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

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

(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

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

TBA

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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

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

$A_1(X)$ 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.

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 *F**σ*. 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

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.

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.

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.

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

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

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.

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.

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.

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

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

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

poniedziałek, 10-10-2016 - 16:15, 605

Boole'owskie obrazy przestrzeni spójnych zwartych

poniedziałek, 06-06-2016 - 16:15, 604

Przestrzenie homeomorficzne ze swoimi hiperprzestrzeniami.

poniedziałek, 30-05-2016 - 16:15, 604

Fraisse theory and homogeneity of the Cantor set

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

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.

poniedziałek, 21-03-2016 - 16:15, 604

Uniwersalne potoki minimalne i teoria Ramseya (cykl wykładów)

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