\mathbb{Z}_p),
%\label{eq:IFS-GF}
\\
\widehat{\mathcal{G}} =
(X; \;\; g^k\circ f_i \;\;: i\in I, k\in\mathbb{Z}_p).
%\label{eq:IFS-Gdoubletilde}
\end{eqnarray*}

Moreover, the attractor \(A_{\widetilde{\mathcal{G}}}\) of a contractive IFS \(\widetilde{\mathcal{G}} = (X; f_i\circ g^j: i\in I, j\in\mathbb{Z}_p)\) is a smaller copy of \(A^{\flat}\): \(A_{\mathcal{F}} \subset A_{\widetilde{\mathcal{G}}} \subset A^{\flat} = \bigcup_{k=0}^{p-1} g^k(A_{\widetilde{\mathcal{G}}})\).

Remark: \(\widehat{\mathcal{G}}\) appears in Symmetry in Chaos by Field & Golubitsky, cf. http://larryriddle.agnesscott.org/ifskit/IFShelp/howtoCreateSymmetricFractal.html by L.R. Riddle

The question whether the disjunctive chaos game algorithm is valid for the enriched IFS \(\widetilde{\mathcal{F}}\) leads to interesting problems in combinatorics on words. Finally, to allow for similar results in case \(g\) is a non-periodic isometry, or \(\mathcal{F}\) is enriched by more than one isometry, one needs to employ infinite IFSs (F. Strobin, 2021).]]>Talks2022-05-19T14:44:34+02:00https://settheory.pwr.edu.pl/blog/files/19c72b1f45abe35f1957100f8fe8dd7c-151.php#unique-entry-id-151https://settheory.pwr.edu.pl/blog/files/19c72b1f45abe35f1957100f8fe8dd7c-151.php#unique-entry-id-151
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: David Chodounsky (Czech Academy of Sciences)

Title: Sacks indestructible ultrafilters and reaping families

Abstract: Preservation of reaping families and especially ultrafilters on countable sets is a well studied theme in set theory of the reals. A. Miller proved that if an ultrafilter remains a reaping family in some forcing extension, then it has to be also Sacks indestructible. The existence of Sacks indestructible ultrafilters in ZFC is an open question. A related problem is Sacks indestructibility of reaping families which are complements of ideals. We prove that complements of most classical ideals are indestructible with one notable exception, the ideal of sets asymptotic density zero.

The presented results are from an upcoming paper with O. Guzman and M. Hrusak.]]>Talks2022-05-13T08:27:28+02:00https://settheory.pwr.edu.pl/blog/files/b5e92c358a9126998257f5b2e5c0fd67-150.php#unique-entry-id-150https://settheory.pwr.edu.pl/blog/files/b5e92c358a9126998257f5b2e5c0fd67-150.php#unique-entry-id-150
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Mirna Dzamonja (Université deParis-Cité)

Title: Reasonable structures of size \(\aleph_1\)

Abstract: We are interested to develop a theory of structures of size \(\aleph_1\) which are ’tame’ in the sense that they in some sense or other preserve the nice properties that we are used to seeing on the countable structures.
We explain the aim of the programme and then discuss a joint work with Wiesław Kubiś on a specific way of constructing structures of size \(\aleph_1\) using finite approximations, namely by organising the approximations along a simplified morass. We demonstrate a connection with Fraïssé limits and show that the naturally obtained structure of size \(\aleph_1\) is homogeneous. We give some examples of interesting structures constructed, such as a homogeneous antimetric space of size \(\aleph_1\). Finally, we comment on the situation when one Cohen real is added.]]>Talks2022-05-08T20:10:31+02:00https://settheory.pwr.edu.pl/blog/files/52deb580bc2291d6eaf7aa3176f15b88-149.php#unique-entry-id-149https://settheory.pwr.edu.pl/blog/files/52deb580bc2291d6eaf7aa3176f15b88-149.php#unique-entry-id-149
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Damina Sobota (KGRC, Vienna)

Title: On sequences of finitely supported measures on products of compact spaces

Abstract: Cembranos, Freniche, and Khurana (all independently) proved that for every two infinite compact spaces \(K\) and \(L\) the Banach space \(C(K\times L)\) contains a complemented copy of the space \(c_0\). To obtain this copy all the three proofs utilize in some way the Josefson-Nissenzweig theorem which more or less asserts that there is a sequence \((mu_n)\) of normalized signed Radon measures on \(K\times L\) such that \(mu_n(f)\) converges to \(0\) for every \(f\) from \(C(K\times L)\). Since most (if not all) of the known proofs of the J-N theorem are non-constructive, it follows that the (known to me) proofs of Cembranos et al. are also non-constructive. During my talk I'll show a generalization of the theorem of Cembranos et al. whose proof uses a direct construction of a sequence \((mu_n)\) of finitely supported measures on \(K\times L\) as above. I'll also discuss the case of pseudocompact spaces \(K\) and \(L\) and pose some questions.]]>Talks2022-05-04T19:27:34+02:00https://settheory.pwr.edu.pl/blog/files/34f70628cadf485f4246b5337777d12f-148.php#unique-entry-id-148https://settheory.pwr.edu.pl/blog/files/34f70628cadf485f4246b5337777d12f-148.php#unique-entry-id-148
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Ondrej Zindulka (Czech Technical University, Prague)

Title: Microscopic sets, Hausdorff measures and their cardinal invariants

Abstract: A set in a metric space is microscopic it admits, for every \(\varepsilon>0\), a cover \(\{E_n\}\) such that the diameter of each \(E_n\) is at most \(\varepsilon^n\). The notion was introduced 21 years ago and since then a number of people contributed to the theory. I will provide a brief account of the state of art and present new results and in particular the so far overlooked relation to Hausdorff measures. Attention will be paid to cardinal invariants of the ideal of microscopic sets and sets of Hausdorff measure zero in metric spaces and Polish groups.]]>Talks2022-04-24T15:37:19+02:00https://settheory.pwr.edu.pl/blog/files/319e8dda93aecaf18a9d3863a23259f0-147.php#unique-entry-id-147https://settheory.pwr.edu.pl/blog/files/319e8dda93aecaf18a9d3863a23259f0-147.php#unique-entry-id-147
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.]]>Talks2022-04-09T15:36:52+02:00https://settheory.pwr.edu.pl/blog/files/3af8ddf49b1e3566df253cad54f4aead-146.php#unique-entry-id-146https://settheory.pwr.edu.pl/blog/files/3af8ddf49b1e3566df253cad54f4aead-146.php#unique-entry-id-146
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Konrad Królicki (Hungarian Academy of Sciences)

Title: Nonsingular hyperfinite actions of groups

Abstract: Any action of a finitely generated group on a standard probability space induces a measurable Schreier graph.When the action is nonsingular, i.e. it preserves the measure class, the measurable graph is called a measured graphing. We say that a measured graphing is hyperfinite if for any \(\varepsilon >0\), one can remove a part of measure at most \(\varepsilon\) in such a way that the components of the remainder are finite. I will define the notion of local convergence for measured graphs, i.e. finite Schreier graphs with a probability measure on their vertices, and how their limits may be represented with nonsingular actions. The objective of the talk is to present one part of the nonsingular theorem of Schramm: if a sequence of measured graphs is hyperfinite, then the limit graphing is hyperfinite as well. Joint work with Gabor Elek.]]>Talks2022-04-02T17:44:22+02:00https://settheory.pwr.edu.pl/blog/files/8a6fb34d194a06bc4b13a034b2fb78f4-145.php#unique-entry-id-145https://settheory.pwr.edu.pl/blog/files/8a6fb34d194a06bc4b13a034b2fb78f4-145.php#unique-entry-id-145
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Adam Bartos (Czech Academy of Sciences)

Title: Hereditarily indecomposable continua as Fraïssé limits

Abstract: Irwin and Solecki introduced projective Fraïssé theory and showed that the Fraïssé limit of the projective class of finite linear graphs is a pre-space of the pseudo-arc. This allowed to characterize the pseudo-arc as the unique approximatively projectively homogeneous arc-like continuum. We introduce a framework for Fraïssé theory where the pseudo-arc itself is a Fraïssé limit, and apply the framework to obtain similar characterizations for P-adic pseudo-solenoids. This is joint work with Wiesław Kubiś.]]>Talks2022-03-27T21:03:00+02:00https://settheory.pwr.edu.pl/blog/files/3ea0af2c4ca26f885948509319c7a114-144.php#unique-entry-id-144https://settheory.pwr.edu.pl/blog/files/3ea0af2c4ca26f885948509319c7a114-144.php#unique-entry-id-144
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Maciej Korpalski

Title: Continuous discrete extension of double arrow spaces

Abstract: Double arrow space is a separable linearly ordered compact space. By adding a discrete countable set in a special way we can extend those spaces so that extension is still compact. We will talk about some properties of those extensions and see counterexamples to them.]]>Talks2022-03-21T14:54:24+01:00https://settheory.pwr.edu.pl/blog/files/e0def867c2822f708a8c180f9117831e-143.php#unique-entry-id-143https://settheory.pwr.edu.pl/blog/files/e0def867c2822f708a8c180f9117831e-143.php#unique-entry-id-143
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Szymon Żeberski

Title: Remarks on Eggleston theorem

Abstract: We will discuss possible variants and generalizations of Eggleston theorem about inscribing big rectangles into big subsets of the plane. We will focus mainly on product of two Cantor spaces and comeager sets. ]]>Talks2022-03-13T18:05:20+01:00https://settheory.pwr.edu.pl/blog/files/62e28e6d5de3ea1d315c2b00c3efbcd5-142.php#unique-entry-id-142https://settheory.pwr.edu.pl/blog/files/62e28e6d5de3ea1d315c2b00c3efbcd5-142.php#unique-entry-id-142
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Robert Rałowski

Title: On \(T_1\)- and \(T_2\)-productable compact spaces

Abstract: We prove that if there exists a continuous surjection from a metric compact space \(X\) onto a product \(X\times T\) where \(T\) is a \(T_1\) second countable topological space which has the cardinality of the continuum then there exists a surjection from \(X\) onto the product \(X\times [0, 1]\) where the interval \([0, 1]\) is equipped with the usual Euclidean topology.]]>Talks2022-02-27T11:07:55+01:00https://settheory.pwr.edu.pl/blog/files/3e229a540ddccffa3ffa4f185beec4f4-141.php#unique-entry-id-141https://settheory.pwr.edu.pl/blog/files/3e229a540ddccffa3ffa4f185beec4f4-141.php#unique-entry-id-141
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Agnieszka Widz

Title: Almost disjoint magic sets

Abstract: Given a family of real functions F we say that a set M ⊆ ℝ is magic for F if for all f, g ∈ F we have f [M ] ⊆ g[M ] ⇒ f = g. This notion was introduced by Diamond, Pomerance and Rubel in 1981. Recently some results about magic sets were proved by Halbeisen, Lischka and Schumacher. Inspired by their work I constructed two families of magic sets one of them being almost disjoint and the other one being independent. During my talk I will sketch the background and present the proof for the almost disjoint family, which uses a Kurepa tree.]]>Talks2022-01-21T08:51:39+01:00https://settheory.pwr.edu.pl/blog/files/436bac418339778d66a10dd146e87e99-140.php#unique-entry-id-140https://settheory.pwr.edu.pl/blog/files/436bac418339778d66a10dd146e87e99-140.php#unique-entry-id-140
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Sebastian Jachimek

Title: A Banach space induced by a compact family

Abstract: In the talk we will present an example of a Banach space induced (in some particular way) by some compact family of subsets of natural numbers. In particular, we will prove that this space is \(l_1\)-saturated and does not have the Schur property.]]>Talks2022-01-14T18:31:39+01:00https://settheory.pwr.edu.pl/blog/files/0d4ce5d3b1fe76b29aa2c6d008b9b0b4-139.php#unique-entry-id-139https://settheory.pwr.edu.pl/blog/files/0d4ce5d3b1fe76b29aa2c6d008b9b0b4-139.php#unique-entry-id-139
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Grzegorz Plebanek

Title: A complemented subspace of a C(K)-space which is not a C(K)-space

Abstract: We present a construction of two separable compacta K and L such that C(L) is a direct sum of C(K) and some Banach space X which is not isomorphic to a space of continuous functions. Joint work with Alberto Salguero Alarcon (Badajoz).]]>Talks2021-12-02T13:44:54+01:00https://settheory.pwr.edu.pl/blog/files/977c5624f2500af8fa60019e00df9bf9-138.php#unique-entry-id-138https://settheory.pwr.edu.pl/blog/files/977c5624f2500af8fa60019e00df9bf9-138.php#unique-entry-id-138
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Łukasz Mazurkiewicz

Title: Possible modifications of Lusin analytic set

Abstract: In the last talk we breathly mentioned an example of a complete analytic set created by Lusin. This time we will prove that it is a complete analytic set, which is not an element of \(Bor[K_\sigma]\). Then we will investigate some possible modifications of this example in order to decide, which partial orders make this set complete analytic.]]>Talks2021-11-25T08:28:15+01:00https://settheory.pwr.edu.pl/blog/files/b03f11bc449653050182496c3486bb63-137.php#unique-entry-id-137https://settheory.pwr.edu.pl/blog/files/b03f11bc449653050182496c3486bb63-137.php#unique-entry-id-137
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Arturo Martinez-Celis

Title: Michael Spaces and Selective Ultrafilters

Abstract: Lindelöf space X is Michael if it has a non-Lindelöf product with the space of irrational numbers. The existence of these kinds of spaces using only the standard axioms of ZFC is still unknown. We will look into some of the combinatorics related to this problem and discuss its relationships with Selective Ultrafilters.]]>Talks2021-11-22T13:00:11+01:00https://settheory.pwr.edu.pl/blog/files/0fb41e54a9e55cc0fd2afc8642665099-136.php#unique-entry-id-136https://settheory.pwr.edu.pl/blog/files/0fb41e54a9e55cc0fd2afc8642665099-136.php#unique-entry-id-136
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Łukasz Mazurkiewicz

Title: Families of sets closed under Suslin operation

Abstract: In the talk we discuss some extensions of classical results regarding closure of measurable sets and sets with Baire property under Suslin operation. This will lead us to the theory of analytic sets, where the example of an analytic set created by Lusin will be considered.]]>Talks2021-11-14T09:06:58+01:00https://settheory.pwr.edu.pl/blog/files/af1bcc597b965bdef40fd4a6d70460f2-135.php#unique-entry-id-135https://settheory.pwr.edu.pl/blog/files/af1bcc597b965bdef40fd4a6d70460f2-135.php#unique-entry-id-135
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Maciej Korpalski

Title: Combinatorial characterization of null set covering

Abstract: In this talk we will recall a result from Bartoszynski regarding partial characterization of covering coefficient of ideal formed by sets with Lebesgue measure equal to zero. This is done in terms of slaloms and small sets. This theorem's proof had some problems along the way and we will see how to fix them.]]>Talks2021-11-08T16:45:11+01:00https://settheory.pwr.edu.pl/blog/files/ff6e6b11c4f1caf8222391029672e847-134.php#unique-entry-id-134https://settheory.pwr.edu.pl/blog/files/ff6e6b11c4f1caf8222391029672e847-134.php#unique-entry-id-134
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Grzegorz Plebanek

Title: Generalized Corson compacta and calibers of measures

Abstract: We consider compact spaces which can be embedded into a product of real lines so that the support of every element is of size \(<\kappa\); here \(\kappa\) is a fixed cardinal number. We discuss measure-theoretic properties of such spaces and related properties of Banach spaces of continuous functions. ]]>Talks2021-10-22T09:30:26+02:00https://settheory.pwr.edu.pl/blog/files/b0e4547dc51ddb565b34dd3dc7d0d92f-133.php#unique-entry-id-133https://settheory.pwr.edu.pl/blog/files/b0e4547dc51ddb565b34dd3dc7d0d92f-133.php#unique-entry-id-133
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Piotr Borodulin-Nadzieja


