O seminarium

Zbigniew Lipecki
Alicja Samulewicz
Michał Morayne
Robert Rałowski
Marcin Michalski
Aleksandra Kwiatkowska
Arturo Martinez Celis
Sebastian Jachimek
Maciej Korpalski
Agnieszka Widz
Alek Cieślak
Łukasz Mazurkiewicz

Terminy i tematyka spotkań

wtorek, 11-06-2024 - 17:00, room A.4.1 C-19 (Politechnika Wrocławska)
On Q- and selective measures
Jadwiga Świerczyńska (Uniwersytet Wrocławski)
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 $\ell^{\infty}$.
wtorek, 04-06-2024 - 20:10, room A.4.1 C-19 (Politechnika Wrocławska)
Finitely additive measures on Boolean algebras: freeness and integration
Andres Uribe-Zapata (TU Wien)
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 \emph{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. [1]https://shorturl.at/sHY59.
wtorek, 23-04-2024 - 17:00, A.4.1 C-19 (Politechnika Wrocławska)
The Nikodym property and filters on $\omega$. Part II
Tomasz Żuchowski (UWr)
In this talk we will continue studying the family $\mathcal{AN}$ of ideals on $\omega$ presented in the Part I. Recall that $\mathcal{I}\in\mathcal{AN}$ iff there exists a density submeasure $\varphi$ on $\omega$ such that $\varphi(\omega)=\infty$ and $\mathcal{I}\subseteq Exh(\varphi)$. We will present several conditions for a density ideal $\mathcal{I}$ equivalent to the fact that $\mathcal{I}\in\mathcal{AN}$. Next, we will make an analysis of the cofinal structure of the family $\mathcal{AN}$ ordered by the Katetov order $\leq_K$. We will prove that there is a family of size $\mathfrak{d}$ which is $\leq_K$-dominating in $\mathcal{AN}$, but there are no $\leq_K$-maximal elements in $\mathcal{AN}$.
wtorek, 16-04-2024 - 17:00, room A.4.1 C-19, Politechnika Wrocławska
Function spaces on Corson-like compacta
Krzysztof Zakrzewski (MIM UW)
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 Cp(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 Cp(K) and other related results.
wtorek, 09-04-2024 - 17:00, room A.4.1 C-19 (Politechnika Wrocławska)
Topological properties of compact spaces K that are preserved by isomorphisms of C(K)
Jakub Rondos (Uni Wien)
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.
wtorek, 26-03-2024 - 17:00, room A.4.1 C-19 Politechnika Wrocławska
The Nikodym property and filters on $\omega$. Part I
Tomek Żuchowski
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.
wtorek, 19-03-2024 - 17:00, room A.4.1 C-19 Politechnika Wroclawska
Perfectly meager sets in the transitive sense and the Hurewicz property
Piotr Szewczak (UKSW)
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
wtorek, 12-03-2024 - 17:00, room A.4.1 C-19 (Politechnika Wrocławska)
Forcing extensions of models of determinacy
Grigor Sargsyan (IMPAN)
We will give an overview of what has been recently forced over models of determinacy. In particular, we will explain how to obtain combinatorially rich ZFC extensions by forcing over a model of determinacy axioms. Part of this work are joint with Paul Larson and Douglas Blue.
wtorek, 27-02-2024 - 17:00, room A.4.1 C-19 (Politechnika Wroclawska)
Aftermath of the Winterschool
Grzegorz Plebanek
We shall discuss two problems on measures on compact spaces posed by Jiri Spurny.
wtorek, 09-01-2024 - 17:15, 601
Fams on omega
Piotr Borodulin-Nadzieja
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.
wtorek, 19-12-2023 - 17:00, 601
Antichain numbers and other cardinal invariants of ideals
Alek Cieślak (Politechnika Wrocławska)
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 add*J and 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.
wtorek, 05-12-2023 - 17:15, 601
Non-meager filters
Daria Perkowska
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}.
wtorek, 28-11-2023 - 17:15, 601
Zoo of ideal Schauder bases
Jarosław Swaczyna (Politechnika Łódzka)
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 consider also coordinate functionals e_n^\star. About ten years ago Kadets asked if those functionals are necessarily continuous at least for some nice ideals, eg ideal of sets of density zero. During my talk I will present 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.
wtorek, 21-11-2023 - 17:15, 601
Ultrafilters and finitely additive measures in forcing theory
Diego Mejia (Shizuoka University)
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.
wtorek, 07-11-2023 - 17:00, 601
A countably tight P(K) space admitting a nonseparable measure
Zdenek Silber (IM PAN)
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 L1(mu). It was proven by Fremlin that, under Martin's axiom and negation of continuum hypothesis, for a compact Hausdorff space K the existence of a Radon probability of uncountable type is equivalent to the exitence of a continuous surjection from K onto [0,1]^omega1. 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(KxK) 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.
wtorek, 31-10-2023 - 17:15, 601
Cofinalities of tree ideals and Shrinking Property
Alek Cieślak (Politechnika Wrocławska)
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 $\omega$.
wtorek, 24-10-2023 - 17:15, 601
Straightening almost chains into barely altenating ones
Maciej Korpalski
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 $A_x$.
wtorek, 13-06-2023 - 17:15, room A.4.1 C-19 (Politechnika Wrocławska)
Remarks and questions on hyperspaces of knots: Borel complexity and local contractibility
Paweł Krupski (Politechnika Wrocławska)
wtorek, 06-06-2023 - 17:15, A.4.1 C-19 (Politechnika Wrocławska)
Totally imperfect Menger sets
Piotr Szewczak (UKSW)
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
wtorek, 30-05-2023 - 17:15, room A.4.1 C-19 Politechnika Wrocławska
How noncompact is the space of Lebesgue measurable functions?
