Date
 Speaker
 Title

11Jan2021 
O.Maslyuchenko (Chernivtsi) 
Compact subspaces of the space of separately continuous functions
We characterize compact subspace of the space S(X x Y) of separately continuous realvalued functions defined on the product of two infinite compact Hausdorff spaces X,Y. The space S(X x Y) is endowed with the topology of sectional uniform convergence. The main result states that a compact Hausdorff space K is homeomorphic to a subspace of S(X x Y) if and only if w(K) < max{c^{♯}(X),c^{♯}(Y)} where c^{♯}(X)=sup{U:U is a disjoint family of open sets in X} is the sharp cellularity of X.

28Dec2020 
Serhiy Maksymenko (Kyiv) 
Smooth approximations and their applications to homotopy types
Let M, N the be smooth manifolds, C ^{r}(M,N)
the space of C ^{r} maps endowed with weak C ^{r}
Whitney topology, and B⊂C ^{r}(M,N)
an open subset. It is proved that for 0≤ r<s≤∞
the inclusion B ∩ C ^{s}(M,N) ⊆ B
is a weak homotopy equivalence.
It is also established a parametrized variant of such a result.
In particular, it is shown that for a compact manifold M,
the inclusion of the space of C ^{s} isotopies
[0,1] x M → M fixed near {0,1} x M into the space of loops
Ω(D ^{r}(M), id _{M}) of the group of
C ^{r} diffeomorphisms of M at id _{M}
is a weak homotopy equivalence.
This result can be found in the paper:
O. Khokhliuk, S. Maksymenko,
Smooth approximations and their applications to homotopy types,
Proceedings of the International Geometry Center, 12:2 (2020) 68108
arXiv:2008.11991, doi: 10.15673/tmgc.v13i2.1781
You can watch the talk on

21Dec2020 
Taras Banakh (Lviv) 
A semigroup is finite if and only if it is chainfinite and antichainfinite
Using the famous Ramsey Theorem (several times)
we shall prove that a semigroup S is finite if and only if S
contains no infinite chains and no infinite antichains.
A subset A of S is called a chain (resp. antichain) if
for any (distinct) elements x,y ∈ A the product xy does (not)
belong to the doubleton {x,y}.
As an exercise, you can try to prove this characterization yourself
(maybe it is simple??)
You can watch the talk on

14Dec2020 
Mykhailo Popov (IvanoFrankivskChernivtsi) 
Some open problems about complemented subspaces of L_{1}
The talk has been mainly prepared for the International Conference
dedicated to 70th anniversary of Professor Oleh Lopushanski,
September 1619, 2019,
IvanoFrankivsk (Ukraine), but unfortunately has not ever been
presented. In the talk I will focus on
a long standing open problem due to Lindenstrauss and Rosenthal (1969), asking
of whether every complemented infinitedimensional subspace of L _{1}
is isomorphic to either L _{1} or l _{1}. The first part of the talk contains statements of partial results and discussion
of some natural ideas to solve the problem. The second part is devoted to presentation
of Rosenthal's biased coin convolution operator, which seems to be discovered
to solve the complementation problem. In the third part, I gonna pose my two recent
related problems.
You can watch the talk on

7Dec2020 
Taras Vasylyshyn (IvanoFrankivsk) 
Algebras of symmetric functions on some Banach spaces
При дослідженні топологічних алгебр функцій важливо мати описи спектрів (множин нетривіальних неперервних лінійних мультиплікативних функціоналів (характерів)) цих алгебр, оскільки спектр алгебри функцій є природною областю визначення для елементів алгебри. Для деяких підалгебр алгебри Фреше аналітичних функцій на банаховому просторі, які складаються з функцій із деякими додатковими властивостями, можна отримати певну додаткову інформацію про спектр або навіть отримати повний опис спектра. Однією із таких властивостей є інваріантність функцій відносно дії деяких фіксованих лінійних операторів на банаховому просторі, на якому задані функції. Якщо цей простір є переставноінваріантним банаховим простором вимірних функцій на деякій множині A ⊂ R, то природно розглядати функції на цьому просторі, які є інваріантними відносно композиції їхнього аргументу з кожною бієкцією множини A, яка зберігає міру. Такі функції називають симетричними. У даній доповіді буде розглянуто алгебри симетричних аналітичних функцій на деяких просторах вимірних за Лебегом функцій, а також, на декартових степенях цих просторів.