Title: On P-measures in random model

Abstract: We consider compact spaces which can be embedded into a product of real lines so that the support of every element is of size \(<\kappa\); here \(\kappa\) is a fixed cardinal number. We discuss measure-theoretic properties of such spaces and related properties of Banach spaces of continuous functions. ]]>Talks2021-10-15T07:17:43+02:00https://settheory.pwr.edu.pl/blog/files/6afed00423eb39a2a8ce353cbd2936a4-132.php#unique-entry-id-132https://settheory.pwr.edu.pl/blog/files/6afed00423eb39a2a8ce353cbd2936a4-132.php#unique-entry-id-132
Location: room 605, Mathematical Institute, University of Wroclaw

Speaker: Alberto Salguero Alarcón (Universidad de Extremadura, Badajoz, Spain)


Title: A twisted sum of \(C(K)\)-spaces not isomorphic to any \(C(K)\)-space

Abstract: A twisted sum of two Banach spaces \(X\) and \(Y\) is another space \(Z\) containing \(Y\) as a closed subspace so that \(Z/Y=X\). In this talk we focus on twisted sums of spaces of continuous functions on compact spaces. It has been known for some time that a twisted sum of two \(C(K)\)-spaces does not need to be isomorphic to a \(C(K)\)-space. We will focus on one recent and singular construction which serves as an example: a twisted sum of \(c_0\) and \(c_0(\mathfrak c)\) which is not isomorphic to any \(C(K)\)-space. This is part of a joint work with Grzegorz Plebanek. ]]>Talks2021-10-10T21:34:40+02:00https://settheory.pwr.edu.pl/blog/files/702e6a3a7ba70139778a6f96b060e1d3-131.php#unique-entry-id-131https://settheory.pwr.edu.pl/blog/files/702e6a3a7ba70139778a6f96b060e1d3-131.php#unique-entry-id-131
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.]]>Talks2021-06-03T09:27:17+02:00https://settheory.pwr.edu.pl/blog/files/697a6054a0f8f2acaac81e09325dc8a7-130.php#unique-entry-id-130https://settheory.pwr.edu.pl/blog/files/697a6054a0f8f2acaac81e09325dc8a7-130.php#unique-entry-id-130
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Maciej Malicki (Polish Academy of Sciences)

Title: Infinitary continuous logic and descriptive set theory

Abstract: There are deep connections between model theory of the infinitary logic and descriptive set theory: Scott analysis, the López-Escobar theorem or the Suzuki theorem are well known examples of this phenomenon. In this talk, I will present results of a research devoted to generalizing these connections to the setting of continuous infinitary logic and Polish metric structures. In particular, I will discuss a continuous counterpart of a theorem of Hjorth and Kechris characterizing essential countability of the isomorphism relation on a given Borel class of countable structures. As an application, I will give a short model-theoretic proof of a result of Kechris saying that orbit equivalence relations induced by continuous actions of locally compact Polish groups are essentially countable. This is joint work with Andreas Hallbäck and Todor Tsankov.
]]>Talks2021-05-19T18:30:48+02:00https://settheory.pwr.edu.pl/blog/files/655f6d6fb73586fc3269c34255db7a4d-129.php#unique-entry-id-129https://settheory.pwr.edu.pl/blog/files/655f6d6fb73586fc3269c34255db7a4d-129.php#unique-entry-id-129
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Udayan Darji (University of Louisville)

Title: Local Entropy and Descriptive Complexity

Abstract: Blanchard introduced the concepts of Uniform Positive Entropy (UPE) and Complete Positive Entropy (CPE) as topological analogues of K-automorphism. He showed that UPE implies CPE, and that the converse is false. A flurry of recent activities study the relationship between these two notions. For example, one can assign a countable ordinal which measures how complicated a CPE system is. Recently, Barbieri and Gracia-Ramos constructed Cantor CPE system at every level of CPE. Westrick showed that natural rank associated to CPE systems is actually a $\Pi^1_1\)-rank. More importantly, she showed that the collection of CPE \(Z_2\) SFT's is a \(\Pi^1_1\)-complete set. In this talk, we discuss some results, where UPE and CPE coincide and others where we show that the complexity of certain classes of CPE systems is \(\Pi^1_1\)-complete. This is joint work with Garica-Ramos.]]>Talks2021-05-13T15:11:53+02:00https://settheory.pwr.edu.pl/blog/files/466c5140a7a67fa5abc6f32098fcfe56-128.php#unique-entry-id-128https://settheory.pwr.edu.pl/blog/files/466c5140a7a67fa5abc6f32098fcfe56-128.php#unique-entry-id-128
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Matteo Viale (University of Torino)

Title: Absolute model companionship, forcibility, and the Continuum Problem

Abstract: Absolute model companionship (AMC) is a strengthening of model companionship defined as follows:
For a theory \(T\) , \(T_{\exists\lor\forall}\) denotes the logical consequences of \(T\) which are boolean combinations of universal sentences. \(T\) is the AMC of \(T^*\) if it is model complete and \(T_{\exists\lor\forall}=T^*_{\exists\lor\forall}\).

The \(\{+, ·, 0, 1\}\)-theory ACF of algebraically closed field is the model companion of the theory of Fields but not its AMC as \(\exists x (x^2+1=0)\in ACF_{\exists\lor\forall}\setminus Fields_{\exists\lor\forall}\).

We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) \(2^{\aleph_0} = \aleph_2\) is the unique solution to the continuum problem which can be in the AMC of a partial Morleyization of the \(\in\)-theory
ZFC+there are class many supercompact cardinals. We also show that (assuming large cardinals) forcibility overlaps with the apparently stronger notion of consistency for any mathematical problem \(\varphi\) expressible as a \(\Pi_2\) -sentence of a (very large fragment of) third order arithmetic (CH, the Suslin hypothesis, the Whitehead conjecture for free groups, are a small sample of such problems \(\varphi\)).

Partial Morleyizations can be described as follows: let \(Form_{\tau}\) be the set of first order \(\tau\)-formulas; for a subset A of \(Form_{\tau}\), \(\tau_A\) is the expansion of \(\tau\) adding atomic relation symbols \(R_\varphi\) for all formulas \(\varphi\in A\) and \(T_{\tau,A}\) is the \(\tau_A\)-theory asserting that each \(\tau\)-formula \(\varphi(x)\in A\) is logically equivalent to the corresponding atomic formula \(R_\varphi (x\sim x)\). For a \(\tau\)-theory T, \(T + Ti_{\tau,A}\) is the partial Morleyization of T induced by \(A\subseteq F_\tau\).]]>Talks2021-05-06T18:05:33+02:00https://settheory.pwr.edu.pl/blog/files/462eec857773b15f69ffcc80c99050f0-127.php#unique-entry-id-127https://settheory.pwr.edu.pl/blog/files/462eec857773b15f69ffcc80c99050f0-127.php#unique-entry-id-127
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Adam Kwela (University of Gdańsk)

Title: Unboring ideals

Abstract: We say that a space \(X\) is \(FinBW(I)\) (\(I\) is an ideal on the set of natural numbers), if for each sequence \((x_n)\) in \(X\) one can find a set \(A\) not belonging to \(I\) such that \((x_n)_{n\in A}\) converges in \(X\). Thus, the classical Bolzano-Weierstrass theorem states that every compact subset of the real line is \(FinBW(Fin)\) (\(Fin\) is the ideal of all finite subsets of naturals). During my talk I will present new results concerning \(FinBW(I)\) spaces and discuss relationship between the studied notions and the Katetov order on ideals. In particular, under \(MA\) I will characterize for all \(\Pi^0_4\) ideals when \(FinBW(I)\) and \(FinBW(J)\) differ.]]>Talks2021-04-27T04:53:12+02:00https://settheory.pwr.edu.pl/blog/files/eea74278471e3ef71860ee959e91228f-126.php#unique-entry-id-126https://settheory.pwr.edu.pl/blog/files/eea74278471e3ef71860ee959e91228f-126.php#unique-entry-id-126
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Szymon Żeberski

Title: Applications of non-measurable unions

Abstract: Using a game-theoretic approach (Set-Cover game) we obtain a generalization of the classical result of Brzuchowski, Cichoń, Grzegorek and Ryll-Nardzewski on non-measurable unions. We will present applications of this result to establishing some countability and continuity properties of measurable functions and homomorphisms between topological groups.

It is a joint work with Taras Banakh and Robert Rałowski https://arxiv.org/abs/2011.11342.]]>Talks2021-04-14T11:44:53+02:00https://settheory.pwr.edu.pl/blog/files/9a17f1ce05e8575c869a2385e8483069-125.php#unique-entry-id-125https://settheory.pwr.edu.pl/blog/files/9a17f1ce05e8575c869a2385e8483069-125.php#unique-entry-id-125
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Aristotelis Panagiotopoulos (University of Munster)

Title: The definable content of (co)homological invariants: Cech cohomology

Abstract: In this talk we will develop a framework for enriching various classical invariants of homological algebra and algebraic topology with additional descriptive set-theoretic information. The resulting "definable invariants" can be used for much finer classification than their purely algebraic counterparts. We will illustrate how these ideas apply to the classical Cech cohomology invariants to produce a new "definable cohomology theory" which, unlike its classical counterpart, it provides a complete classification to homotopy classes of mapping telescopes of d-tori, and for homotopy classes of maps from mapping telescopes of d-tori to spheres. In the process, we will develop several Ulam stability results for quotients of Polish abelian non-archimedean groups G by Polishable subgroups H. A special case of these rigidity results answer a question of Kanovei and Reeken regarding quotients of the \(p\)-adic groups.

This is joint work with Jeffrey Bergfalk and Martino Lupini.]]>Talks2021-04-12T17:02:18+02:00https://settheory.pwr.edu.pl/blog/files/5ff7637d5e7c3f9884de87a774c65bd2-124.php#unique-entry-id-124https://settheory.pwr.edu.pl/blog/files/5ff7637d5e7c3f9884de87a774c65bd2-124.php#unique-entry-id-124
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Witold Marciszewski (University of Warsaw)

Title: On zero-dimensional subspaces of Eberlein compacta

Abstract: Let us recall that a compact space K is Eberlein compact if it can be embedded into some Banach space X equipped with the weak topology. Our talk will be devoted to the known problem of the existence of nonmetrizable compact spaces without nonmetrizable zero-dimensional closed subspaces. Several such spaces were obtained using some additional set-theoretic assumptions. Recently, P. Koszmider constructed the first such example in ZFC. We investigate this problem for the class of Eberlein compact spaces. We construct such Eberlein compacta, assuming the existence of a Luzin set. We also show that it is consistent with ZFC that each Eberlein compact space of weight greater than \(\omega_1\) contains a nonmetrizable closed zero-dimensional subspace.

The talk is based on the paper "On two problems concerning Eberlein compacta": http://arxiv.org/abs/2103.03153]]>Talks2021-03-25T02:47:55+01:00https://settheory.pwr.edu.pl/blog/files/aaaa9595359b767948e1de1935cd1a65-123.php#unique-entry-id-123https://settheory.pwr.edu.pl/blog/files/aaaa9595359b767948e1de1935cd1a65-123.php#unique-entry-id-123
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Gonzalo Martinez Cervantes (University of Murcia)

Title: L-orthogonal sequences versus L-orthogonal elements

Abstract: Let \(X\) be a Banach space. We say that a sequence \(\{x_n\}_n\) in the sphere of a Banach space \(X\) is an L-orthogonal sequence if the norm of \(x+x_n\) converges to \(1+\|x\|\) for every \(x\) in \(X\). On the other hand, we say that an element \(x^{**}\) in the sphere of \(X^{**}\) is L-orthogonal to \(X\) if the norm of \(x^{**}+x\) is equal to \(1+\|x\|\) for every \(x\) in \(X\). In this talk we will recall some results due to G. Godefroy, N. J. Kalton, B. Maurey, V. Kadets, V. Shepelska and D.Werner relating these concepts to the containment of an isomorphic copy of \(\ell_1\). It is natural to conjecture that the weak*-closure of an L-orthogonal sequence always contains L-orthogonal elements in the bidual. Indeed, this is the case for separable Banach spaces. We will see that this conjecture is independent of ZFC. Namely, we provide an affirmative answer under the existence of selective ultrafilters, whereas a counterexample can be constructed if no Q-point exists.

This is a joint work (in progress) with Antonio Avilés and Abraham Rueda Zoca.]]>Talks2021-03-17T09:33:21+01:00https://settheory.pwr.edu.pl/blog/files/0db871fc19ceeb698a4e4659a3ed8ad2-122.php#unique-entry-id-122https://settheory.pwr.edu.pl/blog/files/0db871fc19ceeb698a4e4659a3ed8ad2-122.php#unique-entry-id-122
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Władysław Wilczyński (University of Łódź)

Title: Convergence with respect to measure and category

Abstract: D. Fremlin in 1975 has proved that if \((X,S,m)\) is a probability space, then a sequence of measurable functions on \(X\) either has a subsequence convergent a.e., or there exists a subsequence without measurable pointwise cluster point. His proof is based upon the properties of weak convergent sequences in square integrable functions. The weaker form of the theorem was proved by Bucchioni and Goldman in1978. Their proof uses only some properties of the pair (family of measurable subsets of \([0,1]\), family of null sets). The pair (family of subsets of \([0,1]\) having the Baire property, family of sets of the first category) behaves similarly , so it was possible to obtain similar result for the convergence in category considered by E. Wagner in 1978.

Some lemmas similar to that in the paper of Bucchioni were used earlier to prove the equivalence of the convergence in category and the Cauchy condition for this type of convergence.]]>Talks2021-03-10T15:24:46+01:00https://settheory.pwr.edu.pl/blog/files/7c3293f059d7e6b143657e9733d0aff5-121.php#unique-entry-id-121https://settheory.pwr.edu.pl/blog/files/7c3293f059d7e6b143657e9733d0aff5-121.php#unique-entry-id-121
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Arturo Antonio Martínez Celis Rodríguez (University of Wroclaw)

Title: Rosenthal Families

Abstract: A collection of infinite subsets of the natural numbers is a Rosenthal family if it can replace the family of all infinite subsets in a classical Lemma by Rosenthal concerning sequences of measures on pairwise disjoint sets. In this talk we will show that every ultrafilter is a Rosenthal family and that the minimal size of a Rosenthal family is the reaping number. We will also try to show some connections to functional analysis.]]>Talks2021-03-03T17:25:24+01:00https://settheory.pwr.edu.pl/blog/files/579cf70b8580789ee710bacc7d7abc41-120.php#unique-entry-id-120https://settheory.pwr.edu.pl/blog/files/579cf70b8580789ee710bacc7d7abc41-120.php#unique-entry-id-120
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Grigor Sargsyan (Rutgers & IMPAN)

Title: The exact strength of Sealing

Abstract: Shoenfield's celebrated absoluteness theorem says that no \(\Sigma^1_2\) fact \(\phi\) can be shown to be independent of the axioms of ZFC via the method of forcing. A set of reals is universally Baire if its continuous preimages have the Baire property in all topological spaces. Can there be independence results about such sets?

Sealing is a generic absoluteness statement which was introduced by Woodin. First given a generic object \(g\), let \(\Gamma^\infty_g\) be the set of universally Baire sets of \(V[g]\) and \(R_g\) be the set of reals of \(V[g]\).
Sealing (essentially) says that for all \(V\)-generic \(g\) and all \(V[g]\)-generic \(h\) there is an embedding
\(j: L(\Gamma^\infty_g, R_g)\to L(\Gamma^\infty_g*h, R_g*h).\)

Thus, in a way, Sealing says that there cannot be independence results about universally Baire sets, and as such it is a generalization of Shoenfield's absoluteness theorem.