Zbigniew Lipecki (IMPAN)
The space in question is the space $\textfrak M$ of Lebesgue measurable subsets of the unit interval equipped with the usual Fr'echet--Nikodym (semi)metric. We show that there exists a sequence of elements of $\textfrak M$ 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.
wtorek, 23-05-2023 - 17:15, room A.4.1 C-19, Politechnika Wrocławka
A tool to avoid some technical forcing arguments when working with the Hechler forcing
Barnabas Farkas (TU, Wiedeń)
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.
wtorek, 25-04-2023 - 17:15, A.4.1 C-19, Politechnika Wrocławska
Incomparable Borel linear subspaces of $\mathbb{R}$ (over $\mathbb{Q}$)
Sławomir Solecki (Cornell University)
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.
wtorek, 18-04-2023 - 17:15, room A.4.1 C-19, Politechnika Wrocławska
On nwd-MAD families
Jonathan Cancino (Czech Academy of Sciences)
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.
wtorek, 04-04-2023 - 17:15, room A.4.1 C-19 (Politechnika Wrocławska)
On Delta-spaces
Arkady Leiderman (Ben-Gurion University)
wtorek, 28-03-2023 - 17:15, A.4.1 C-19 Politechnika Wrocławska
Katětov order on Borel ideals
Tomasz Żuchowski (University of Wrocław)
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.
wtorek, 21-03-2023 - 17:15, *A.4.1* C-19, Politechnika Wrocławska
Combinatorial Banach spaces
Sebastian Jachimek (Uniwersytet Wrocławski)
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.
wtorek, 14-03-2023 - 17:15, A.2.22 C-19, Politechnika Wrocławska
Countable extensions of compact lines
Grzegorz Plebanek
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 N. Joint work with Maciej Korpalski.
wtorek, 07-03-2023 - 17:15, A.2.22 C-19, Politechnika Wrocławska
Trees and Cohen reals
Aleksander Cieślak (Politechnika Wrocławska)
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.
wtorek, 24-01-2023 - 17:00, 3.11 C-11, Politechnika Wrocławska
Descriptive complexity in number theory and dynamics
Bill Mance (Uniwersytet Adama Mickiewicza)
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.
wtorek, 17-01-2023 - 17:00, room 3.11 C-11, Politechnika Wrocławska
Ideal analytic sets
Łukasz Mazurkiewicz (Politechnika Wrocławska)
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.
wtorek, 10-01-2023 - 17:00, 3.11 C-11, Politechnika Wrocławska
Ultrafilters avoiding measures
Artsiom Ranchynski (Uniwersytet Wrocławski)
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 $\beta N$.
wtorek, 13-12-2022 - 17:15, 3.11 C-11 Wrocław University of Science and Technology
Between Cantor and Smulian: the intersection theorem and its applications
Jacek Jachymski (Łódź University of Technology)
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.
wtorek, 06-12-2022 - 17:15, 3.11 C-11, Politechnika Wrocławska
Kappa-pseudocompactness and uniform homeomorphisms of function spaces
Mikołaj Krupski (Uniwersytet Warszawski)
A Tychonoff space X is called \kappa-pseudocompact if for every continuous mapping f of X into 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.
wtorek, 29-11-2022 - 17:15, 3.11 C-11, Politechnika Wrocławska
Some combinatorics related to the Michael space problem III
Arturo Martinez-Celis
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.
wtorek, 22-11-2022 - 17:15, 3.11 C-11, Politechnika Wrocławska
Some combinatorics related to the Michael space problem II
Arturo Martinez Celis (Uniwersytet Wrocławski)
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 ε≥0) is enough to construct a Michael space. If the time allows it, we will show a model of ZFC without Michael ultrafilters. We will occasionally check the result of the World Cup match between Poland and Mexico (which will take place in the time of the seminar).
wtorek, 15-11-2022 - 17:15, room 3.11 C-11, Politechnika Wrocławska
Blocking properties of the diagonal in Cartesian product
Daria Michalik (UJK)
(join work with A. Illanes, V. Martı́nez-de-la-Vega, and J. M. Martı́nez-Montejano) Bobok, Pyrih and Vejnar presented 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.
wtorek, 08-11-2022 - 17:00, C11-3.11 (Politechnika Wrocławska)
A Banach space C(K) reading the dimension of K
Damian Głodkowski (MIM UW)
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
wtorek, 25-10-2022 - 17:15, pokój 3.11 w C-11, Politechnika Wrocławska
P-measures in models without P-points
Adam Morawski (Uniwersytet Wrocławski)
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.
wtorek, 18-10-2022 - 17:00, sala 3.11 C-11 Politechnika Wrocławska
Some combinatorics related to the Michael space problem
Arturo Martinez Celis (Uniwersytet Wrocławski)
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 ε≥0) is enough to construct a Michael space.
wtorek, 11-10-2022 - 17:00, 3.11 C-11, Politechnika Wrocławska
On the hyperspace of simple closed curves in the plane
Paweł Krupski (Politechnika Wrocławska)
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.
wtorek, 14-06-2022 - 17:00, 601
From the Steinhaus property to the Laczkovich one
Eliza Jabłońska (AGH)
wtorek, 07-06-2022 - 17:00, 601
Ideal independent families and ultrafilters
Jonathan Cancino (Czech Academy of Sciences)
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 $\mathfra{s}{mm}$. 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.
wtorek, 31-05-2022 - 17:00, 601
Does there exist a Hindman space which is not a van der Waerden space?