23Nov2020 
Mykhailo Popov (IvanoFrankivskChernivtsi) 
Theorems on representations of linear and orthogonally additive operators
Ми підсилюємо та узагальнюємо відомі теореми,
встановлені раніше різними авторами, зокрема,
О.Маслюченком, В.Михайлюком і М.Поповим у 2009 р.,
про представлення регулярних операторів на векторних та
банахових ґратках. Основний наш результат стверджує,
що кожний ортогонально адитивний оператор T з векторної ґратки
E у порядково повну векторну ґратку F, яка є ідеалом деякої
порядково неперервної банахової ґратки G, допускає єдине
подання у вигляді T = T^{a} + T^{c}, де
T^{a} є сумою абсолютно порядково сумовною сім'єю
операторів, що зберігають диз'юнктність, та T^{c} 
порядково вузький ортогонально адитивний оператор.
Якщо ж, крім того, E має головну проективну властивість,
то таке ж саме представлення має довільний регулярний
лінійний оператор.

16Nov2020 
Serhiy Bardyla (Vienna) 
Regular countably compact spaces admitting only constant continuous realvalued maps
We construct a few (consistent) examples of such spaces and resolve a few problems of Tzannes, Banakh, Ravsky and the speaker.
You can watch the talk on

9Nov2020 
Alex Ravsky (Lviv) 
Bounds on the extent of a topological space
The extent e(X) of a space X is a supremum of sizes of closed
discrete subspaces of X.
Assuming that X belongs to some class of spaces, we bound
e(X) by other cardinal characteristics of X, for instance Lindelof number,
spread or density.
You can watch the talk on

2Nov2020 
All (LvivIvanoFrankivskChernivtsi) 
Divertissement
Participants of newly created WestUkrainian Mathematical Seminar
will discuss their recent results, open problems and plans for future.
You can watch the talk on

14Sep2020 
Andrzej Katunin (Gliwice) 
Fractals and fractal sets in Euclidean, hypercomplex, and multicomplex spaces
The seminar talk is devoted to recent activities in studies on fractals and
fractal sets, which is divided to two main parts:
polytopebased fractals, and multi and hypercomplex fractal sets.
In the first part the presenter will discuss his early studies on
determination of polyhedra and higherdimensional polytopes which are
suitable to construct fractals based on them. This part will also
include the methods of visualization of fractals based on regular convex
4polytopes. In the second part the complex fractal sets
(Julia and Mandelbrot sets) will be discussed and their modifications
with respect to a power of a classical quadratic recursive function
z^{2}+c with an analysis of a
possibility to generate fractal sets using these modified functions
will be presented. Next, the
construction of fractal sets in quaternionic and octonionic number
spaces as well as within the
derivatives of mentioned complex and hypercomplex number spaces
will be discussed. Finally, the
multicomplex and multihypercomplex number spaces will be discussed
in the light of possibilities of
construction of fractal sets within these spaces.
The talk will be concluded with a short summary of
research done todate within the topic of the talk together with a
definition of the open problems.

13Jul2020 
Катерина Максимик (Львів) 
Про локально компактнi групи з нулем
Досліджуються алгебраїчні умови на групу G,
при виконанні яких локально компактна трансляційно неперервна
топологія на дискретній групі G з приєднаним нулем є або компактною,
або дискретною. Введено електорально гнучкі та електорально стійкі
групи та вивчаються їх властивості. Зокрема, доведено, що кожна
група, яка містить нескінченну циклічну підгрупу нескінченного
індексу та кожна незліченна комутативна група є електорально
гнучкими, а також, що кожна зліченна локально скінченна група є
електорально стійкою.
Основним результатом роботи є таке твердження: якщо
G  дискретна електорально гнучка нескінченна група, то
кожна гаусдорфова трансляційно неперервна локально компактна
топологія на G^{0} є або дискретною, або компактною.
На довільній нескінченній віртуально циклічній групі
(а отже, на електорально стійкій групі) з приєднаним нулем
побудовано недискретну некомпактну локально компактну
трансляційно неперервну топологію з єдиною неізольованою точкою.

29Jun2020 
Анатолій Савчук (Львів) 
Коскінченні часткові ізометрії зліченних метричних просторів
(за матеріалами кандидатської дисертації)

8Jun2020 
Taras Banakh (Lviv) 
Characterizing the category of sets
We discuss conditions that characterize the category of sets uniquely
up to equivalence or isomorphism.
More details can be found in Section 58 of
this book.

1Jun2020 
Mykhailo Zarichnyi (Lviv) 
Universal k_{ω}spaces related to property C
P. Borst characterized compact metrizable spaces that have property C in
the sense of D.F. Addis and J.H. Gresham as those for which dim_{C},
a transfinite extension of the covering dimension, is determined.
T. Radul proved that for an uncountable set of countable ordinals
β there exist absorbing sets in the class of compact metric spaces
of dim_{C} less than β.
In the talk, we are looking for a counterpart of Radul's result in
the category of k_{ω}spaces.