It is an open problem if large cardinals imply Sealing. No canonical inner model can satisfy it, and so if some large cardinal implies it then its inner model theory must be significantly different than the current theory we have. Surprisingly, Woodin showed that if there are proper class of Woodin cardinals and delta is a supercompact then collapsing \(2^{2^\delta}\) to be countable forces Sealing. Because of its impact on the inner model problem and because of Woodin's result, it seemed that the set theoretic strength of Sealing must be at the level of supercompact cardinals. However, the speaker and Nam Trang showed that it is weaker than a Woodin cardinal that is a limit of Woodin cardinals (which are significantly smaller than supercompact cardinals). We will exposit this theorem and will also explain its consequences on the inner model problem. ]]>Talks2021-02-25T11:07:40+01:00https://settheory.pwr.edu.pl/blog/files/64f8a8c1715bcb1f6dba9975f955e01b-119.php#unique-entry-id-119https://settheory.pwr.edu.pl/blog/files/64f8a8c1715bcb1f6dba9975f955e01b-119.php#unique-entry-id-119
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Benjamin Vejnar (Charles University, Prague)

Title: Complexity of some classes of metrizable compacta up to homeomorphism

Abstract: There is a general framework called Invariant Descriptive Set Theory (IDST) which can be used to measure the complexities of classification problems. We follow the framework IDST when studying the complexity of compact metrizable spaces, continua, absolute retracts, rim-finite continua, dendrites, or rim-finite compacta up to homeomorphism. Using the tools of IDST we show that there is no compact metrizable space such that every continuum is homeomorphic to exactly one component of this space. This can be used to answer a question by P. Minc.]]>Talks2021-01-28T11:27:39+01:00https://settheory.pwr.edu.pl/blog/files/9812db2a193d5e64a3e68181161ec5d9-118.php#unique-entry-id-118https://settheory.pwr.edu.pl/blog/files/9812db2a193d5e64a3e68181161ec5d9-118.php#unique-entry-id-118
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Marton Elekes (Alfred Renyi Institute)

Title: Games, their values, and Baire class 1 functions

Abstract: We consider interesting descriptive set-theoretic problems emerging from theoretical economics. First, we investigate a certain two-player game coming from gambling theory. Then, as a by-product, we obtain a novel game that characterizes the Baire class 1 functions. Finally, we determine the exact complexity of the so-called value of the above game, which turns out to be a less well-known class, namely analytic-inductive.]]>Talks2021-01-20T16:57:00+01:00https://settheory.pwr.edu.pl/blog/files/3a5bb7692a12bd948f61be1b82f79dd5-117.php#unique-entry-id-117https://settheory.pwr.edu.pl/blog/files/3a5bb7692a12bd948f61be1b82f79dd5-117.php#unique-entry-id-117
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Grzegorz Plebanek

Title: A connected version of Kunen's compact L-space

Abstract: Modifying Kunen's construction from 1981, we show that under CH there is a compact connected space K that carries a regular normal probability measure (normal = `all Borel sets with empty interior have measure zero'). Then we show that the Banach space C(K) of continuous functions is isomorphic to no space of the form C(L) with L compact and zero-dimensional.
]]>Talks2021-01-12T10:08:02+01:00https://settheory.pwr.edu.pl/blog/files/14bccf57112dd386e5ef13d674c8839a-116.php#unique-entry-id-116https://settheory.pwr.edu.pl/blog/files/14bccf57112dd386e5ef13d674c8839a-116.php#unique-entry-id-116
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Michael Hrusak (National Autonomous University of Mexico)

Title: Invariant Ideal Axiom

Abstract: We shall introduce a consistent set-theoretic axiom IIA which has a profound impact on convergence properties in topological groups. As an application we show that consistently (consequence of IIA) every countable sequential group is either metrizable or \(k_\omega\).
]]>Talks2020-12-10T19:55:10+01:00https://settheory.pwr.edu.pl/blog/files/6c2d2bc8b6d2e1673e63a49197de6053-115.php#unique-entry-id-115https://settheory.pwr.edu.pl/blog/files/6c2d2bc8b6d2e1673e63a49197de6053-115.php#unique-entry-id-115
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Dana Bartošová (University of Florida)

Title: Attempts to understand the universal minimal flow of \(\mathbb{Z}\times\mathbb{Z}\)

Abstract: Every \(\mathbb{Z}\)-flow on a compact Hausdorff space \(X\) can be interpreted as a homeomorphism \(f : X \to X\) and its forward and backward iterates. A flow is minimal if every orbit is dense. The universal minimal flow \(M(\mathbb{Z})\) maps continuously onto every minimal flow while preserving the action, and it is unique up to isomorphism. The purpose of this project is to understand \(M(\mathbb{Z} \times \mathbb{Z})\) in terms of \(M(\mathbb{Z})\). We will start with the few results that are out there about the connection between the corresponding Čech-Stone compactifications \(\beta (\mathbb{Z}\times\mathbb{Z})\) and \(\beta (\mathbb{Z})\) by Hindman, Blass, and Blass and Moche, that are useful in our considerations. This is a joint work with Ola Kwiatkowska.
]]>Talks2020-12-03T11:17:56+01:00https://settheory.pwr.edu.pl/blog/files/50dfafcb6cad577cc3d2f9848dd7a9e4-114.php#unique-entry-id-114https://settheory.pwr.edu.pl/blog/files/50dfafcb6cad577cc3d2f9848dd7a9e4-114.php#unique-entry-id-114
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Włodzimierz J. Charatonik (Missouri University of Science and Technology)

Title: Projective Fraïssé limits of trees

Abstract: We continue study of projective Fraïssé limit developed by Irvin, Panagiotopoulos and Solecki. We modify the ideas of monotone, confluent, or retraction from continuum theory as well as several properties of continua so as to apply to topological graphs. As the topological realizations of the Fraïssé limits we obtain either some known continua, for example the dendrite \(D_3\) or the Cantor fan, or quite new, interesting ones for which we do not yet have topological characterizations.
]]>Talks2020-11-25T11:05:15+01:00https://settheory.pwr.edu.pl/blog/files/3a452cf6dcc0b642d6f84dfdc24250be-113.php#unique-entry-id-113https://settheory.pwr.edu.pl/blog/files/3a452cf6dcc0b642d6f84dfdc24250be-113.php#unique-entry-id-113
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Andrzej Rosłanowski (University of Nebraska Omaha)

Title: Borel sets without perfectly many overlapping translations

Abstract: For a perfect Abelian Polish group H we force a Borel set B which has many translations with pairwise intersections of size at least k, but does not have a perfect set of such translations. This is joint work with Saharon Shelah.
]]>Talks2020-11-19T11:38:02+01:00https://settheory.pwr.edu.pl/blog/files/6cf673683abb30f07f86b34f0f17b81e-112.php#unique-entry-id-112https://settheory.pwr.edu.pl/blog/files/6cf673683abb30f07f86b34f0f17b81e-112.php#unique-entry-id-112
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Jerzy Krzempek (Silesian University of Technology)

Title: End points of chainable continua

Abstract: Answering a question posed by R. Adikari and W. Lewis, I shall prove that for every zero-dimensional separable metric space G there is a Suslinian chainable continuum whose end points form a set homeomorphic to G. I will discuss some structural properties of such continua.
]]>Talks2020-11-13T23:14:35+01:00https://settheory.pwr.edu.pl/blog/files/64c8c04f97673d9f118fb03fa27f4e4a-111.php#unique-entry-id-111https://settheory.pwr.edu.pl/blog/files/64c8c04f97673d9f118fb03fa27f4e4a-111.php#unique-entry-id-111
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Sławomir Solecki (Cornell University)

Title: Random continuum and Brownian motion

Abstract: We describe a probabilistic model involving iterated Brownian motion for constructing a random chainable continuum. We show that this random continuum is indecomposable. We use our probabilistic model to define a Wiener-type measure on the space of all chainable continua. This is joint work with Viktor Kiss.
]]>Talks2020-11-04T18:48:21+01:00https://settheory.pwr.edu.pl/blog/files/ac8ad8d754ddd958036a20ee6f3821f6-110.php#unique-entry-id-110https://settheory.pwr.edu.pl/blog/files/ac8ad8d754ddd958036a20ee6f3821f6-110.php#unique-entry-id-110
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Mirna Dzamonja (CNRS & Pantheon-Sorbonne University & Czech Academy of Sciences)

Title: On wide Aronszajn trees

Abstract: Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size \(\omega_1\) but with no uncountable branches play an important role by being clocks of Ehrenfeucht--Fraïssé games that measure similarity of model of size \(\aleph_1\). We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that map T into T’ while preserving the strict order \(<_T\). This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wide Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would be under MA(\(\omega_1\)).

In our upcoming JSL paper with Saharon Shelah we prove that this is not the case: under MA(\(\omega_1\)) there is no universal wide Aronszajn tree.

The talk will discuss that paper. The paper is available on the arxiv and on line at JSL in the preproof version DOI: 10.1017/jsl.2020.42 ]]>Talks2020-10-27T16:02:56+01:00https://settheory.pwr.edu.pl/blog/files/be927d60e68c6db97aed6cef629ea167-109.php#unique-entry-id-109https://settheory.pwr.edu.pl/blog/files/be927d60e68c6db97aed6cef629ea167-109.php#unique-entry-id-109
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Paweł Krupski

Title: The complexity of homogeneous continua

Abstract: I will show that the family of all homogeneous continua in the hyperspace of all subcontinua of the cube \(I^n, n=2,3,\ldots ,\omega\), is analytic and contains a topological copy of the linear space \(c_0=\{(x_k)\in {\mathbb{R}}^\omega: \lim x_k=0\}\) as a closed subset. A historical background will also be sketched.]]>Talks2020-10-21T15:10:06+02:00https://settheory.pwr.edu.pl/blog/files/ca5bbbd6271f5fab9abb1fbd6bb4745f-108.php#unique-entry-id-108https://settheory.pwr.edu.pl/blog/files/ca5bbbd6271f5fab9abb1fbd6bb4745f-108.php#unique-entry-id-108
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Antonio Aviles (University of Murcia)

Title: Amalgamation of measures and Banach lattices

Abstract: Given two measures that coincide on the intersection of their domains, can we find a measure that is a common extension of those two? Kellerer's results on marginal measures constitute an important partial positive answer. We will see how this is connected to some basic properties of the category of Banach lattices, like amalgamation and existence of injective objects. Joint work with Pedro Tradacete.]]>Talks2020-10-14T07:59:41+02:00https://settheory.pwr.edu.pl/blog/files/2daed34469780c327e505af4f85adff9-107.php#unique-entry-id-107https://settheory.pwr.edu.pl/blog/files/2daed34469780c327e505af4f85adff9-107.php#unique-entry-id-107
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Jan van Mill (University of Amsterdam)

Title: Splitting Tychonoff cubes into homeomorphic and homogeneous parts (and more)

Abstract: We prove (among other things) that if \(X\) is the Tychonoff cube of weight \(\tau\), where \(\tau\) is uncountable, and \(\mathcal{E}\) is a cover of \(X\) by subspaces each homeomorphic to a topological group, then \(|\mathcal{E}|\ge \tau^+\).]]>Talks2020-10-07T19:24:07+02:00https://settheory.pwr.edu.pl/blog/files/fb0e322ab1deb7368772afb396151c51-106.php#unique-entry-id-106https://settheory.pwr.edu.pl/blog/files/fb0e322ab1deb7368772afb396151c51-106.php#unique-entry-id-106
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Robert Rałowski

Title: Nonmeasurable unions with respect to analytic families

Abstract: We say that metric \(\rho\) is analytic on Hausdorff topological space if
1) \((X,\rho)\) is separable metric space,
2) identity \(id:(X,\rho) \to X\) is continuous,
2) every \(\rho\)-Cauchy sequence is converged in \(X.\)

Family \(\mathcal A \subseteq P(X)\) is analytic if
1) \(X\in \mathcal A\)
2) \(\mathcal A\) is closed on intersections
3) each \(A\in{\mathcal A}\) has analytic metric \(\rho\) and for any \(\epsilon>0\) there is a countable cover \(\mathcal U \subseteq \mathcal A\) of \(\mathcal A\) with \(\epsilon\) \(\rho\)-diameter.

We present a theorem that generalizes the well known result obtained by Brzuchowski, Cichoń, Grzegorek and Ryll-Nardzewski about nonmeasurable unions.

Theorem. Let \({\mathcal A}\) be an analytic family of Hausdorff space \(X\), any \(I\) \(\sigma\)-ideal of \(X.\) If \( J\subseteq I\) is point-finite family such that \(\bigcup J \notin I\) then there is a subfamily \(J' \subseteq J\) and \(A\in {\mathcal A}\) such that
1) \(A\cap \bigcap J' \notin I\)
2) for every \(A' \in {\mathcal A}\) if \(A' \subseteq A\cap \bigcup J'\) then \(A' \in I.\)

We show that the above Theorem implies the Theorem on nomeasurabie unions with respect to tree ideals like Marczeski ideal \(s_0\) for example.
Moreover, the above Theorem implies theorem on nonmeasurable unions with respect to \(\sigma\)-ideals which has Marczewski-Burstin representation.

The last mentioned result gives a theorem about nonmeasurable unions with respect to the ideal of Ramsey-null set in Ramsey space with Ellentuck topology.

The talk is based on a joint work with Taras Banakh and Szymon Żeberski.]]>Talks2020-06-18T10:43:36+02:00https://settheory.pwr.edu.pl/blog/files/329d561403a3b7ca8e2ed4e0db7d14ef-105.php#unique-entry-id-105https://settheory.pwr.edu.pl/blog/files/329d561403a3b7ca8e2ed4e0db7d14ef-105.php#unique-entry-id-105
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Sakae Fuchino (Kobe University)

Title: A/the (possible) solution of the Continuum Problem

Abstract. In this talk, I examine the following trichotomy which holds under the requirement that a sufficiently strong natural reflection principle should hold:

The continuum (\(=2^{\aleph_0}\)) is either 1. \(\aleph_1\) or 2. \(\aleph_2\) or 3. fairly large.

Here, the fair largeness of the continuum can be expressed either in terms of weak mahloness and/or some other ``large'' cardinal notions compatible with the continuum, or even in terms of existence of some saturated ideals.

The reflection principles we consider here can be formulated as the following type of Downward Löwenheim-Skolem Theorems:

1'. For any structure A of countable signature, there is an elementary substructure B of A of cardinality \(<\aleph_2\) in terms of stationary logic.

2'. For any structure A of countable signature, there is an elementary substructure B of A of cardinality \(<2^{\aleph_0}\) in terms of stationary logic but only for formulas without free second order variables.

3'. For any structure A of countable signature, there is an elementary substructure B of A of cardinality \(<2^{\aleph_0}\) in terms of PKL logic (a variant of the stationary logic) in weak interpretation.

The reflection points \(<\aleph_2\) and \(<2^{\aleph_0}\) can be considered to be natural/necessary since the reflection down to \(<\aleph_2\) declares that \(\aleph_1\) strongly represents the situation of uncountability; the reflection down to \(<2^{\aleph_0}\) can be interpreted in the way that the reflection manifests that the continuum is very "rich".
The Downward Löwenheim-Skolem Theorems in terms of stationary logics can be also regarded as very natural principles: They can be characterized in terms of Diagonal Reflection Principles of Sean Cox.

Analyzing these three scenarios, we obtain the notion of Laver-generically large cardinals.
Existence of a Laver-generically supercompact cardinal

1''. for \(\sigma\)-closed pos implies 1'.;

2''. for proper pos implies 2'.; while the existence of a Laver-generically supercompact cardinal

3''. for ccc pos implies 3'.

The symmetry of the arguments involved suggests the possibility that the trichotomy might be a set-theoretic multiversal necessity.

If time allows, I shall also discuss about the reflection of non-metrizability of topological spaces, Rado's Conjecture and Galvin's Conjecture in connection with the reflection properties in 1., 2. and 3.