Rafał Filipów (University of Gdańsk)
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 N. This result is from our joint paper with K. Kowitz, A. Kwela nad J. Tryba (Proc. AMS 150(2022), no. 2)
wtorek, 24-05-2022 - 17:00, 601
Sacks indestructible ultrafilters and reaping families
David Chodounsky (Czech Academy of Sciences)
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. About 15 minutes before the seminar we invite you for coffee and a chat.
wtorek, 17-05-2022 - 17:00, 601
Reasonable structures of size aleph_1
Mirna Dzamonja (Université deParis-Cité)
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 ℵ1 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 ℵ1 is homogeneous. We give some examples of interesting structures constructed, such as a homogeneous antimetric space of size ℵ1. Finally, we comment on the situation when one Cohen real is added. About 15 minutes before the seminar we invite you for coffee and a chat.
wtorek, 10-05-2022 - 17:00, 601
On sequences of finitely supported measures on products of compact spaces
Damian Sobota (Kurt Goedel Research Center, Vienna)
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.
poniedziałek, 09-05-2022 - 15:00, 601
Microscopic sets, Hausdorff measures and their cardinal invariants
Ondrej Zindulka (Czech Technical University, Prague)
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. About 15 minutes before the seminar we invite you for coffee and a chat.
wtorek, 26-04-2022 - 17:00, 601
Marczewski ideals of product trees
Aleksander Cieślak (Politechnika Wrocławska)
We investigate Marczewski style ideals associated with the product of two tree-like forcing notions and compare these to original, one dimensional ones. Before the seminar we invite you for coffee and a chat.
wtorek, 12-04-2022 - 17:00, 601
Konrad Królicki (Hungarian Academy of Sciences)
wtorek, 05-04-2022 - 17:00, 601
Hereditarily indecomposable continua as Fraïssé limits
Adam Bartos (Czeska Akademia Nauk)
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. About 15 minutes before the seminar we invite you for coffee and a chat.
wtorek, 29-03-2022 - 17:00, 601
Continuous discrete extension of double arrow
Maciej Korpalski (Uniwersytet Wrocławski)
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. About 15 minutes before the seminar we invite you for coffee and a chat.
wtorek, 22-03-2022 - 17:00, 601
Remarks on Eggleston theorem
Szymon Żeberski (Politechnika Wrocławska)
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.
wtorek, 15-03-2022 - 17:00, 605
On T1- and T2-productable compact spaces
Robert Rałowski (Politechnika Wrocławska)
We prove that if there exists a continuous surjection from a metric compact space X onto a product X × T where T is a T1 second countable topological space which has the cardinality of the continuum then there exists a surjection from X onto the product X × [0, 1] where the interval [0, 1] is equipped with the usual Euclidean topology. About 15 minutes before the seminar we invite you for coffee and a chat.
wtorek, 01-03-2022 - 17:00, 605
Almost disjoint magic sets
Agnieszka Widz (Politechnika  Łódzka)
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 [1]. Recently some results about magic sets were proved by Halbeisen, Lischka and Schumacher [2]. 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.
wtorek, 25-01-2022 - 17:00, 605
A Banach space induced by a compact family
Sebastian Jachimek (University of Wrocław)
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.
wtorek, 18-01-2022 - 17:00, 605
A complemented subspace of a C(K)-space which is not a C(K)-space
Grzegorz Plebanek
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).
wtorek, 07-12-2021 - 17:00, 605
Possible modifications of Lusin analytic set
Łukasz Mazurkiewicz (Politechnika Wrocławska)
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.
wtorek, 30-11-2021 - 17:00, 605
Michael Spaces and Selective Ultrafilters
Arturo Martinez Celis (University of Wrocław)
A 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.
wtorek, 16-11-2021 - 17:00, 605 (we start in the social room)
Combinatorial characterization of null set covering
Maciej Korpalski (Univeristy of Wroclaw)
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.
wtorek, 09-11-2021 - 17:00, 605 (at 17 coffee in the social room)
Generalized Corson compacta and calibers of measures
Grzegorz Plebanek
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.
wtorek, 26-10-2021 - 17:00, 605
On P-measures in random model
Piotr Borodulin-Nadzieja
We show that there is a measure with approximation property in the classical random model.
wtorek, 19-10-2021 - 17:00, social room / 605
A twisted sum of $C(K)$-spaces not isomorphic to any $C(K)$-space
Alberto Salguero Alarcón (Universidad de Extremadura, Badajoz, Spain)
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 (see for example [1, theorem 3.5.b] or [2]). We will focus on one recent and singular construction which serves as an example: a twisted sum of c0 and c0 (c) which is not isomorphic to any C(K)-space. This is part of a joint work with Grzegorz Plebanek.
wtorek, 12-10-2021 - 17:00, 605
Full-splitting Miller trees and Cohen reals
Aleksander Cieślak (Politechnika Wrocławska)
We will investigate tree ideal fm_0 related to certain widening of Miller tree*s. This - so called - full Miller trees con*sist 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.
wtorek, 08-06-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Infinitary continuous logic and descriptive set theory
Maciej Malicki (IMPAN)
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.
wtorek, 25-05-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Local Entropy and Descriptive Complexity
Udayan Darji (University of Louisville)
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 Π^1_1-rank. More importantly, she showed that the collection of CPE Z_2 SFT's is a Π^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 Π^1_1-complete. This is joint work with Garica-Ramos.
