O seminarium

Topologia
wtorek
17:00
18:00
zoom
Uczestnicy: 
Zbigniew Lipecki
Alicja Samulewicz
Michał Morayne
Robert Rałowski
Marcin Michalski
Aleksandra Kwiatkowska
Arturo Martinez Celis
Andrzej Rosłanowski
Jan van Mill
Mirna Dzamonja
Jan van Mill
Aleksander Błaszczyk
Sakae Fuchino
Andrzej Kucharski
Włodzimierz Charatonik
Dana Bartosova
Adam Bartos
David Chodounsky
Oleg Gutik
Taras Banakh
Wiesław Kubiś

Terminy i tematyka spotkań

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)
UNDER THE CONTINUUM HYPOTHESIS, THE PAROVICHENKO THEOREM IMPLIES THAT THE BOOLEAN ALGEBRA $P(\OMEGA)/FIN$ IS THE UNCOUNTABLE FRAïSSé LIMIT OF THE CATEGORY OF COUNTABLE BOOLEAN ALGEBRAS. WE WILL DISCUSS WHAT HAPPENS IN OTHER MODELS OF SET THEORY.
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
GONZALO MARTINEZ CERVANTES (Murcia)
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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