Most of the results to be presented here are obtained in a joint work with Hiroshi Sakai and André Ottenbreit Maschio Rodrigues.]]>Talks2020-06-11T22:59:10+02:00https://settheory.pwr.edu.pl/blog/files/9d6e1e4358f309d6c4a4b0e7109931cd-104.php#unique-entry-id-104https://settheory.pwr.edu.pl/blog/files/9d6e1e4358f309d6c4a4b0e7109931cd-104.php#unique-entry-id-104
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Gianluca Basso (Université de Lausanne & Torino)

Title: The universal minimal flow of topological groups beyond Polish

Abstract. When \(G\) is a Polish group, one way of knowing that it has "nice" dynamics is to show that \(M(G)\), the universal minimal flow of \(G\), is metrizable. For non-Polish groups, this is not the relevant dividing line: the universal minimal flow of \( \mathrm{Sym}(\kappa) \) is the space of linear orders on \(\kappa\)—not a metrizable space, but still "nice"—, for example. In this talk, we present a set of equivalent properties of topological groups which characterize having "nice" dynamics. We show that the class of groups satisfying such properties is closed under some topological operations and use this to compute the universal minimal flows of some concrete groups, like \(\mathrm{Homeo}(\omega_{1})\). This is joint work with Andy Zucker.]]>Talks2020-06-03T20:45:29+02:00https://settheory.pwr.edu.pl/blog/files/0bc0ec2394a5296dad7df9f63a2b1d57-103.php#unique-entry-id-103https://settheory.pwr.edu.pl/blog/files/0bc0ec2394a5296dad7df9f63a2b1d57-103.php#unique-entry-id-103
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Wiesław Kubiś (Czech Academy of Sciences)

Title: Uniform homogeneity

Abstract. A mathematical structure is called homogeneous if every isomorphism between its small substructures extends to an automorphism. Typically, "small" means "finite" or "finitely generated". A stronger variant, which we call "uniform homogeneity" requires that for each small substructure there is a suitable extension operator. We shall present examples of homogeneous but uniformly homogeneous structures. The talk is based on two works: one joint with S. Shelah (https://arxiv.org/abs/1811.09650), another one joint with B. Kuzeljevic (https://arxiv.org/abs/2004.13643).]]>Talks2020-05-27T11:18:30+02:00https://settheory.pwr.edu.pl/blog/files/36441aac52b0de4fb2848ad4c398b09c-102.php#unique-entry-id-102https://settheory.pwr.edu.pl/blog/files/36441aac52b0de4fb2848ad4c398b09c-102.php#unique-entry-id-102
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Witold Marciszewski (Uniwersytet Warszawski)

Title: On countable dense homogeneous topological vector spaces

Abstract. Recall that a topological space X is countable dense homogeneous (CDH) if X is separable, and given countable dense subsets D,E of X, there is an autohomeomorphism of X mapping D onto E. This is a classical notion tracing back to works of Cantor, Frechet and Brouwer. The canonical examples of CDH spaces include the Cantor set, the Hilbert cube, and all separable Banach spaces. All Borel, but not closed linear subspaces of Banach spaces are not CDH. By \(C_p(X)\) we denote the space of all continuous real-valued functions on a Tikhonov space X, endowed with the pointwise topology. V. Tkachuk asked if there exists a nondiscrete space X such that \(C_p(X)\) is CDH. Last year R. Hernandez Gutierrez gave the first consistent example of such a space X. He has asked whether a metrizable space X must be discrete, provided \(C_p(X)\) is CDH. We answer this question in the affirmative. Actually, combining our theorem with earlier results, we prove that, for a metrizable space X, \(C_p(X)\) is CDH if and only if X is discrete of cardinality less than pseudointersection number \(\mathfrak p\). We also prove that every CDH topological vector space X is a Baire space. This implies that, for an infinite-dimensional Banach space E, both spaces (E,w) and (E*,w*) are not CDH. We generalize some results of Hrusak, Zamora Aviles, and Hernandez Gutierrez concerning countable dense homogeneous products.

This is a joint work with Tadek Dobrowolski and Mikołaj Krupski. The preprint containing these results can be found here: https://arxiv.org/abs/2002.07423
]]>Talks2020-05-22T06:37:22+02:00https://settheory.pwr.edu.pl/blog/files/9840e742be0825462ca5bcd7bc36c1f8-101.php#unique-entry-id-101https://settheory.pwr.edu.pl/blog/files/9840e742be0825462ca5bcd7bc36c1f8-101.php#unique-entry-id-101
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Lyubomyr Zdomskyy (KGHR, Vienna)

Title: Menger and Hurewicz spaces: products and applications to forcing

Abstract. This talk will be devoted to (products of) Menger and Hurewicz spaces and their connections to forcing and mad families. In particular, we shall show that in the Laver model, each mad family can be destroyed by a ccc poset preserving the ground model reals unbounded and splitting. It is an important open problem whether the same follows from CH.]]>Talks2020-05-14T18:23:14+02:00https://settheory.pwr.edu.pl/blog/files/1aaf46c37c36c8621a16e1d14f945eb9-100.php#unique-entry-id-100https://settheory.pwr.edu.pl/blog/files/1aaf46c37c36c8621a16e1d14f945eb9-100.php#unique-entry-id-100
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Włodzimierz Charatonik

Title: Degree of non local connectedness

Abstract. For a given continuum \(X\) we assign a cardinal number or a symbol \(\infty\)  \(\tau(X)\) called degree of non local connectedness. The number \(\tau(X)\) cannot be increased by a continuous image; we show theorems about cartesian products, hyperspaces etc. Based on an article by Janusz J. Charatonik and Włodzimierz J. Charatonik]]>Talks2020-05-14T18:20:31+02:00https://settheory.pwr.edu.pl/blog/files/0a1f2fc07f1e7183c6a2866ab09a9f8c-99.php#unique-entry-id-99https://settheory.pwr.edu.pl/blog/files/0a1f2fc07f1e7183c6a2866ab09a9f8c-99.php#unique-entry-id-99
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Włodzimierz Charatonik

Title: Zero-dimensional compact metric spaces \(X\) whose squares \(X^2\) are homeomorphic to \(X\)

Abstract. We construct a family of cardinality \(\omega_1\) of (non homeomorphic) countable compact metric spaces \(X\) such that \(X\) is homeomorphic to \(X^2\). Based on an article by Włodzimierz J. Charatonik and Sahika Sahan.]]>Talks2020-05-05T21:56:55+02:00https://settheory.pwr.edu.pl/blog/files/e60e1f348643438ef3b289f6f9e82897-98.php#unique-entry-id-98https://settheory.pwr.edu.pl/blog/files/e60e1f348643438ef3b289f6f9e82897-98.php#unique-entry-id-98
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Taras Banakh

Title: Set Theoretic Problems in Large-Scale Topology

Abstract. We survey some set-theoretic problems appearing in large-scale topology.
More details can be found in the preprints (written jointly with Igor Protasov):
https://arxiv.org/abs/2004.01979
https://arxiv.org/abs/2002.08800]]>Talks2020-04-30T14:59:48+02:00https://settheory.pwr.edu.pl/blog/files/c57f5e9cd2b4b306f2b58c1dd5085495-97.php#unique-entry-id-97https://settheory.pwr.edu.pl/blog/files/c57f5e9cd2b4b306f2b58c1dd5085495-97.php#unique-entry-id-97
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Ziemowit Kostana (University of Warsaw)

Title: Cohen-like poset for adding Fraisse limits

Abstract. There exist a natural forcing notion which turns given countable set into a Fraisse limit of a given Fraisse class. This long-known phenomenon provided a rough intuition that Fraisse limits, as "generic structures", have some connections with forcing. The goal of the talk is to look at some particular instances and possible applications of this idea.]]>Talks2020-04-26T12:57:07+02:00https://settheory.pwr.edu.pl/blog/files/2a945d8c9d1e09e6b19043c5b11836c0-96.php#unique-entry-id-96https://settheory.pwr.edu.pl/blog/files/2a945d8c9d1e09e6b19043c5b11836c0-96.php#unique-entry-id-96
Location: Zoom.us: if you want to participate please contact organizers

Speaker: Aleksandra Kwiatkowska

Title: Simplicity of the automorphism groups of homogeneous structures

Abstract. We prove simplicity for the automorphism groups of order and tournament expansions of homogeneous structures like the bounded Urysohn space and the random graph. In particular, we will show that the automorphism group of the linearly ordered random graph is a simple group. The talk will be based on a preprint https://arxiv.org/pdf/1908.05249.pdf joint with Filippo Calderoni and Katrin Tent.]]>Talks2020-04-20T12:30:18+02:00https://settheory.pwr.edu.pl/blog/files/799edfa2ad2236228691cc4c8aee0c1b-95.php#unique-entry-id-95https://settheory.pwr.edu.pl/blog/files/799edfa2ad2236228691cc4c8aee0c1b-95.php#unique-entry-id-95
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.
]]>Talks2020-01-16T07:51:05+01:00https://settheory.pwr.edu.pl/blog/files/2ec54cb000dc91b6d5928d56e294d8ea-94.php#unique-entry-id-94https://settheory.pwr.edu.pl/blog/files/2ec54cb000dc91b6d5928d56e294d8ea-94.php#unique-entry-id-94
Room: D1-215

Speaker: Grzegorz Plebanek

Title: Baire Category Theorem in \(\mathbb{R}^\kappa\)

Abstract. A topological space X is BAIRE if it satisifes the theorem in the title. The space X is hereditary Baire if every closed subspace of X is Baire. We discuss a question which products of the real lines are hereditary Baire.
]]>Talks2019-11-27T21:51:58+01:00https://settheory.pwr.edu.pl/blog/files/f6770134bf128e9275379b18d8ced998-93.php#unique-entry-id-93https://settheory.pwr.edu.pl/blog/files/f6770134bf128e9275379b18d8ced998-93.php#unique-entry-id-93
Room: D1-215

Speaker: Olena Hryniv, Ivan Franko National University of Lviv

Title: A parallel metrization theorem

Abstract. Two non-empty sets \(A, B\) of a metric space \( (X , d)\) are called parallel if \( d(a, B) = d(A, B) = d(A, b) \) for any points \( a \in A \) and \( b \in B.\) Answering a question posed on mathoverflow.net, we prove that for a cover \( \mathcal{C}\) of a metrizable space \(X\) by compact subsets, the following conditions are equivalent:
(i) the topology of \(X\) is generated by a metric d such that any two sets \(A, B\) of \(\mathcal{C}\) are parallel;
(ii) the cover \( \mathcal{C}\) is disjoint, lower semicontinuous and upper semicontinuous.
]]>Talks2019-11-04T17:11:27+01:00https://settheory.pwr.edu.pl/blog/files/a80bf2977068a121f28cbc63051f228b-92.php#unique-entry-id-92https://settheory.pwr.edu.pl/blog/files/a80bf2977068a121f28cbc63051f228b-92.php#unique-entry-id-92
Room: D1-215

Speaker: Robert Rałowski

Title: Mycielski among trees - nonstandard proofs, part 2

Abstract. We present proofs of Mycielski like Theorem for sigma ideal of meager subsets of Baire space and Egglestone like Theorem. In both proofs we use Schoenfield Absolutness Theorem. In Mycielski Theorem we replace the term perfect set by slalom perfect set what is some strengthen of the classical version. Results are from common paper with Marcin Michalski and Szymon Żeberski.]]>Talks2019-10-23T22:48:01+02:00https://settheory.pwr.edu.pl/blog/files/678548e9807f66547cf38a1b6f011d48-91.php#unique-entry-id-91https://settheory.pwr.edu.pl/blog/files/678548e9807f66547cf38a1b6f011d48-91.php#unique-entry-id-91
Room: D1-215

Speaker: Robert Rałowski

Title: Mycielski among trees - nonstandard proofs

Abstract. We present proofs of Mycielski like Theorem for sigma ideal of meager subsets of Baire space and Egglestone like Theorem. In both proofs we use Schoenfield Absolutness Theorem. In Mycielski Theorem we replace the term perfect set by slalom perfect set what is some strengthen of the classical version. Results are from common paper with Marcin Michalski and Szymon Żeberski.]]>Talks2019-05-19T22:16:00+02:00https://settheory.pwr.edu.pl/blog/files/530236bbea98d7a4b10fa34a0a530ed1-90.php#unique-entry-id-90https://settheory.pwr.edu.pl/blog/files/530236bbea98d7a4b10fa34a0a530ed1-90.php#unique-entry-id-90
Room: D1-215

Speaker: Grzegorz Plebanek

Title: Small almost disjoint families with applications

Abstract. We consider almost disjoint families that cannot be divided into n separated parts (for a fixed n). The basic question is what is the possible size of such a family. Those families are applicable to some problems on spaces of continuous functions.]]>Talks2019-04-04T15:37:57+02:00https://settheory.pwr.edu.pl/blog/files/af2b1a19143a06289b7ef5f072b88558-89.php#unique-entry-id-89https://settheory.pwr.edu.pl/blog/files/af2b1a19143a06289b7ef5f072b88558-89.php#unique-entry-id-89
Room: D1-215

Speaker: Szymon Żeberski

Title: Mycielski theorem and Miller trees

Abstract. The classical Mycielski theorem says that for comeager \(A\subseteq [0,1]^2\) one can find a perfect set \(P\) such that \(P\times P\subseteq A\cup\Delta\). (The same is true if we start with \(A\) of measure 1.)

We will discuss how far this can be generalized if we replace perfect set by superperfect set, i.e a body of a Miller tree.

It turns out that there is a comeager \(A\subseteq (\omega^\omega)^2\) such that \(A\cup \Delta\) does not contain any set of the form \(M\times M\), where \(M\) is superperfect.

However, for comeager \(A\subseteq [0,1]^2\) one can find a perfect set \(P\) and a superperfect set \(M\supseteq P\) such that \(P\times M\subseteq A\cup\Delta\).

We will also discuss measure case, where results are slightly different.]]>Talks2019-03-23T04:04:47+01:00https://settheory.pwr.edu.pl/blog/files/9d487528be15ce0355adeb0e2477894d-88.php#unique-entry-id-88https://settheory.pwr.edu.pl/blog/files/9d487528be15ce0355adeb0e2477894d-88.php#unique-entry-id-88
Room: D1-215

Speaker: Damian Sobota (University of Viena)

Title: Josefson-Nissenzweig theorem for C(K)-spaces

Abstract. The Josefson-Nissenzweig theorem is a powerful tool in Banach space theory. Its special version for Banach spaces of continuous functions reads as follows: for a given infinite compact space K there exists a sequence \((\mu_n)\) of normalized signed Radon measures on K such that the integrals \(\mu_n(f)\) converge to 0 for any function f in \(C(K)\). During my talk I will investigate when the sequence \((\mu_n)\) can be chosen in such a way that every \(\mu_n\) is just a finite linear combination of Dirac point measures (in other words, \(\mu_n\) has finite support). This will appear to have connections with the Grothendieck property of Banach spaces and complementability of the space \(c_0\). In particular, I'll present a very elementary proof that \(c_0\) is always complemented in a space \(C(K\times K)\).]]>Talks2019-02-23T07:54:41+01:00https://settheory.pwr.edu.pl/blog/files/883c996ccd023949ee8497755e0a796e-87.php#unique-entry-id-87https://settheory.pwr.edu.pl/blog/files/883c996ccd023949ee8497755e0a796e-87.php#unique-entry-id-87
Room: D1-215

Speaker: Barnabas Farkas (TU Wien)

Title: Degrees of destruction

Abstract. I'm going to present a survey on our results (joint with L. Zdomskyy) about the following strong notion of destroying Borel ideals: We say that the forcing notion \(\mathbb{P}\) \(+\)-destroys the Borel ideal \(\mathcal{I}\) if \(\mathbb{P}\) adds an \(\mathcal{I}\)-positive \(\dot{X}\) which has finite intersection with every \( A \in \mathcal{I}\cap V\). I will talk about the following:

1. Examples when usual destruction (that is, when \(\dot{X}\) required to be infinite only) implies \(+\)-destruction, and when it does not.
   
   
2. Characterization of those Borel ideals which can be \(+\)-destroyed, in particular, we will see that if \(\mathcal{I}\) can be \(+\)-destroyed then the associated Mathias-Prikry forcing \(+\)-destroys it.
   
   
3. Characterization of those analytic P-ideals which are \(+\)-destroyed by the associated Laver-Prikry forcing.
   

]]>Talks2019-01-08T21:21:40+01:00https://settheory.pwr.edu.pl/blog/files/3e7e0318bdd84d6d3c61106b9b74b8bc-86.php#unique-entry-id-86https://settheory.pwr.edu.pl/blog/files/3e7e0318bdd84d6d3c61106b9b74b8bc-86.php#unique-entry-id-86
Room: D1-215

Speaker: Daria Michalik

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 [1] 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 \(n\ge 2\).
   