wtorek, 18-05-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Absolute model companionship, forcibility, and the Continuum Problem
Matteo Viale (Università di Torino)
Absolute model companionship (AMC) is a strengthening of model companionship defined as follows: For a theory T , T_∃∨∀ 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_∃∨∀ = T*_∃∨∀. The {+, ·, 0, 1}-theory ACF of algebraically closed field is the model companion of the theory of Fields but not its AMC as ∃x(x2 + 1 = 0) ∈ ACF_∃∨∀ \ Fields_∃∨∀. 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 ∈-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 ψ 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 ψ). Partial Morleyizations can be described as follows: let Form_τ be the set of first order τ -formulae; for a subset A of Form_τ , τ_A is the expansion of τ adding atomic relation symbols R_φ for all formulae φ in A and T_τ,A is the τ_A -theory asserting that each τ -formula φ(x) ∈ A is logically equivalent to the corresponding atomic formula R_φ (x~x). For a τ -theory T T + Ti_τ,A is the partial Morleyization of T induced by A ⊆ F_τ.
wtorek, 11-05-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Unboring ideals
Adam Kwela (University of Gdańsk)
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.
wtorek, 27-04-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Applications of non-measurable unions
Szymon Żeberski (Politechnika Wrocławska)
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.
wtorek, 20-04-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
The definable content of (co)homological invariants: Cech cohomology
Aristotelis Panagiotopoulos (Universität Münster)
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.
wtorek, 13-04-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
On zero-dimensional subspaces of Eberlein compacta
Witold Marciszewski (University of Warsaw)
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":
wtorek, 30-03-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
L-orthogonal sequences versus L-orthogonal elements
Gonzalo Martinez Cervantes
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.
wtorek, 23-03-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Convergence in measure and in category.
Władysław Wilczyński (University of Łódź)
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.
wtorek, 16-03-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Rosenthal families
Arturo Antonio Martínez Celis Rodríguez (University of Wrocław)
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.
wtorek, 09-03-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
The exact strength of Sealing
Grigor Sargsyan (Rutgers & IMPAN)
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)-> 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.
wtorek, 02-03-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Complexity of some classes of metrizable compacta up to homeomorphism
Benjamin Vejnar (Charles University, Prague)
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.
wtorek, 02-02-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Games, their values, and Baire class 1 functions
Márton Elekes (Alfréd Rényi Institute of Mathematics, Budapest)
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 characterises 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.
wtorek, 26-01-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
A connected version of Kunen's compact L-space
Grzegorz Plebanek
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.
wtorek, 19-01-2021 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Invariant Ideal Axiom
Michael Hrusak (Universidad Nacional Autónoma de México)
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 $k_\omega$.
wtorek, 15-12-2020 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Attempts to understand the universal minimal flow of ZxZ
Dana Bartošová (University of Florida)
Every Z-flow on a compact Hausdorff space X can be interpreted as a homeomorphism f : X → X and its forward and backward iterates. A flow is minimal if every orbit is dense. The universal minimal flow M (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 (Z × Z) in terms of M (Z). We will start with the few results that are out there about the connection between the corresponding Čech-Stone compactifications β(Z×Z) and β(Z) by Hindman, Blass, and Blass and Moche, that are useful in our considerations. This is a joint work with Ola Kwiatkowska.
wtorek, 08-12-2020 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Projective Fraisse limits of trees
Włodzimierz Charatonik (Missouri University of Science and Technology)
We continue study of projective Fra\"{\i}ss\'e 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 Fra\"{\i}ss\'e 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. This is a joint work with Robert P. Roe.
środa, 25-11-2020 - 18:30, zoom.us (contact pborod@math.uni.wroc.pl)
Borel sets without perfectly many overlapping translations
Andrzej Rosłanowski (University of Nebrasca Omaha)
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.
wtorek, 24-11-2020 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
End-points of chainable continua
Jerzy Krzempek (Silesian University of Technology)
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.
wtorek, 17-11-2020 - 17:00, zoom.us (ask pborod@math.uni.wroc.pl)
Random continuum and Brownian motion
Sławomir Solecki (Cornell)
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.
wtorek, 10-11-2020 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
On wide Aronszajn trees
Mirna Dzamonja (Université Panthéon Sorbonne, Paris)
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 ω1 but with no uncountable branches play an important role by being clocks of Ehrenfeucht--Fraïssé games that measure similarity of model of size ℵ1. 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(ω1). In our upcoming JSL paper with Saharon Shelah we prove that this is not the case: under MA(ω1) 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
wtorek, 03-11-2020 - 16:30, zoom.us (contact pborod@math.uni.wroc.pl)
Complexity of homogeneous continua
Paweł Krupski (Politechnika Wrocławska)
I will show that the family of all homogeneous continua in the hyperspace of all subcontinua of the cube I^n, n=2,3,...,\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.
wtorek, 27-10-2020 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Amalgamation of measures and Banach lattices
Antonio Aviles (Murcia)
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.
wtorek, 20-10-2020 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Splitting Tychonoff cubes into homeomorphic and homogeneous parts (and more)
Jan van Mill (University of Amsterdam)
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 $|\mathcal{E}|\ge \tau^+$.