[1] A. Illanes, V. Martinez-de-la-Vega, Symmetric products as cones, Topology Appl. 228 (2017), 36–46.

]]>Talks2018-12-06T21:12:54+01:00https://settheory.pwr.edu.pl/blog/files/f1bd99939302b2eecc4abaf61cf00a42-85.php#unique-entry-id-85https://settheory.pwr.edu.pl/blog/files/f1bd99939302b2eecc4abaf61cf00a42-85.php#unique-entry-id-85
Room: D1-215

Speaker: Sakae Fuchino

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.]]>Talks2018-11-26T10:01:07+01:00https://settheory.pwr.edu.pl/blog/files/fbde8823d674b9abb70a8f89d8b434f6-84.php#unique-entry-id-84https://settheory.pwr.edu.pl/blog/files/fbde8823d674b9abb70a8f89d8b434f6-84.php#unique-entry-id-84
Room: D1-215

Speaker: Serhii Bardyla

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.]]>Talks2018-11-07T06:43:13+01:00https://settheory.pwr.edu.pl/blog/files/0c5c48e8871b1c198f41b833c925d5f9-83.php#unique-entry-id-83https://settheory.pwr.edu.pl/blog/files/0c5c48e8871b1c198f41b833c925d5f9-83.php#unique-entry-id-83
Room: D1-215

Speaker: Robert Rałowski

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.]]>Talks2018-05-23T11:52:36+02:00https://settheory.pwr.edu.pl/blog/files/588dac20cfd70b073f1039efc7e6f16e-82.php#unique-entry-id-82https://settheory.pwr.edu.pl/blog/files/588dac20cfd70b073f1039efc7e6f16e-82.php#unique-entry-id-82
Room: D1-215

Speaker: Borisa Kuzeljevic

Title: P-ideal dichotomy and versions of the Suslin Hypothesis

Abstract. The talk will be about the relationship of P-ideal dichotomy with the statement that all Aronszajn trees are special. This is joint work with Stevo Todorcevic.]]>Talks2018-05-09T17:24:44+02:00https://settheory.pwr.edu.pl/blog/files/eb3494eb655d4c77237ee544764fad13-81.php#unique-entry-id-81https://settheory.pwr.edu.pl/blog/files/eb3494eb655d4c77237ee544764fad13-81.php#unique-entry-id-81
Room: D1-215

Speaker: Andrzej Starosolski

Title: The Rudin-Keisler ordering of P-points under b=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 \(\mathfrak b =\mathfrak  c\).]]>Talks2018-04-22T07:56:21+02:00https://settheory.pwr.edu.pl/blog/files/cbb3d1e5e7dbc398864ea3b5cebfbe33-80.php#unique-entry-id-80https://settheory.pwr.edu.pl/blog/files/cbb3d1e5e7dbc398864ea3b5cebfbe33-80.php#unique-entry-id-80
Room: D1-215

Speaker: Marek Bienias

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.]]>Talks2018-04-13T14:36:37+02:00https://settheory.pwr.edu.pl/blog/files/68b6c39594273f4458965730d777e935-79.php#unique-entry-id-79https://settheory.pwr.edu.pl/blog/files/68b6c39594273f4458965730d777e935-79.php#unique-entry-id-79
Room: D1-215

Speaker: Piotr Borodulin-Nadzieja

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.]]>Talks2018-03-20T21:43:30+01:00https://settheory.pwr.edu.pl/blog/files/0d9d6ecf26ab08ea045bfebf2885441f-78.php#unique-entry-id-78https://settheory.pwr.edu.pl/blog/files/0d9d6ecf26ab08ea045bfebf2885441f-78.php#unique-entry-id-78
Room: D1-215

Speaker: Grzegorz Plebanek

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 \(V=L\). ]]>Talks2018-03-07T21:13:57+01:00https://settheory.pwr.edu.pl/blog/files/b96b5324697d23c7eacf336a92896d9f-77.php#unique-entry-id-77https://settheory.pwr.edu.pl/blog/files/b96b5324697d23c7eacf336a92896d9f-77.php#unique-entry-id-77
Room: D1-215

Speaker: Grzegorz Plebanek

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. ]]>Talks2018-02-26T22:57:41+01:00https://settheory.pwr.edu.pl/blog/files/6b59ae1c7468156a111284441e37c359-76.php#unique-entry-id-76https://settheory.pwr.edu.pl/blog/files/6b59ae1c7468156a111284441e37c359-76.php#unique-entry-id-76
Room: D1-215

Speaker: Jacek Tryba

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.]]>Talks2017-12-18T15:17:05+01:00https://settheory.pwr.edu.pl/blog/files/653ec447a2fdc4a8d81ca3ef6f2d4714-75.php#unique-entry-id-75https://settheory.pwr.edu.pl/blog/files/653ec447a2fdc4a8d81ca3ef6f2d4714-75.php#unique-entry-id-75
Room: D1-215

Speaker: Olena Karlova

Title: Extension of Borel maps and Borel retracts of topological spaces

Abstract. We will discuss the problem of extension of (dis)continuous maps between topological spaces. Concepts of Baire and Borel retracts of topological spaces will be introduced. Some open problems will be considered.]]>Talks2017-11-22T10:49:28+01:00https://settheory.pwr.edu.pl/blog/files/c9c0a57d42af579ca891f862c80b814d-74.php#unique-entry-id-74https://settheory.pwr.edu.pl/blog/files/c9c0a57d42af579ca891f862c80b814d-74.php#unique-entry-id-74
Room: D1-215

Speaker: Marcin Michalski

Title: Bernstein, Luzin and Sierpiński meet trees

Abstract. In [2] we have proven that if \(\mathfrak{c}\) is a regular cardinal number, then the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to Marczewski ideal \(s_0\). We will generalize this result for other tree ideals - \(m_0\) and \(l_0\) - using some lemmas on special kind of fusion sequences for trees of respective type.


Let us introduce a following notion. Let \(\mathbb{X}\) be a set of trees.

Definition. We call a set \(B\) a \(\mathbb{X}\)-Bernstein set, if for each \(X\in\mathbb{X}\) we have \([X]\cap B\neq\emptyset\).

We shall explore this notion for various set of trees, including Sacks, Miller and Laver trees, with the support of technics developed in [1].


[1] Brendle J., Strolling through paradise, Fundamenta Mathematicae, 148 (1995), pp. 1-25.

[2] Michalski M., Żeberski Sz., Some properties of I-Luzin, Topology and its Applications, 189 (2015), pp. 122-135.]]>Talks2017-11-19T22:01:59+01:00https://settheory.pwr.edu.pl/blog/files/f4e4a3b51e7594f15debc99f6553c816-73.php#unique-entry-id-73https://settheory.pwr.edu.pl/blog/files/f4e4a3b51e7594f15debc99f6553c816-73.php#unique-entry-id-73
Room: D1-215

Speaker: Sakae Fuchino

Title: Downward Löwenheim-Skolem Theorems in stationary logic
]]>Talks2017-11-09T23:38:23+01:00https://settheory.pwr.edu.pl/blog/files/8aef31526afb5b36ec7f4ffa83158016-72.php#unique-entry-id-72https://settheory.pwr.edu.pl/blog/files/8aef31526afb5b36ec7f4ffa83158016-72.php#unique-entry-id-72
Room: D1-215

Speaker: Tomasz Natkaniec

Title: Perfectly everywhere surjective but not Jones functions

Abstract. Given a function \(f:\mathbb{R}\to\mathbb{R}\) we say that

1. \(f\) is perfectly surjective (\(f\in \mathrm{PES}\)) if \(f[P]=\mathbb{R}\) for every perfect set \(P\);


2. \(f\) is a Jones function (\(f\in\mathrm{J}\)) if \(C\cap f\neq\emptyset\) for every closed \(C\subset\mathbb{R}^2\) with \(\mathrm{dom}(C)\) of size \(\mathfrak{c}\).
   

M. Fenoy-Munoz, J.L. Gamez-Merino, G.A. Munoz-Fernandez and E. Saez-Maestro in the paper A hierarchy in the family of real surjective functions [Open Math. 15 (2017), 486--501] asked about the lineability of the set \(\mathrm{PES}\setminus\mathrm{J}\).
Answering this question we show that the class \(\mathrm{PES}\setminus\mathrm{J}\) is \(\mathfrak{c}^+\)-lineable. Moreover, if
\(2^{<\mathfrak{c}}=\mathfrak{c}\) then \(\mathrm{PES}\setminus\mathrm{J}\) is \(2^\mathfrak{c}\)-lineable. We prove also that the additivity number
\(A(\mathrm{PES}\setminus\mathrm{J})\) is between \(\omega_1\) and \(\mathfrak{c}\). Thus \(A(\mathrm{PES}\setminus\mathrm{J})=\mathfrak{c}\) under CH,
however this equality can't be proved in ZFC, because the Covering Property Axiom CPA implies \(A(\mathrm{PES}\setminus\mathrm{J})=\omega_1<\mathfrak{c}\).

The talk is based on the joint paper:
K.C.Ciesielski, J.L. Gamez-Merino, T. Natkaniec, and J.B.Seoane-Sepulveda, On functions that are almost continuous and perfectly everywhere surjective but not Jones. Lineability and additivity, submitted.]]>Talks2017-11-06T21:20:15+01:00https://settheory.pwr.edu.pl/blog/files/280f4b2a34d78d34f78c3506fd439f13-71.php#unique-entry-id-71https://settheory.pwr.edu.pl/blog/files/280f4b2a34d78d34f78c3506fd439f13-71.php#unique-entry-id-71
Room: D1-215

Speaker: Barnabas Farkas (TU Wien)

Title: Cardinal invariants versus towers in analytic P-ideals / An application of matrix iteration

Abstract. I will present two models concerning interactions between the existence of towers in analytic P-ideals and their cardinal invariants. It is trivial to see that if there is no tower in \(\mathcal{I}\), then \(\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})\). I will prove that this implication cannot be reversed no matter the value of \(\mathrm{non}^*(\mathcal{I})\). More precisely, let \(\mathcal{I}\) be an arbitrary tall analytic P-ideal, I will construct the following two models:

Model1 of \(\mathrm{non}^*(\mathcal{I})=\mathfrak{c}\), there is a tower in \(\mathcal{I}\), and \(\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})\). Method: Small filter iteration.

Model2 of \(\mathrm{non}^*(\mathcal{I})<\mathfrak{c}\), there is a tower in \(\mathcal{I}\), and \(\mathrm{add}^*(\mathcal{I})<\mathrm{cov}^*(\mathcal{I})\). Method: Matrix iteration.

This is a joint work with J. Brendle and J. Verner.]]>Talks2017-10-11T23:05:48+02:00https://settheory.pwr.edu.pl/blog/files/81148cd0b31c50614c69f776fe4e9599-70.php#unique-entry-id-70https://settheory.pwr.edu.pl/blog/files/81148cd0b31c50614c69f776fe4e9599-70.php#unique-entry-id-70
Room: D1-215

Speaker: Ziemowit Kostana

Title: Non-measurabity of algebraic sum

Abstract. Consider following problems:

1. If \(A\) is meagre (null) subset of real line, does there necessarily exist set \(B\) such that algebraic sum \(A+B\) doesn't have Baire property (is non-measurable)?


2. If \(A\) is meagre (null) subset of real line, does there necessarily exist non-meagre (non-null) additive subgroup, disjoint with some translation of \(A\)?

It is not hard to prove that positive answer to 2. implies positive answer to 1, both for measure and category. We answer 2. affirmatively for category, while version for measure turns out to be independent of ZFC. The latter was essentially proved last year by A. Rosłanowski and S. Shelah. Both results holds for Cantor space with coordinatewise addition mod. 2 as well.]]>Talks2017-10-10T05:57:56+02:00https://settheory.pwr.edu.pl/blog/files/ac197d55f731521790ca3d5095b02448-69.php#unique-entry-id-69https://settheory.pwr.edu.pl/blog/files/ac197d55f731521790ca3d5095b02448-69.php#unique-entry-id-69
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.]]>Talks2017-06-12T09:19:15+02:00https://settheory.pwr.edu.pl/blog/files/e249a2b1e2511f402a1fb96330b83c55-68.php#unique-entry-id-68https://settheory.pwr.edu.pl/blog/files/e249a2b1e2511f402a1fb96330b83c55-68.php#unique-entry-id-68
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.]]>Talks2017-05-18T10:55:40+02:00https://settheory.pwr.edu.pl/blog/files/8f61ee34fbf1af55f259409230e85163-67.php#unique-entry-id-67https://settheory.pwr.edu.pl/blog/files/8f61ee34fbf1af55f259409230e85163-67.php#unique-entry-id-67
Room: D1-215

Speaker: Jarosław Swaczyna

Title: Haar-small sets

Abstract. In locally compact Polish groups there is a very natural \(\sigma\)-ideal of null sets with respect to Haar-measure. In non locally compact groups there is no Haar measure, however Christensen introduced a notion of Haar-null sets which is an analogue of locally compact case. In 2013 Darji introduced a similar notion of Haar-meager sets. During my talk I will present some equivalent definition of Haar-null sets which leads us to joint generalization of those notions. This is joint work with T. Banakh, Sz. Głąb and E. Jabłońska.]]>Talks2017-05-05T22:15:23+02:00https://settheory.pwr.edu.pl/blog/files/addecf0665aa97f3287a3a9d6c08b3fe-66.php#unique-entry-id-66https://settheory.pwr.edu.pl/blog/files/addecf0665aa97f3287a3a9d6c08b3fe-66.php#unique-entry-id-66
Room: D1-215

Speaker: Joanna Jureczko

Title: Some remarks on Kuratowski partitions, new results

Abstract.K. Kuratowski in 1935 posed the problem whether a function \(f \colon X \to Y\) from a completely metrizable space \(X\) to a metrizable space \(Y\) is continuous apart from a meager set. This question is equivalent to the question about the existence of so called a Kuratowski partition, i. e. a partition \(\mathcal{F}\) of a space \(X\) into meager sets such that \(\bigcup \mathcal{F}'\) for any \(\mathcal{F}' \subset \mathcal{F}\). With any Kuratowski partition we may associate a \(K\)-ideal, i.e. an ideal of the form

\(I_{\mathcal{F}} = \{A \subset \kappa \colon \bigcup_{\alpha \in A} F_\alpha \textrm{ is meager }, F_\alpha \in \mathcal{F}\}.\)

It would seem that the information about \(I_{\mathcal{F}}\) would give us full information about the ideal and the world in which it lives.
My talk is going to show that it is big simplification and localization technique from a Kuratowski partition cannot be omitted but the proof can be much simplier. During the talk I will show among others a new proof of non-existence of a Kuratowski partition in Ellentuck topology and a new combinatorial proof of Frankiewicz - Kunen Theorem (1987) on the existence of measurable cardinals. ]]>Talks2017-04-24T08:28:47+02:00https://settheory.pwr.edu.pl/blog/files/e74f64b9979d5c296422dfaefea6f75d-65.php#unique-entry-id-65https://settheory.pwr.edu.pl/blog/files/e74f64b9979d5c296422dfaefea6f75d-65.php#unique-entry-id-65
Room: D1-215

Speaker: Marcin Michalski

Title: Luzin's theorem

Abstract. In 1934 Nicolai Luzin proved that each subset of the real line can be decomposed into two full subsets with respect to ideal of measure or category. We shall present the proof of this result partially decoding his work and we will also briefly discuss possible generalizations.]]>Talks2017-04-10T08:50:31+02:00https://settheory.pwr.edu.pl/blog/files/86f99132a81ee8d91becc1461bc92be8-64.php#unique-entry-id-64https://settheory.pwr.edu.pl/blog/files/86f99132a81ee8d91becc1461bc92be8-64.php#unique-entry-id-64
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.]]>Talks2017-03-22T09:21:54+01:00https://settheory.pwr.edu.pl/blog/files/55e6c0aca68e716f8b4158f355963540-63.php#unique-entry-id-63https://settheory.pwr.edu.pl/blog/files/55e6c0aca68e716f8b4158f355963540-63.php#unique-entry-id-63
Room: D1-215

Speaker: Judyta Bąk

Title: Domain theory and topological games

Abstract. Domain is a partially ordered set, in which there was introduced some specific relation. We say that a space is domain representable if it is homeomorphic to a space of maximal elements of some domain. In 2015 W. Fleissner and L. Yengulalp introduced a notion of \(\pi\)-domain representable space, which is analogous of domain representable. We prove that a player \(\alpha\) has a winning strategy in the Banach--Mazur game on a space \(X\) if and only if \(X\) is countably \(\pi\)-domain representable. We give an example of countably \(\pi\)-domain representable space, which is not \(\pi\)-domain representable.]]>Talks2017-03-10T08:01:32+01:00https://settheory.pwr.edu.pl/blog/files/969074b1c4fb1c783faa4c955d136236-62.php#unique-entry-id-62https://settheory.pwr.edu.pl/blog/files/969074b1c4fb1c783faa4c955d136236-62.php#unique-entry-id-62
Room: D1-215

Speaker: Piotr Szewczak

Title: The Scheepers property and products of Menger spaces

Abstract. A topological space \(X\) is Menger if for every sequence of open covers \(\mathcal{O}_1, \mathcal{O}_2,\ldots \) of the space \(X\), there are finite subfamilies \(\mathcal{F}_1\subseteq \mathcal{O}_1,\ \mathcal{F}_2\subseteq\mathcal{O}_2,\ldots \) such that their union is a cover of \(X\). If, in addition, for every finite subset \(F\) of \(X\) there is a natural number \(n\) with \(F\subseteq\bigcup\mathcal{F}_n\), then the space \(X\) is Scheepers. The above properties generalize \(\sigma\)-compactness, and Scheepers’ property is formally stronger than Menger’s property. It is consistent with ZFC that these properties are equal.

One of the open problems in the field of selection principles is to find the minimal hypothesis that the above properties can be separated in the class of sets of reals. Using purely
combinatorial approach, we provide examples under some set theoretic hypotheses. We apply obtained results to products of Menger spaces

This a joint work with Boaz Tsaban (Bar-Ilan University, Israel) and Lyubomyr Zdomskyy (Kurt Godel Research Center, Austria).
]]>Talks2016-12-18T11:14:26+01:00https://settheory.pwr.edu.pl/blog/files/7f3a573560c813522fd80b6c4d3872d1-61.php#unique-entry-id-61https://settheory.pwr.edu.pl/blog/files/7f3a573560c813522fd80b6c4d3872d1-61.php#unique-entry-id-61
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".]]>Talks2016-12-13T00:07:02+01:00https://settheory.pwr.edu.pl/blog/files/a9ecf368390f083b6ab24998673fd12c-60.php#unique-entry-id-60https://settheory.pwr.edu.pl/blog/files/a9ecf368390f083b6ab24998673fd12c-60.php#unique-entry-id-60
Room: D1-215

Speaker: Marcin Michalski

Title: Decomposing the real line into Borel sets closed under addition

Abstract. We will show some results proved by M. Elekes and T.Keleti in "Decomposing the real line into Borel sets closed under addition".
]]>Talks2016-11-23T21:35:36+01:00https://settheory.pwr.edu.pl/blog/files/75947ad1d2f29229760f10cbca9f1850-59.php#unique-entry-id-59https://settheory.pwr.edu.pl/blog/files/75947ad1d2f29229760f10cbca9f1850-59.php#unique-entry-id-59
Room: D1-215

Speaker: Artur Bartoszewicz

Title: On the sets of subsums of series

Abstract. The first observations connected with sets of subsums of series (s.c. achievement sets) belong to Kakeya and are over 100 years old. In my lecture I want to present the story of the studies of the problem and the results obtained by my cooperators and me quite recently. These results concern the series generating Cantorvals, connections between the achievement sets of series and the atractors of affine IFS's and achievement sets of conditionally convergent series in the plane.
]]>Talks2016-11-16T09:14:01+01:00https://settheory.pwr.edu.pl/blog/files/ddd0f867b9fcc5de565db3007b3bb798-58.php#unique-entry-id-58https://settheory.pwr.edu.pl/blog/files/ddd0f867b9fcc5de565db3007b3bb798-58.php#unique-entry-id-58
Room: D1-215

Speaker: Daria Michalik

Title: Degree of homogeneity of connes over locally connected curves

Abstract. \(\mathcal{H}(X)\) denotes the group of self-homeomorphisms of \(X\). An orbit of a point \(x_0\) in \(X\)is the set: \(\mathcal{O}_X(x_0) = \{h(x_0) : h\in\mathcal{H}(X)\}.\)

\(X\) is \(1/n\)-homogeneous if \(X\) has exactly \(n\) orbits. In such a case we say that the degree of homogeneity of \(X\) equals \(n\). P. Pellicer Covarrubias, A. Santiago-Santos calculated the degree of homogeneity of connes over local dendrites depending on the degree of homogeneity of their bases. We will generalize above result on connes over locally connected curves.
]]>Talks2016-11-10T15:49:22+01:00https://settheory.pwr.edu.pl/blog/files/0cf67425c427ee7b4c1d7d1228f23b7d-57.php#unique-entry-id-57https://settheory.pwr.edu.pl/blog/files/0cf67425c427ee7b4c1d7d1228f23b7d-57.php#unique-entry-id-57
Room: D1-215

Speaker: Szymon Głąb

Title: Dense free subgroups of automorphism groups of homogeneous partially ordered sets

Abstract. Let \(1 \le n \le\omega\). Let \(A_n\) be a set of natural numbers less than \(n\). Define \(<\) on \(A_n\) so that for no \(x, y \in A_n\) is \(x < y\). Let \(B_n = A_n \times\mathbb{Q}\) where \(\mathbb{Q}\) is the set of rational numbers. Define \(<\) on \(B_n\) so that \((k, p) < (m, q)\) iff \(k = m\) and \(p < q\). Let \(C_n = B_n\) and define \(<\) on \(C_n\) so that \((k, p) < (m, q)\) iff \(p < q\). Finally, let \((D, <)\) be the universal countable homogeneous partially ordered set, that is a Fraisse limit of all finite partial orders.

A structure is called ultrahomogeneous, if every embedding of its finitely generated substructure can be extended to an automorphism. Schmerl showed that there are only countably many, up to isomorphism, ultrahomogeneous countable partially ordered sets. More precisely he proved the following characterization:

Let \((H, <)\) be a countable partially ordered set. Then \((H, <)\) is ultrahomogeneous iff it is isomorphic to one of the following:

1. \((A_n, <)\) for \(1 \le n \le\omega\);


2. \((B_n, <)\) for \(1 \le n \le\omega\);


3. \((C_n, <)\) for \(1 \le n \le\omega\);


4. \((D, <)\).

Moreover, no two of the partially ordered sets listed above are isomorphic. Consider automorphisms groups \(Aut(A_\omega) = S_\infty\), \(Aut(B_n) \), \(Aut(C_n)\) and \(Aut(D)\). We prove that each of these groups contains two elements f, g such that the subgroup generated by f and g is free and dense. By Schmerl’s Theorem the automorphism group of a countable infinite partially ordered set is freely topologically 2-generated.
]]>Talks2016-11-04T16:15:48+01:00https://settheory.pwr.edu.pl/blog/files/73f87ab5a4d69e3248faad02981c2729-56.php#unique-entry-id-56https://settheory.pwr.edu.pl/blog/files/73f87ab5a4d69e3248faad02981c2729-56.php#unique-entry-id-56
Room: D1-215

Speaker: Marcin Michalski

Title: Universal sets for bases of \(\sigma\)-ideals

Abstract. We shall construct universal sets of possibly low Borel rank for classic \(\sigma\)-ideals of sets: \(\mathcal{N}\)-family of measure zero sets, \(\mathcal{M}\)-family of meager sets, \(\mathcal{M}\cap\mathcal{N}\) and \(\mathcal{E}\). We will also discuss briefly cases of other ideals.
]]>Talks2016-10-20T09:19:57+02:00https://settheory.pwr.edu.pl/blog/files/025c89f6eee769b806630c2a1d124d2e-55.php#unique-entry-id-55https://settheory.pwr.edu.pl/blog/files/025c89f6eee769b806630c2a1d124d2e-55.php#unique-entry-id-55
Room: D1-215

Speaker: David Chodounsky

Title: Combinatorial properties of the Mathias-Prikry forcing

Abstract. I will review basic fact and results about the Mathias-Prikry forcing and I will present and prove sufficient condition for genericity of reals with respect to this poset. Time permitting, further connections of parameters of the forcing with its properties will be explored.]]>Talks2016-10-17T22:50:16+02:00https://settheory.pwr.edu.pl/blog/files/83a4ab779bc075e685a541177d68d26e-54.php#unique-entry-id-54https://settheory.pwr.edu.pl/blog/files/83a4ab779bc075e685a541177d68d26e-54.php#unique-entry-id-54
Room: D1-215

Speaker: Marcin Michalski

Title: On some properties of sigma-ideals

Abstract. We shall consider a couple of properties of \(\sigma\)-ideals and relations between them. Namely we will prove that \(\mathfrak c\)-cc \(\sigma\)-ideals are tall, Weaker Smital Property implies that every Borel \(\mathcal{I}\)-positive set contains a witness for non(\(\mathcal{I}\)) as well, as satisfying ccc and Fubini Property. We give also a characterization of nonmeasurability of \(\mathcal{I}\)-Luzin sets and prove that the ideal \([\mathbb R]^{\leq\omega}\) does not posses the Fubini Property using some interesting lemma about perfect sets.]]>Talks2016-10-10T11:06:01+02:00https://settheory.pwr.edu.pl/blog/files/17f9d3a3d51886fb44334869abda5eba-53.php#unique-entry-id-53https://settheory.pwr.edu.pl/blog/files/17f9d3a3d51886fb44334869abda5eba-53.php#unique-entry-id-53
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.]]>Talks2016-09-14T21:32:19+02:00https://settheory.pwr.edu.pl/blog/files/933ee4dcb64e3f1add200172cf081bb8-52.php#unique-entry-id-52https://settheory.pwr.edu.pl/blog/files/933ee4dcb64e3f1add200172cf081bb8-52.php#unique-entry-id-52
Room: D1-215

Speaker: Shashi Srivastava

Title: Some Applications of Descriptive Set Theory to Transition Probabilitie

Abstract. We use measurable selection theorems and prove several results on extensions and existence of transition probabilities with prescribed domain. This is part of joint work with E. E. Doberkat. The remaining part of the work will be presented at Mathematical Institute, University of Wroclaw on 21 September 2016.
]]>Talks2016-05-25T10:50:03+02:00https://settheory.pwr.edu.pl/blog/files/f8d62186501a39cbe2af62808cb98642-51.php#unique-entry-id-51https://settheory.pwr.edu.pl/blog/files/f8d62186501a39cbe2af62808cb98642-51.php#unique-entry-id-51
Room: D1-215

Speaker: Wiesław Kubiś

Title: Abstract Banach-Mazur game

Abstract. We will discuss an infinite game in which two players alternately choose some objects (structures) from a given class. The only rule is that at each move the structure chosen by the player should extend the one chosen in the previous move by the opponent. One of the players wins if the limit of the chain of structures resulting from the play is isomorphic to some concrete (fixed in advance) object. We will show some basic results and relevant examples concerning the existence of winning strategies.
]]>Talks2016-05-11T10:28:21+02:00https://settheory.pwr.edu.pl/blog/files/488704141e2f1d017879dc88af6b8fd4-50.php#unique-entry-id-50https://settheory.pwr.edu.pl/blog/files/488704141e2f1d017879dc88af6b8fd4-50.php#unique-entry-id-50
Room: D1-215

Speaker: Andrzej Kucharski

Title: \(\kappa\)-metrizable spaces

Abstract. We introduce a new supclass of \(\kappa\)-metrizable spaces, namely \(\omega\) \(\kappa\)-metrizable spaces. We show that  \(\kappa\)-metrizable spaces form a proper subclass of \(\omega\) \(\kappa\)-metrizable spaces. On the other hand,  for pseudocompact spaces the new class coincides with \(\kappa\)-metrizable spaces.]]>Talks2016-04-26T14:02:43+02:00https://settheory.pwr.edu.pl/blog/files/1248f1e27baeda2d0468fee3c38269d1-49.php#unique-entry-id-49https://settheory.pwr.edu.pl/blog/files/1248f1e27baeda2d0468fee3c38269d1-49.php#unique-entry-id-49
Room: D1-215

Speaker: Barnabas Farkas

Title: Towers in filters and related problems

Abstract. I am going to present a survey on my recently finished joint work with J. Brendle and J. Verner. In this paper we investigated which filters can contain towers, that is, a \(\subseteq^*\)-decreasing sequence in the filter without any pseudointersection (in \([\omega]^\omega\)). I will present Borel examples which contain no towers in \(\mathrm{ZFC}\), and also examples for which it is independent of \(\mathrm{ZFC}\). I will prove that consistently every tower generates a non-meager filter, in particular (consistently) Borel filters cannot contain towers. And finally, I will present the "map'' of logical implications and non-implications between (a) the existence of a tower in a filter \(\mathcal{F}\), (b) inequalities between cardinal invariants of \(\mathcal{F}\), and (c) the existence of a peculiar object, an \(\mathcal{F}\)-Luzin set of size \(\geq\omega_2\).]]>Talks2016-04-20T15:22:04+02:00https://settheory.pwr.edu.pl/blog/files/a854d7ed3de39c33ceb76b383e492ef3-48.php#unique-entry-id-48https://settheory.pwr.edu.pl/blog/files/a854d7ed3de39c33ceb76b383e492ef3-48.php#unique-entry-id-48Monday, April 25, 2016 17:15

Room: 604 IM

Speaker: Magdalena Nowak

Title: Counterexamples for IFS-attractors

Abstract. I deal with the part of Fractal Theory related to finite families of (weak) contractions, called iterated function systems (IFS). An attractor is a compact set which remains invariant for such a family. Thus, I consider spaces homeomorphic to attractors of either IFS or weak IFS, as well, which I will refer to as Banach and topological fractals, respectively. I present a collection of counterexamples in order to show that all the presented definitions are essential, though they are not equivalent in general.]]>Talks2016-03-18T19:01:58+01:00https://settheory.pwr.edu.pl/blog/files/2ac0674f9a5e0f9a8a37d2a550a3ebb0-47.php#unique-entry-id-47https://settheory.pwr.edu.pl/blog/files/2ac0674f9a5e0f9a8a37d2a550a3ebb0-47.php#unique-entry-id-47
Room: D1-215

Speaker: Aleksandra Kwiatkowska

Title: Universal flows and Ramsey theory

Abstract. The subject lies on the crossroad of topological dynamics, topology, topological groups and Ramsey theory. We will present Kechris-Pestov-Todorcevic theorem about connections between structural Ramsey theory, extremely amenable groups and universal minimal flows. We will show some examples. Next, we will focus on groups of homeomorphisms (Cantor set, Lelek fan, pseudoarc, Hilbert cube). We will recall known results and ask some questions.]]>Talks2016-03-14T18:57:41+01:00https://settheory.pwr.edu.pl/blog/files/31db54afee7d9a1573431fce33a252dc-46.php#unique-entry-id-46https://settheory.pwr.edu.pl/blog/files/31db54afee7d9a1573431fce33a252dc-46.php#unique-entry-id-46
Room: D1-215

Speaker: Tomasz Żuchowski

Title: Nonseparable growth of omega supporting a strictly positive measure

Abstract. We will construct in ZFC a compactification \(\gamma\omega\) of \(\omega\) such that its remainder \(\gamma\omega\backslash\omega\) is not separable and carries a strictly positive measure, i.e. measure positive on nonempty open subsets. Moreover, the measure on our space is defined by the asymptotic density of subsets of \(\omega\).