wtorek, 13-10-2020 - 17:00, zoom.us
Nonmeasurable unions with respect to analytic families
Robert Rałowski (Politechnika Wrocławska)
poniedziałek, 14-09-2020 - 17:15, HS
Dynamical obstructions for classification by (co)homology and other TSI-group invariants
Aristotelis Panagiotopoulos (Uniwersytet w Münster)
One of the leading questions in many mathematical research programs is whether a certain classification problem admits a “satisfactory” solution. Hjorth's theory of turbulence provides conditions under which such a classification problem cannot be solved using only isomorphism types of countable structures as invariants. In the same spirit we will introduce "unbalancedness": a new dynamical obstruction to classification by orbits of a Polish group which admits a two-side invariant metric (TSI). We will illustrate how unbalancedness can be used for showing that a classification problem cannot be solved by classical homology and cohomology theories and how to apply this result to Morita equivalence of continuous-trace C∗-algebras or to the isomorphism problem of Hermitian line bundles. This is joint work with Shaun Allison
wtorek, 23-06-2020 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
A/the (possible) solution of the Continuum Problem
Sakae Fuchino (Kobe)
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.
wtorek, 16-06-2020 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
The universal minimal flow of topological groups beyond Polish
Gianluca Basso (Lozanna, Turyn)
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.
wtorek, 09-06-2020 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Uniform homogeneity
Wiesław Kubiś (Czech Academy of Sciences)
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).
wtorek, 02-06-2020 - 17:00, zoom.us (kontat pborod@math.uni.wroc.pl)
On countable dense homogeneous topological vector spaces
Witold Marciszewski (UW)
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 Cp(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 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
wtorek, 26-05-2020 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Menger and Hurewicz spaces: products and applications to forcing.
Lyubomyr Zdomskyy (KGRC, Wiedeń)
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.
wtorek, 12-05-2020 - 17:00, zoom.us (contact pborod@math.uni.wroc.pl)
Set Theoretic Problems in Large-Scale Topology
Taras Banakh (Lviv)
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
wtorek, 05-05-2020 - 17:00, zoom.us (conact pborod@math.uni.wroc.pl)
Cohen-like poset for adding Fraisse limits
Ziemowit Kostana (MIMUW)
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.
wtorek, 28-04-2020 - 17:15, Zoom, kontakt: pborod@math.uni.wroc.pl
Simplicity of the automorphism groups of homogeneous structures
Aleksandra Kwiatkowska
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.
wtorek, 21-04-2020 - 17:00, Zoom (po link należy zgłosić się do organizatorów seminarium)
Forcing with wider Silver
Aleksander Cieślak (Nokia)
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.
poniedziałek, 10-02-2020 - 17:00, 604
Boolean algebras that resemble uncountable Fraïssé limits. $P(\OMEGA)/FIN$ and its relatives.
Antonio Aviles (Murcia)
poniedziałek, 13-01-2020 - 17:15, 604
Almost disjoint families and spaces of continuous functions.
Grzegorz Plebanek
Given an almost disjoint family A, we consider K_A, the compact space defined by A. We discuss the number of nonisomorphic Banach spaces of the form C(K_A), of continuous functions.
poniedziałek, 25-11-2019 - 17:15, 604
Generalized inverse limits
Włodzimierz J. Charatonik (Missouri University of Science and Technology)
The notion of inverse limits was generalized by Ingram and Mahavier to multivalued settings. We investigate topological properties that are preserved by those generalized inverse limits. We have +theorems about local connectedness, trivial shape, arc-likeness, tree-likeness, dimension etc. The talk is illustrated by many examples.
poniedziałek, 21-10-2019 - 17:15, 604
Compactifiable classes of compacta
Adam Bartoš
poniedziałek, 14-10-2019 - 17:15, 604
Compactifiable classes of compacta
Adam Bartoš (Uniwersytet Wrocławski)
Two classes of topological spaces are \emph{equivalent} if every member of one class has a homeomorphic copy in the other class and vice versa. We say that a class of metrizable compacta $\mathcal{C}$ is \emph{compactifiable} if there is a continuous map $q\colon A \to B$ between metrizable compacta such that the family $\{q^{-1}(b): b \in B\}$ is equivalent to $\mathcal{C}$. I will present several results from the joint work with J. Bobok, J. van Mill, P. Pyrih, and B. Vejnar arxiv.1801.01826.
poniedziałek, 27-05-2019 - 17:15, 604
Some twisting around the Cantor space
Alberto Salguero Alarcón (Universidad de Extramadura)
A twisted sum of Banach spaces $X$ and $Y$ is another space $Z$ containing $Y$ as a subspace such that $Z/Y = X$. In this talk we study the behaviour of twisted sums in which $X$ is a $C(K)$-space, using techniques from Topology and Functional Analysis. Most of the action will take place in the space of continuous functions on the Cantor space.
poniedziałek, 13-05-2019 - 17:15, 604
A_1(X), the space of compacta in X with one accumulation point
Krzysztof Omiljanowski
$A_1(X)$ is contractible for each locally connected continuum $X$. $A_1(C)$ is homeomorphic to $Q^\omega$, where $C$ is the Cantor set.
poniedziałek, 29-04-2019 - 17:15, 604
Banach spaces and analytic P-ideals generated by compact sets.
Piotr Borodulin-Nadzieja

With families of finite subsets of ω we can associate in a natural way a Banach space (in a way in which e.g. Schreier space is defined) and an analytic P-ideal. I will present several examples of such families with Banach spaces and ideals induced by them. I will show that non-trivial ideals generated by compact families cannot be Fσ. As a corollary we obtain certain strengthening of Ptak's lemma and Mazur's lemma.