Our remainder is a Stone space of a Boolean subalgebra of Lebesgue measurable subsets of \(2^{\omega}\) containing all clopen sets.]]>Talks2016-03-08T10:16:56+01:00https://settheory.pwr.edu.pl/blog/files/fc93d9f755daef855b00e377bbd6f2d5-45.php#unique-entry-id-45https://settheory.pwr.edu.pl/blog/files/fc93d9f755daef855b00e377bbd6f2d5-45.php#unique-entry-id-45
Room: D1-215

Speaker: Piotr Borodulin-Nadzieja

Title: Mathias forcings for slaloms

Abstract. We will show an example of a Boolean algebra which is not sigma-centered but sigma-n-linked. Moreover, it has property (*) of Fremlin. Such examples were known before. We will construct our algebra using Mathias forcing for something resembling the density filter. ]]>Talks2016-02-25T15:41:42+01:00https://settheory.pwr.edu.pl/blog/files/54fb67f0e78f514a0ca1778fb6ee95a0-43.php#unique-entry-id-43https://settheory.pwr.edu.pl/blog/files/54fb67f0e78f514a0ca1778fb6ee95a0-43.php#unique-entry-id-43
Room: D1-215

Speaker: Robert Rałowski

Title: Bernstein set and continuous functions

Abstract. Alexander V. Osipov asked "It is true that for any Bernstein subset \(B\subset \mathbb{R}\) there are countable many continous functions from \(B\) to \(\mathbb{R}\) such that the union of images of \(B\) is a whole real line \(\mathbb{R}\)". We give the positive answer for this question, but we show that this result is not true for a \(T_2\) class of functions.

We show some consistency results for completely nonmeasurable sets with respect to \(\sigma\)-ideals of null sets and meager sets on the real line.

These results was obtained commonly with Jacek Cichoń, Michał Morayne and me.]]>Talks2016-02-19T18:43:55+01:00https://settheory.pwr.edu.pl/blog/files/7b1c2714f2ea32f7ac3c2f631e9899cb-42.php#unique-entry-id-42https://settheory.pwr.edu.pl/blog/files/7b1c2714f2ea32f7ac3c2f631e9899cb-42.php#unique-entry-id-42
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. ]]>Talks2016-01-03T19:24:58+01:00https://settheory.pwr.edu.pl/blog/files/4b930f5c9591317c47863abf46914c79-41.php#unique-entry-id-41https://settheory.pwr.edu.pl/blog/files/4b930f5c9591317c47863abf46914c79-41.php#unique-entry-id-41
Room: D1-215

Speaker: Jan Stary

Title: Coherent ultrafilters

Abstract. The notion of a P-ultrafilter on \( \omega \)can be stranghtened in a natural way to the notion of a coherent P-ultrafilter on a complete ccc Boolean algebra. These ultrafilters exist generically under the condition isolated by Ketonen, namely \( \mathfrak c = d\). Similarly, under the Canjar condition \( {\mathfrak c }= cov(Meager)\), coherently Ramsey ultrafilters can be shown to exist. Existence of "coherent" versions of other traditional objects is an ongoing programme. The coherent ultrafilters are relevant in an old topological question: a coherent P-ultrafilter on an algebra B is an untouchable point in the Stone space of B, witnessing its non homogeneity.]]>Talks2015-12-03T20:07:10+01:00https://settheory.pwr.edu.pl/blog/files/6f559b70013e96beb0e1fd17258b6230-40.php#unique-entry-id-40https://settheory.pwr.edu.pl/blog/files/6f559b70013e96beb0e1fd17258b6230-40.php#unique-entry-id-40
Room: D1-215

Speaker: David Chodounsky

Title: Y-cc and Y-proper forcing notions

Abstract. In our recent joint work J. Zapletal, we explored a new type of properties of forcing notions, of which the Y-cc and Y-proper are the most prolific examples. These two notions can be seen as a generalization of the notion of \(\sigma\)-centered. While keeping similar consequences a larger and better behaved class of posets is cover by these notions. I will give an introduction to this topic with the focus on understating the basic techniques and ideas.]]>Talks2015-11-18T18:41:06+01:00https://settheory.pwr.edu.pl/blog/files/0239e910de3f7de1f38e86b8ec24955b-39.php#unique-entry-id-39https://settheory.pwr.edu.pl/blog/files/0239e910de3f7de1f38e86b8ec24955b-39.php#unique-entry-id-39
Room: D1-215

Speaker: Piotr Szewczak (Cardinal Stefan Wyszyński University in Warsaw); Coauthor: Boaz Tsaban (Bar-Ilan University, Israel)

Title: Products of Menger spaces

Abstract. A topological space \(X\) is Menger if for every sequence of open covers \(O_1, O_2, \ldots\) there are finite subfamilies \(F_1\) of \(O_1\), \(F_2\) of \(O_2\), . . . such that their union is a cover of \(X\). The above property generalizes sigma-compactness.

One of the major open problems in the field of selection principles is whether there are, in ZFC, two Menger sets of real numbers whose product is not Menger. We provide examples under various set theoretic hypotheses, some being weak portions of the Continuum Hypothesis, and some violating it. The proof method is new.]]>Talks2015-11-16T16:59:18+01:00https://settheory.pwr.edu.pl/blog/files/48efdc891db5b4a94ee9594e2f32640f-38.php#unique-entry-id-38https://settheory.pwr.edu.pl/blog/files/48efdc891db5b4a94ee9594e2f32640f-38.php#unique-entry-id-38
Room: D1-215

Speaker: Marcin Michalski

Title: A generalized version of the Rothberger theorem

Abstract. We call a set \(X\) a generalized Luzin set if \(|L\cap M|<|L|\) for every meager set \(M\). Dually, if we replace meager set with a null set, we obtain a definition of a generalized Sierpiński set.

We will show that if \(2^\omega\) is a regular cardinal then for every generalized Luzin set \(L\) and every generalized Sierpiński set \(S\) an algebraic sum \(L+S\) belongs to the Marczewski ideal \(s_0\) (i.e. for every perfect set \(P\) there exists a perfect set \(Q\) such that \(Q\subseteq P\) and \(Q\cap (L+S)=\emptyset\)). To prove the theorem we shall prove and use a generalized version of the Rothberger theorem.

We will also formulate a series of results involving algebraic, topological and measure structure of the real line, that emerged during searching for a proof of the above theorem.]]>Talks2015-10-27T17:13:06+01:00https://settheory.pwr.edu.pl/blog/files/9666708084dcdc268aa34cea886976a6-37.php#unique-entry-id-37https://settheory.pwr.edu.pl/blog/files/9666708084dcdc268aa34cea886976a6-37.php#unique-entry-id-37
Room: D1-215

Speaker: Julia Wódka

Title: Comparison of some families of real functions in sense of porosity

Abstract. We consider set \(\mathbb{R}^\mathbb{R}\) with uniform convergence metric, i.e:
\[\rho(f,g)=\min\{1,\sup\limits_{x\in\mathbb{R}}|f(x)-g(x)|\}\quad \text{for \(f,g\in\mathbb{R}^\mathbb{R}\)}\]
and the following subsets of \(\mathbb{R}^\mathbb{R}\):

1. Darboux functions (\(f\in\mathscr{D}\) if whenever \(a < b\) and \(y\) is a number between \(f(a)\) and \(f(b)\), there exists an \(x_0\in(a, b)\) such that \(f(x_0) = y\)).


2. quasi-continuous functions (\(f\in\mathscr{Q}\) if it is quasi-continuous at any point \(x\in\mathbb{R}\)).
   
   Function \(f\) is quasi-continuous at \(x \in \mathbb{R}\) if for any open interval \(I\ni x\) and each \(\varepsilon>0\) there exists a nontrivial interval \(J\subset I\) such that \({\rm diam} (f[J\cup \{x\}]) <\varepsilon\).


3.   Świątkowski functions ( \(f \in \mathscr{\acute S}\) if for all \( a < b \) with \(f(a) \ne f(b)\), there is a \(y\) between \(f(a)\) and \(f(b)\) and an \(x\in(a,b) \cap \mathcal{C}(f)\) such that \(f(x)=y\), where \(\mathcal{C}(f)\) denotes the set of all continuity points of function \(f\)).


4. Świątkowski functions (\(f\in\mathscr{\acute S}_s\) if for all \(a < b\) and each \(y\) between \(f(a)\) and \(f(b)\) there is an \(x\in(a,b) \cap \mathcal{C}(f)\) such that \(f(x)=y\)).

The aim of this is to compare this sets in terms of porosity.

Let \((X,d)\) be a metric space, \(x\in A\subset M\), and \(r\in\mathbb{R}_+\). We define
\[\gamma(x,r,M)=\sup\{{t\geq 0}:\ \exists_{z\in M} B(z,t)\subset B(x,r)\setminus M\}\]
and
\[p^u(M, x)=2\limsup\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.\]
\[p_l(M, x)=2\liminf\limits_{t\to r^+}\frac{\gamma(x,r,M)}{r}.\]
Quantity \(p^u(M,x)\) is called upper porosity of \(M\) at the point \(x\). We say that \(M\) is upper \(p-\)porous if \(p=\inf\{p^u(M,x):\ x\in M\}>0\).
Analogously we define lower porosity.]]>Talks2015-10-21T21:14:23+02:00https://settheory.pwr.edu.pl/blog/files/121f735c465fc7314edf4b6a00533c82-36.php#unique-entry-id-36https://settheory.pwr.edu.pl/blog/files/121f735c465fc7314edf4b6a00533c82-36.php#unique-entry-id-36
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.]]>Talks2015-10-12T15:58:35+02:00https://settheory.pwr.edu.pl/blog/files/8b15bea2faab68f7119ee26f4f2e7c1e-35.php#unique-entry-id-35https://settheory.pwr.edu.pl/blog/files/8b15bea2faab68f7119ee26f4f2e7c1e-35.php#unique-entry-id-35
Room: D1-215

Speaker: Antonio Aviles (University of Murcia)

Title: Boolean algebras obtained by push-out iteration

Abstract. We discuss the notion of push-out in the category of Boolean algebras, and we describe a method of constructing Boolean algebras by transfinite iterative push-outs. Under CH and in a model obtained by adding \(\aleph_2\) Cohen reals to a model of CH, \(P(\omega)/fin\) is such an algebra.]]>Talks2015-10-12T11:21:36+02:00https://settheory.pwr.edu.pl/blog/files/5f53266255016d3b389be47f1796a671-34.php#unique-entry-id-34https://settheory.pwr.edu.pl/blog/files/5f53266255016d3b389be47f1796a671-34.php#unique-entry-id-34
Room: D1-215

Speaker: Damian Sobota

Title: The Nikodym property and cardinal invariants of the continuum

Abstract. A Boolean algebra \(\mathcal{A}\) is said to have the Nikodym property if every sequence \((\mu_n)\) of measures on \(\mathcal{A}\) which is elementwise bounded (i.e. \(\sup_n|\mu_n(a)|<\infty\) for every \(a\in\mathcal{A}\)) is uniformly bounded (i.e. \(\sup_n\|\mu_n\|<\infty\)). The property is closely related to the classical Banach-Steinhaus theorem for Banach spaces.

My recent study concerns the problem how (and whether at all) we can describe the structure of the class of Boolean algebras with the Nikodym property in terms of well-known objects occuring inside \(\wp(\omega)\) or \(\omega^\omega\), e.g. countable Boolean algebras, dominating families, Lebesgue null sets etc. During my talk I will present an attempt to obtain such a description via families of antichains in countable subalgebras of \(\wp(\omega)\) having some special measure-theoretic properties.]]>Talks2015-06-08T22:02:42+02:00https://settheory.pwr.edu.pl/blog/files/de30c38e101a2bdffba5478f1a72b709-33.php#unique-entry-id-33https://settheory.pwr.edu.pl/blog/files/de30c38e101a2bdffba5478f1a72b709-33.php#unique-entry-id-33June 16, 2015 17:15

Room: D1-215

Speaker: Magdalena Nowak

Title: Zero-dimensional spaces as topological and Banach fractals

Abstract. A topological space \(X\) is called a topological fractal if \(X=\bigcup_{f\in\mathcal{F}}f(X)\) for a finite system \(\mathcal{F}\) of continuous self-maps of \(X\), which is topologically contracting in the sense that for every open cover \(\mathcal{U}\) of \(X\) there is a number \(n\in\mathbb{N}\) such that for any functions \(f_1,\dots,f_n\in \mathcal{F}\), the set \(f_1\circ\dots\circ f_n(X)\) is contained in some set \(U\in\mathcal{U}\). If, in addition, all functions \(f\in\mathcal{F}\) have Lipschitz constant \(<1\) with respect to some metric generating the topology of \(X\), then the space \(X\) is called a Banach fractal. It is known that each topological fractal is compact and metrizable. We prove that a zero-dimensional compact metrizable space \(X\) is a topological fractal if and only if \(X\) is a Banach fractal if and only if \(X\) is either uncountable or \(X\) is countable and its scattered height \(\hbar(X)\) is a successor ordinal.]]>Talks2015-05-27T21:38:34+02:00https://settheory.pwr.edu.pl/blog/files/a7ee1ca325d3a4a493fb09fe8159a4d1-32.php#unique-entry-id-32https://settheory.pwr.edu.pl/blog/files/a7ee1ca325d3a4a493fb09fe8159a4d1-32.php#unique-entry-id-32
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.]]>Talks2015-05-21T17:01:02+02:00https://settheory.pwr.edu.pl/blog/files/cbbddfdd9f820c1dac7ac8d7dc9bfca2-31.php#unique-entry-id-31https://settheory.pwr.edu.pl/blog/files/cbbddfdd9f820c1dac7ac8d7dc9bfca2-31.php#unique-entry-id-31May 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.]]>Talks2015-05-20T10:19:07+02:00https://settheory.pwr.edu.pl/blog/files/e24dc3ef44733bea421bb6a3bda2720f-30.php#unique-entry-id-30https://settheory.pwr.edu.pl/blog/files/e24dc3ef44733bea421bb6a3bda2720f-30.php#unique-entry-id-30
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.]]>Talks2015-05-15T19:36:58+02:00https://settheory.pwr.edu.pl/blog/files/8576f23763a2aa8322c8dbd349592005-29.php#unique-entry-id-29https://settheory.pwr.edu.pl/blog/files/8576f23763a2aa8322c8dbd349592005-29.php#unique-entry-id-29
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.]]>Talks2015-05-08T08:24:37+02:00https://settheory.pwr.edu.pl/blog/files/593773b318bee2a9db38488310e3e5e7-28.php#unique-entry-id-28https://settheory.pwr.edu.pl/blog/files/593773b318bee2a9db38488310e3e5e7-28.php#unique-entry-id-28
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.]]>Talks2015-04-24T13:09:07+02:00https://settheory.pwr.edu.pl/blog/files/90f47050387f718a1442eb0493f9e36f-27.php#unique-entry-id-27https://settheory.pwr.edu.pl/blog/files/90f47050387f718a1442eb0493f9e36f-27.php#unique-entry-id-27
Room: D1-215

Speaker: Wojciech Bielas

Title: An example of a rigid \(\kappa\)-superuniversal metric space

Abstract. For an uncountable cardinal \(\kappa\) a metric space \(X\) is called to be \(\kappa\)-superuniversal if for every metric space \(Y\) with \(|Y | < \kappa\) every partial isometry from a subset of \(Y\) into \(X\) can be extended over the whole space \(Y\). It is easy to prove that if a \(\kappa\)-superuniversal metric space is of cardinality \(\kappa\), then it is also \(\kappa\)-homogeneous, i.e. every isometry of a subspace \(Y\) of the space with \(|Y | < \kappa\) can be extended to an isometry of the whole space. I will discuss an example of a \(\kappa\)-superuniversal metric space which has exactly one isometry.]]>Talks2015-04-23T13:53:18+02:00https://settheory.pwr.edu.pl/blog/files/8e2ab2d9ed4608572d7832b2a38f96ba-26.php#unique-entry-id-26https://settheory.pwr.edu.pl/blog/files/8e2ab2d9ed4608572d7832b2a38f96ba-26.php#unique-entry-id-26
Room: D1-215

Speaker: Filip Strobin

Title: Spaceability of particular family of continuous functions with nowhere continuous inverses

Abstract. We will show that the family of continuous injections \(T:l_p\to l_p\) (where \(p>0\)) with nowhere continuous inverses (together with zero function) contains isometric copy of \(l_p\). In particular, it means that the set of such functions is spaceable.]]>Talks2015-04-11T09:52:16+02:00https://settheory.pwr.edu.pl/blog/files/f10a1005fc8b645ca9e2d92b6f972904-25.php#unique-entry-id-25https://settheory.pwr.edu.pl/blog/files/f10a1005fc8b645ca9e2d92b6f972904-25.php#unique-entry-id-25
Room: D1-215

Speaker: Mirna Džamonja

Title: WQOs, FACs and their width

Abstract. A quasi-order is WQO if it has no infinite antichains or infinite decreasing sequences. A partial order is FAC if it has no infinite antichains. These restrictions on the orders mean that there are several naturally defined ordinal valued ranks that can be used to study them, for example, the rank of the tree of antichains, called the width. These ranks have been studied from the point of view of order theory, Ramsey theory, and also the theory of algorithms, since it turns out that a large class of « well structured systems « of algorithms can be modeled using the wqo. We shall present certain structural results connecting FAC and WQO orders and then some calculations of the ranks. The new results presented in the talk come from a collaborative work with Schnoebelen and Schmitz.]]>Talks2015-04-11T09:50:11+02:00https://settheory.pwr.edu.pl/blog/files/115bbfdbedf36b7edc7b7c5b8601c754-24.php#unique-entry-id-24https://settheory.pwr.edu.pl/blog/files/115bbfdbedf36b7edc7b7c5b8601c754-24.php#unique-entry-id-24
Room: D1-215

Speaker: Katarzyna Chrząszcz

Title: On some properties of microscopic sets

Abstract: The notion of microscopic set appeared for the first time in paper 'Insiemi ed operatori “piccoli” in analisi funzionale' (APPELL, J., Rend. Istit. Mat. Univ. Trieste 33 (2001), 127–199).
 