poniedziałek, 11-03-2019 - 17:15, 604
Hyperspaces of infinite compacta with finitely many accumulation points
Pawel Krupski (Technical University of Wroclaw)
The hyperspace of infinite closed subsets of the interval $J=[-1,1]$ which have at most n accumulation points is characterized as an $F_{\sigma\delta}$-absorber in the Hilbert cube $2^J$. Consequently, it is homeomorphic to the linear subspace $c_0$ of all sequences $(x_k)$ of real numbers converging to 0 with the product topology. If X is a nondegenerate compact absolute retract then the hyperspace of infinite closed subsets of $X$ having finitely many accumulation points is an $F_{\sigma\delta\sigma}$-absolute retract.
poniedziałek, 21-01-2019 - 16:20, 604
Reducing heights of covers of topological spaces
Adam Malinowski
For a countable cover $\mathcal{A}$ of a compact (Hausdorff) space $Y$ with closed subsets we define its height, which is a measure of its complexity and generalizes the notion of the Cantor-Bendixson rank. If $X$ is another compact space and $f : X \to Y$ is continuous, the cover can be pulled back to $X$ and its height may drop, but can never increase. We inspect how much the height can be reduced as $Y$ and $\mathcal{A}$ are fixed while $X$ and $f$ vary.
poniedziałek, 14-01-2019 - 16:20, 604
Niemetryczne continua dziedzicznie nierozkładalne.
Włodzimierz J. Charatonik ((Missouri University of Science and Technology, Rolla)
Udowodnimy, ze dla każdego n naturalnego istnieje continuum dziedzicznie nierozkładalne (niemetryczne) mające n kompozant.
poniedziałek, 10-12-2018 - 16:20, 604
The ideal of the strongly porous sets in the real line.
Arturo Martinez Celis
In the literature there are many different notions of porosity and one of them is the notion of strong porosity: Given a completely metrizable space X, a subset A of X is a strongly porous set if there is a positive constant p such that for any open ball B of radius r smaller than 1, there is an open ball B' inside of B of radius rp such that B' evades the set A. In this talk we will study the cardinal invariants of the sigma ideal generated by the strongly porous sets of the real line and the Cantor space, their relation with different notions of porosity and with other notions in set theory. In particular we will show the notion of strong porosity induces a combinatorial property on trees and we will see that these trees have a connection with the Sacks forcing and some cardinal invariants related to Martin's axiom.
poniedziałek, 26-11-2018 - 16:20, 604
On semigroups of partial order isomorphism and co-finite partial homeomorphisms.
Oleg Gutik (Lwów)
We give a short survey on results on semigroups of co-finite partial order isomorphism of posets and co-finite partial homeomorphisms of real line, which were obtained by the author and his colleagues.
poniedziałek, 05-11-2018 - 16:20, 604
Dziedziczna wlasnosc Baire'a w hiperprzestrzeniach kompaktow
Mikołaj Krupski (Uniwersytet Warszawski)
Przestrzen topologiczna X jest Baire'a jesli dowolny przeliczalny przekroj zbiorow otwartych i gestych w X jest gesty w X. Przestrzen X jest dziedzicznie Baire'a jesli kazda domknieta podprzestrzen przestrzeni X jest Baire'a. Niech X bedzie przestrzenia metryczna i osrodkowa. W swoim odczycie zajme sie nastepujacym dosc naturalnym pytaniem: Jaka wlasnosc przestrzeni X jest rownowazna dziedzicznej wlasnosci Baire'a hiperprzestrzeni K(X) niepustych zwartych podzbiorow X, z metryka Hausdorffa. Niedawno Gartside, Medini i Zdomskyy podali odpowiedz na powyzsze pytanie i zauwazyli jego zwiazki z innymi naturalnymi pytaniami dotyczacymi wlasnosci narostow uzwarcen przestrzeni topologicznych. Pokaze jak twierdzenie Gartside'a, Mediniego i Zdomskyy'ego ma sie do pewnego twierdzenia Telgarsky'ego i naszkicuje alternatywny, bardziej elementarny dowod udowodnionego przez nich faktu.
poniedziałek, 23-04-2018 - 16:20, 604
O przekształceniach domkniętych przestrzeni sigma-zwartych i wymiarze.
Elżebieta Pol (MIM UW)
Narostem przestrzeni Hilberta l2 nazywamy przestrzeń homeomorficzną z Z \ l2 , gdzie Z jest metryzowalnym uzwarceniem l2 , przy czym l2 jest gęste w Z. Pokażemy, że dla każdego narostu K przestrzeni Hilberta l2 , każdy niejednopunktowy obraz K przy przekształceniu domkniętym albo zawiera podzbiór zwarty nie mający małego wymiaru indukcyjnego pozaskończonego ind albo zawiera zbiory zwarte dowolnie dużego wymiaru indukcyjnego pozaskończonego ind. Skonstruujemy też, dla dowolnego naturalnego n, σ-zwartą metryzowalną przestrzeń n-wymiarową, której każdy niejednopunktowy obraz przy przekształceniu domkniętym ma wymiar co najmniej n, oraz analogiczne pzykłady dla indukcyjnego wymiaru pozaskończonego ind (co daje silną negatywną odpowiedź na pytanie R.Engelkinga i E.Pol z pracy “Countable-dimensional spaces: a survey”, Diss. Math. 216 (1983)). Preprint zawierający prezentowane wyniki jest dostępny na stronie arXiv: 1706.04398 [math.GN]
poniedziałek, 19-03-2018 - 16:20, 604
Sekretne związki przestrzeni Banacha z analitycznymi P-ideałami
Piotr Borodulin-Nadzieja
Na odczycie przedstawie m. in. nowe przyklady analitycznych P-idealow inspirowane pewnymi przestrzeniami Banacha, jak i nowe (?) przyklady przestrzeni Banacha inspirowane pewnymi analitycznymi P-idealami.
poniedziałek, 26-02-2018 - 16:20, 604
Homeomorphisms groups of Ważewski dendrites
Aleksandra Kwiatkowska (Wrocław/Munster)
Badamy uniwersalne potoki minimalne grup homeomorfizmów dendrytów Ważewskiego W_P, gdzie P\subset {3,4,...,\omega}. W przypadku gdy P jest skończony udowodnimy, że uniwersalny potok minimalny Homeo(W_P) jest metryzowalny i go policzymy. To daje odpowiedź na pytanie B. Duchesnego. Jeśli P jest nieskończony pokażemy, że uniwersalny potok minimalny Homeo(W_P) nie jest metryzowalny. Wówczas Homeo(W_P) okazuje się być źródłem ciekawych przykładów. W szczególności wtedy Homeo(W_P) są przykładami grup topologicznych które są jednocześnie prezwarte w sensie Roelckego oraz mają niemetryzowalny uniwersalny potok minimalny.
poniedziałek, 19-02-2018 - 16:20, 604
Complexity of distances between metric and Banach spaces
Michal Doucha (Prague)
We extend the theory of Borel/analytic equivalence relations and reductions between them to the theory of Borel/analytic pseudometrics and reductions between them. This is in the spirit of model theory for metric structures which aims to generalize discrete notions to their continuous counterparts. We consider several classical distances from functional analysis and metric geometry, such as Banach-Mazur distance, Gromov-Hausdorff distance, Kadets distance, Lipschitz distance, etc., and show how they reduce to each other in a Borel way. It is joint work with Marek Cúth and Ondřej Kurka.
poniedziałek, 22-01-2018 - 16:20, 604
Fixed points of continuous group actions on continua
Benjamin Vejnar (Charles University, Prague)
In the late 60's Boyce and Huneke independently solved a twenty years old question of Isbell by giving an example of a pair of commuting continuous functions of the closed unit interval into itself which do not have a common fixed point. It follows that the action of a free commutative semigroup with two generators needs not to have a fixed point when acting on the closed interval.In this talk we study the conditions under which every continuous action of a topological (semi)group on a continuum (that is usually one-dimensional in its nature) has a fixed point. We are dealing e.g. with commutative or compact (semi)groups and with the classes of continua including dendrites, dendroids, uniquely arcwise connected continua or tree-like continua.+
poniedziałek, 08-01-2018 - 16:15, 604
Mardešić's problem on products of linearly ordered spaces.
Grzegorz Plebanek
Było to pytanie o to czy, w szczególności, produkt dwóch zwartych przestrzeni liniowo uporządkowanych można odwzorować na produkt trzech przestrzeni niemetryzowalnych. Przedstawię rozwiązanie problemu uzyskane wraz z Gonzalo Martinezem. W języku algebr Boole'a wynik daje pewne twierdzenia o zanurzalności produktów wolnych algebr nieprzeliczalnych w produkty algebr interwałowych.
piątek, 03-11-2017 - 13:30, 604
Convergence of measures and cardinal characteristics of the continuum
Damian Sobota (Politechnika Wiedeńska)
poniedziałek, 05-06-2017 - 16:15, 604
Słaba selekcja generuje porządek.
Krzysztof Omijlanowski
poniedziałek, 22-05-2017 - 16:15, 604
On various notions of homogeneity of Borel sigma-ideals of Polish spaces
Piotr Zakrzewski (Uniwersytet Warszawski)
Opowiem o niektórych wynikach ze wspólnej pracy z Romanem Polem ,,On Borel maps, calibrated sigma-ideals and homogeneity", której najnowszą wersję można znaleźć na stronie https://www.mimuw.edu.pl/~piotrzak/publications.html. Przedmiotem badań pracy są m.in. sigma-ideały I_0(\mu) oraz I_f(\mu) borelowskich podzbiorów kompaktu X, które dla danej miary borelowskiej \mu mogą być pokryte za pomocą przeliczalnie wielu +zbiorów zwartych miary zero lub, odpowiednio, miary skończonej. Przyjmując definicję J. Zapletala mówimy, że sigma-ideał I na X jest jednorodny, jeśli dla każdego zbioru borelowskiego E +spoza I istnieje funkcja borelowska f: X --> E taka, że przeciwobrazy zbiorów z I są w I. Okazuje się, że dla pewnych naturalnych miar, niejednorodności sigma-ideałów I_0(\mu) i I_f(\mu) +towarzyszy jednorodność uzupełnień związanych z nimi algebr ilorazowych postaci Borel(X)/I.
poniedziałek, 15-05-2017 - 16:15, 604
Sumowalność i typ potęgowy indeksu Szlenka
Szymon Draga (Uniwersytet Śląski)
Podczas referatu przypomnimy pojęcie indeksu Szlenka oraz jego związki z asymptotyczną geometrią przestrzeni Banacha. Uzasadnimy, że $c_0$-suma prosta przestrzeni z jednakowo sumowalnym indeksem Szlenka ma sumowalny indeks Szlenka, a także podamy wzór na typ potęgowy ogólnej sumy prostej przestrzeni Banacha
poniedziałek, 13-03-2017 - 16:15, 604
Extension operators and double interval.
Grzegorz Plebanek
poniedziałek, 27-02-2017 - 16:15, 604
Extension operators and Hausdorff gaps
Grzegorz Plebanek (Uniwersytet Wrocławski)
poniedziałek, 12-12-2016 - 16:15, sala 604
Boolean algebras with a few independent sequences
Grzegorz Plebanek
Streszczenie. Zamierzam omówić podstawowe własności algebr generowanych przez rodziny, które nie zawierają nieskończonego ciągu niezależnego.Tego typu klasy algebr są związane z tak zwanymi słabymi kompaktami Radona-Nikodyma.
poniedziałek, 05-12-2016 - 16:15, 604
Splitting chains
Piotr Borodulin-Nadzieja
poniedziałek, 28-11-2016 - 16:15, 604
O rozkladzie kostek euklidesowych na dwa punktoksztaltne zbiory borelowskie, wg pracy Elzbiety i Romana Polów.