Def. A set  \(A\subseteq\mathbb{R}\) is called microscopic if for every  \(\varepsilon>0\) there exists a sequence of segments \((I_n)_{n\in\mathbb{N}}\) such that \(A\subseteq\bigcup\limits_{n\in\mathbb{N}} I_n\) and \(|I_n|\leq\varepsilon^n\) for \(n\in\mathbb{N}\).

We will give generalizations of given notion to the case of arbitrary metric space. We will analyze algebraic and set-theoretic properties of the family of microscopic sets.]]>Talks2015-04-08T19:18:57+02:00https://settheory.pwr.edu.pl/blog/files/a28c18386df538fe0d231224a60b94fe-23.php#unique-entry-id-23https://settheory.pwr.edu.pl/blog/files/a28c18386df538fe0d231224a60b94fe-23.php#unique-entry-id-23
Room: D1-215

Speaker: Marek Bienias

Title: General methods in algebrability

Abstract: During last 15 years new idea of measuring sets appeared and become popular.

Def.
Let \(\kappa\) be a cardinal number and let \(\mathcal{L}\) be a commutative algebra. Assume that \(A\subseteq\mathcal{L}\). We say that \(A\) is:

• \(\kappa\)-algegrable if \(A\cup \{0\}\) contains \(\kappa\)-generated algebra \(B\);
• strongly \(\kappa\)-algegrable if \(A\cup \{0\}\) contains \(\kappa\)-generated free algebra \(B\).

In many recent articles authors studied algebrability of sets naturally appering in mathematical analysis. It seems that required results are the general methods of algebrability which can cover known methods and give new constructions.

We will describe two methods:  independent Bernstein sets and exponential like. They let us prove many results concerning algebrability and strong algebrability of subsets of algebras \(\mathbb{R}^\mathbb{R}\), \(\mathbb{C}^\mathbb{C}\), \(\mathbb{R}^\mathbb{N}\), \(C[0,1]\), \(\mathcal{l}_\infty\). Most of presented applications give the best possible result in terms of complication of built algebraic structure and cardinality of set of generators of this structure (in most cases \(\mathfrak c\) or \(2^{\mathfrak c}\)).]]>Talks2015-03-26T17:33:48+01:00https://settheory.pwr.edu.pl/blog/files/71fe68f797618478655f0069c73a7e65-22.php#unique-entry-id-22https://settheory.pwr.edu.pl/blog/files/71fe68f797618478655f0069c73a7e65-22.php#unique-entry-id-22Wednesday, April 1, 2015 17:15

Room: D1-215

Speaker: Oleg Gutik

Title: Around the bicyclic monoid: topological and semigroup views

Abstract. We discuss algebraic and topological properties of (semitopological and topological) semigroups which are close to the bicyclic monoid: the semigroup of matrix units, the semigroup of co-finite partial bijections and the polycyclic monoid. We speak about topologizations of such semigroups as topological or semitopological semigroups, their embeddings into compact-like topological semigroups and their closures in topological semigroups.]]>Talks2015-03-23T20:46:27+01:00https://settheory.pwr.edu.pl/blog/files/4c2810c1900e9ad80dceac5917615db6-21.php#unique-entry-id-21https://settheory.pwr.edu.pl/blog/files/4c2810c1900e9ad80dceac5917615db6-21.php#unique-entry-id-21
Room: D1-215

Speaker: Robert Rałowski

Title: Two point sets, continuation

Abstract. We will continue discussion started a week ago concerning two point sets. We will give another example of a property of two point set which is consistent with ZFC.]]>Talks2015-03-11T17:35:55+01:00https://settheory.pwr.edu.pl/blog/files/b226e20c66322f2781fa0bbb3e3106eb-20.php#unique-entry-id-20https://settheory.pwr.edu.pl/blog/files/b226e20c66322f2781fa0bbb3e3106eb-20.php#unique-entry-id-20
Room: D1-215

Speaker: Robert Rałowski

Title: Cohen indestructible mad families in partial two point sets

Abstract. We discuss on classical construction of Cohen indestructible mad family given by Kenneth Kunen and we apply this method to obtain a partial Cohen indestructible mad family in Baire space as a canonical copy of the real plane.]]>Talks2015-03-09T10:48:17+01:00https://settheory.pwr.edu.pl/blog/files/5d65e661806104eaa7126f94cd23a767-19.php#unique-entry-id-19https://settheory.pwr.edu.pl/blog/files/5d65e661806104eaa7126f94cd23a767-19.php#unique-entry-id-19
Room: D1-215

Speaker: Marcin Michalski

Title: Avoiding rational distances

Abstract. We shall present results obtained by Ashutosh Kumar in paper "Avoiding rational distances". The author showed that for any set of reals X of positive outer measure there exists a subset Y of X such that Y has the same outer measure and the distance between any distinct points of Y is irrational. We will also discuss briefly the case of higher dimensions.]]>Talks2015-02-27T16:07:43+01:00https://settheory.pwr.edu.pl/blog/files/17eb4579d63e80e23c704f7d37ca3914-18.php#unique-entry-id-18https://settheory.pwr.edu.pl/blog/files/17eb4579d63e80e23c704f7d37ca3914-18.php#unique-entry-id-18
Room: D1-215

Speaker: Szymon Żeberski

Title: An example of a capacity for which all positive Borel sets are thick

Abstract. The result is obtained together with Michał Morayne. We will show an example of a capacity on Cantors cube for which all positive Borel sets can be partitioned into continuum many positive Borel sets.]]>Talks2015-01-23T14:44:14+01:00https://settheory.pwr.edu.pl/blog/files/e804290d8b0ec1bbbeeacb1e83d87639-17.php#unique-entry-id-17https://settheory.pwr.edu.pl/blog/files/e804290d8b0ec1bbbeeacb1e83d87639-17.php#unique-entry-id-17
Room: D1-215

Speaker: Marcin Michalski

Title: I-Luzin sets

Abstract. We will present some results obtained with Szymon Żeberski involving I-Luzin sets in Euclidean spaces. We shall construct a quite decent (non-trivial, possesing Borel base and translation invariant) sigma-ideal I of sets such that there exists an I-measurable I-Luzin set. We give also sufficient condition of I-nonmeasurability of I-Luzin sets involving Smital Property (precisely- it's weaker version). We also discuss briefly Stenihaus and Smital Properties of Fubini product of sigma-ideals.]]>Talks2014-12-13T03:14:03+01:00https://settheory.pwr.edu.pl/blog/files/a4493d283880c393aa7640ad8f5df69a-16.php#unique-entry-id-16https://settheory.pwr.edu.pl/blog/files/a4493d283880c393aa7640ad8f5df69a-16.php#unique-entry-id-16
Room: D1-215

Speaker: Grzegorz Plebanek

Title: About a particular measure on the square

Abstract. Assuming the existence of Sierpiński set we construct a measure on some \(\sigma\)-field of subsets of the square which is perfect but not compact. This construction in 2001 answered Fremlin's question. We will describe open problems connected to this field.]]>Talks2014-12-05T18:34:33+01:00https://settheory.pwr.edu.pl/blog/files/cb8aa94551da0d6c3a2db48ab9fca11c-15.php#unique-entry-id-15https://settheory.pwr.edu.pl/blog/files/cb8aa94551da0d6c3a2db48ab9fca11c-15.php#unique-entry-id-15
Room: D1-215

Speaker: Robert Rałowski

Title: On generalized Luzin sets

Abstract. We will show results obtained together with Sz. Żeberski concerning properties of \((I,J)\)-Luzin sets (for \(I, J\) \(\sigma\)-ideals on Polish space). Under some settheoretical assumptions we will construct \(\mathfrak{c}\) many generalized Luzin sets which are not Borel equivalent. We will also examine some forcing notions which do not kill generalized Luzin sets.]]>Talks2014-12-01T01:49:37+01:00https://settheory.pwr.edu.pl/blog/files/94b963707b0b748b1423b180f17121ca-14.php#unique-entry-id-14https://settheory.pwr.edu.pl/blog/files/94b963707b0b748b1423b180f17121ca-14.php#unique-entry-id-14
Room: D1-215

Speaker: Piotr Drygier

Title: Compactifications of \(\omega\) with strictly positive measure

Abstract. Under certain axioms related to cardinal invariants we will show the construction of compactification of natural numbers, which reminder is a non-separable space having a strictly positive measure. In addition, we will discuss consequences of given result to complementarity of \(c_0\) in the space of continuous functions.]]>Talks2014-11-19T17:10:51+01:00https://settheory.pwr.edu.pl/blog/files/1fba99cae3f898a87bb502b704c20902-13.php#unique-entry-id-13https://settheory.pwr.edu.pl/blog/files/1fba99cae3f898a87bb502b704c20902-13.php#unique-entry-id-13
Room: D1-215

Speaker: Wojciech Stadnicki

Title: CPA in Mathias model

Abstract. We will formulate CPA for Mathias model. We will give its consequences and generalizations.]]>Talks2014-11-10T15:07:03+01:00https://settheory.pwr.edu.pl/blog/files/0090c5fcb5d75e10feb9b1308968884d-12.php#unique-entry-id-12https://settheory.pwr.edu.pl/blog/files/0090c5fcb5d75e10feb9b1308968884d-12.php#unique-entry-id-12
Room: D1-215

Speaker: Maciej Malicki

Title: Groups of isometries of Polish ultrametric spaces

Abstract. We will characterize Polish ultrametric spaces whose isometry groups have a neighborhood basis at the identity consisting of open subgroups with ample generics. We will also define Polish ultrametric W-spaces and give a characterization of W-spaces whose isometry groups have uncountable strong cofinality.
]]>Talks2014-10-30T19:44:59+01:00https://settheory.pwr.edu.pl/blog/files/6795d2ea1478c7ff522fa60cb4e85c3b-11.php#unique-entry-id-11https://settheory.pwr.edu.pl/blog/files/6795d2ea1478c7ff522fa60cb4e85c3b-11.php#unique-entry-id-11
Room: D1-215

Speaker: Piotr Borodulin-Nadzieja

Title: Analytic P-ideals and Banach spaces

Abstract. We consider certain generalization of the notion of summability of an ideal (on the natural numbers) which connects theory of analytic P-ideals with the theory of Banach spaces. ]]>Talks2014-10-27T16:02:02+01:00https://settheory.pwr.edu.pl/blog/files/3949e6f97c3469f28a10abd3af521ed8-10.php#unique-entry-id-10https://settheory.pwr.edu.pl/blog/files/3949e6f97c3469f28a10abd3af521ed8-10.php#unique-entry-id-10
Room: D1-215

Speaker: Robert Rałowski

Title: On m.a.d. \(s_0\)-nonmeasurable sets with a small dominating subfamilies

Abstract. We show that \(\mathfrak{d}=\aleph_1\) implies the existence of maximal familiy of eventually different reals on Baire space which forms a nonmeasurable set with respect to an ideals generated by trees (perfect, Laver or Miller trees for example).]]>Talks2014-10-16T19:06:19+02:00https://settheory.pwr.edu.pl/blog/files/0d36a303beb2bc31a3802069fd99ad80-9.php#unique-entry-id-9https://settheory.pwr.edu.pl/blog/files/0d36a303beb2bc31a3802069fd99ad80-9.php#unique-entry-id-9
Room: D1-215

Speaker: Marcin Michalski

Title: Luzin and Sierpiński sets, some nonmeasurable subsets of the plane

Abstract: We shall introduce some nonmeasurable and completely nonmeasurable subsets of the plane with various additional properties, e.g. being Hamel basis, intersecting each line in a strong Luzin/Sierpiński set. Also some additive properties of Luzin and Sierpiński sets and their generalization, \(\mathcal{I}\)-Luzin sets, on the line are investigated.]]>Talks2014-10-09T17:51:23+02:00https://settheory.pwr.edu.pl/blog/files/0feaec2115f42f99d1f7ad299ac3058d-8.php#unique-entry-id-8https://settheory.pwr.edu.pl/blog/files/0feaec2115f42f99d1f7ad299ac3058d-8.php#unique-entry-id-8
Room: D1-215

Speaker: Robert Rałowski

Title: Nonmeasurability with respect to Marczewski ideal

Abstract: Among the others we show relative consistency of ZFC theory with \(\aleph_1< 2^{\aleph_0}\) and there is a nonmesurable (with respect to ideal generated by complete Laver trees) m.a.d. family \(\mathcal{A}\) on Baire space \(\omega^\omega\). In ZFC there is  a subset \(\mathcal{A}’\subseteq \mathcal{A}\) of size \(\aleph_1\) unbounded in \(\omega^\omega\). We show that there is m.a.d. family which is  nonmeasurable with respect to Marczewski ideal.]]>Talks2014-10-02T21:07:33+02:00https://settheory.pwr.edu.pl/blog/files/5b4e7f4f70b61e7893e0b1f08f255b68-7.php#unique-entry-id-7https://settheory.pwr.edu.pl/blog/files/5b4e7f4f70b61e7893e0b1f08f255b68-7.php#unique-entry-id-7
Room: D1-215

Speaker: Szymon Żeberski

Title: \(\sigma\)-ideals invariant under measure-preserving homeomorphisms on Cantor's cube

Abstract: Results were obtained together with Taras Banakh and Robert Rałowski. We will show that there are only four nontrivial sigma-ideals with Borel base invariant under measure preserving homeomorphisms on Cantor's cube. These ideals are: \(\mathscr{E}\), \(\mathscr{M} \cap \mathscr{N}\), \(\mathscr{M}\), \(\mathscr{N}\).
]]>Talks2014-06-30T18:33:57+02:00https://settheory.pwr.edu.pl/blog/files/700272676defddbc073633fdeb208d34-0.php#unique-entry-id-0https://settheory.pwr.edu.pl/blog/files/700272676defddbc073633fdeb208d34-0.php#unique-entry-id-0