Paweł Krupski (Politechnika Wrocławska)
poniedziałek, 21-11-2016 - 16:15, 604
Cones over locally connected curves and the uniqueness problem.
Daria Michalik (UKSW, Warsaw)
poniedziałek, 14-11-2016 - 16:15, 604
O entropii topologicznej dla działań grup
Jakub Gismatullin (Wrocław University)
poniedziałek, 07-11-2016 - 16:15, 604
Metric approximation in groups - continuation
Jakub Gismauttlin (University of Wrocław)
poniedziałek, 24-10-2016 - 16:15, 604
Metric approximation in groups
Jakub Gismatullin (University of Wrocław)
Opowiem o klasycznych hipotezach z dynamiki i topologii dotyczących przesunięć Bernoulliego, (np. Hipoteza Kołmogorowa o izomorfizmie, Hipoteza Gottschalka o surjunktywności, Hipotezy Kaplańskiego) które doprowadziły do badań nad aproksymacją metryczną w grupach, powstania klasy grup soficznych i teorii entropii dla działań grup soficznych na przestrzeniach zwartych. Omówię aktualny stan wiedzy i nowe wyniki.
poniedziałek, 17-10-2016 - 16:15, 604
Boole'owskie obrazy przestrzeni spójnych zwartych
Grzegorz Plebanek
poniedziałek, 10-10-2016 - 16:15, 605
Boole'owskie obrazy przestrzeni spójnych zwartych
Grzegorz Plebanek
poniedziałek, 06-06-2016 - 16:15, 604
Przestrzenie homeomorficzne ze swoimi hiperprzestrzeniami.
Włodzimierz Charatonik
poniedziałek, 30-05-2016 - 16:15, 604
Fraisse theory and homogeneity of the Cantor set
Wiesław Kubiś
I will show how Fraisse theory combined with basic category-theoretic tools gives the result of Knaster and Reichbach saying that every homeomorphism between closed nowhere dense subsets of the Cantor set C extends to an auto-homemorphism of C. I will also discuss possible extensions of this result to generalized Baire spaces and other objects.
poniedziałek, 23-05-2016 - 16:15, 604
Weakly Radon-Nikodým compact spaces
A compact space is said to be weakly Radon-Nikodým if it is homeomorphic to a weak*-compact subset of the dual of a Banach space not containing an isomorphic copy of l_1. In this talk I will show some topological properties of this class of compact spaces and its relation with other classes of compact spaces such as Radon-Nikodým or Corson compacta. Most of the results of this talk are contained in the paper 'On weakly Radon-Nikodým spaces' which is available on arxiv.org.
poniedziałek, 09-05-2016 - 16:15, 604
O powracaniu w parach
Piotr Oprocha (Kraków)
Rozważmy odwzorowanie $T$ działające na przestrzeni zwartej metrycznej $X$. Punkt $x$ jest powracający jeśli pod wpływem działania $T$ jeśli wraca w dowolnie małe swoje otoczenie, oraz ma własność powracania w parach jeśli dla dowolnego punktu $y$ powracającego względem pewnego odwzorowania $S$ para $(x,y)$ jest powracająca względem odwzorowania $T\times S$. Innymi słowy, $x$ ma własność powracania w parach jeśli jego czasy powrotu da się zsynchronizować z czasami powrotu każdego innego punktu powracającego $y$. Jeśli osłabimy założenia odnośnie synchronizacji, np. poprzez dodanie dodatkowych założeń o klasie dopuszczalnych odwzorowań $S$, to wtedy istnieje szansa, że zbiór punktów $x$ o tej nowej własności będzie większy. W referacie przedstawimy znane wyniki oraz pewne problemy otwarte związane z powracaniem w parach.
poniedziałek, 11-04-2016 - 16:15, 604
Mocniejsza forma przeliczalnej gęstej jednorodności płaszczyzny.
Maciej Pietroń
poniedziałek, 21-03-2016 - 16:15, 604
Uniwersalne potoki minimalne i teoria Ramseya (cykl wykładów)
Aleksandra Kwiatkowska
Opis: Niniejszy cykl wykładów będzie z pogranicza dynamiki topologicznej, topologii, grup topologicznych oraz teorii Ramseya. Na początek zaprezentuję twierdzenia Kechrisa-Pestova-Todorcevica mówiące o związkach strukturalnej teorii Ramseya z grupami ekstremalnie średniowalnymi i uniwersalnymi potokami minimalnymi oraz omówię kilka przykładów. W dalszej części skupię się na grupach homeomorfizmów (przestrzeni Cantora, miotełki Lelka, pseudołuku, kostki Hilberta), omówię znane wyniki i przedstawię kilka otwartych pytań. Wyklady odbeda sie w nastepujacych terminach: * poniedzialek 21 marca, godz. 16.15, sala 604 IM, * wtorek 22 marca, godz. 17.15, sala 215 w budynku D1 Politechniki, * sroda 23 marca, godz. 15.15, sala w IM (ta godzina moze jeszcze ulec zmianie)
poniedziałek, 18-01-2016 - 16:15, 604
Brzegi Gromowa z kombinatoryczna wlasnoscia Loewnera
Damian Osajda
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