26May2020 
Conference Geometry in Odesa (Lviv) 
online conference
26.05.2020 (10:5011:30) T. Banakh, Ya Stelmakh
A universal coregular countable secondcountable space
This is a talk at the online conference
Geometry in Odesa  2020.
We shall present a topological characterization of the projective space QP ^{∞},
which has many very strange properties.
In particular, it is secondcountable, countable, semiregular and superconnetced.
More details can be found in this
preprint.
26.05.2020 (14:0014:30)M.ZaricnhyiFunctors and fuzzy metric spaces
This is a talk at the online conference
Geometry in Odesa  2020.
26.05.2020 (14:3014:50)M.RomanskyiКонус, надбудова та джойн в асимптотичних категорiях. Лiпшицева та груба еквiвалентностi деяких функторiальних конструкцiй
Це доповідь (14:3014:50) на онлайн конференції Геометрія в Одесі  2020.
28.05.2020 (15:5016:10)M.KhylynskyiMinimal topologies on semigroup of matrix units
Доповідь на онлайн конференції Геометрія в Одесі  2020.

26May2020 
Mykhailo Zaricnhyi (Lviv) 
Functors and fuzzy metric spaces

26May2020 
M.Romanskyi (Drohobych) 
Конус, надбудова та джойн в асимптотичних категорiях. Лiпшицева та груба еквiвалентностi деяких функторiальних конструкцiй

18May2020 
Taras Banakh (Lviv) 
The completion of the hyperspace of finite subsets, endowed with the l^1metric
Переглянути доповідь можна на

11May2020 
Mykhailo Romanskyi (Lviv) 
On functorial constructions in the asymptotic category
We consider some functorial constructions
(like join, cone, spaces of probability measures with finite supports etc.)
in the Dranishnikov category of metric spaces and asymptotically
Lipschitz maps and focus on the question of equivalence of equivalence of
some obtained spaces in this category.
Переглянути доповідь можна на

4May2020 
Oleksandr Maslyuchenko (ChernivtsiKatowice) 
Betweenness spaces and extension of monotone functions
We shall discuss the structure of betweenness,
which is ternary relation satisfying certain natural axioms.
Переглянути доповідь можна на

27Apr2020 
Lyubomyr Zdomskyy (Vienna) 
Hereditarily Baire hyperspaces of metric spaces
Доповідь присвячена характеризації метричних просторів, чиї гіперпростори компактів з топологією Вієторіса є спадково берівськими, в термінах покриттєвих властивостей їхніх залишків.
Базується на спільній роботі з А. Медіні та П. Ґартсайдом. Переглянути доповідь можна за
посиланням .

23Apr2020 
Mykhailo Popov (Chernivtsi) 
Незалежність Радемахерівського типу в булевих алгебрах
Доповідь відбудеться online на базі відеоконференції Uber
https://www.uberconference.com/room/mishampopov
Початок в 15:00.

16Apr2020 
Mykhailo Popov (Чернівці) 
Горизонтальна властивість Єгорова векторних граток
Семінар відбудеться в режимі відеоконференції на
https://www.uberconference.com/room/mishampopov
Початок семінару в четвер 16 квітня 2020 о 15:00.
Запрошуються всі бажаючі.

13Apr2020 
Taras Banakh (Lviv) 
In the von Neumann–Bernays–Gödel axiomatic system, the Axiom of Transposition can be simplified
We shall survey and compare two most important axiomatic systems of Set Theory:
ZFC
and NBG
and also discuss their advantages and disadvantages. Also we shall prove that
the standard list of NBG axioms can be simplified, replacing the Axiom of Transposition
(saying that for any class X there exists a class {(x,y,z):(x,z,y)∈ X}) by the
more natural Axiom of Inversion (saying that for any class X the class
X ^{1}={(x,y):(y,x)∈X} exists). The detail proof of this fact can be
found on Mathoverflow
Slides

8Apr2020 
Mykhailo Zarichnyi (Lviv) 
MinkowskiRådströmHörmander space of convex subsets
We are going to discuss some algebraic properties of the MinkowskiRådströmHörmander construction of linear space of closed convex bounded subsets as well as some its variations.

2Mar2020 
Taras Radul (LNU) 
Transfinite extensions of asymptotic dimension
We consider recent results about existence of metric spaces with given transfinite dimension.

24Feb2020 
Я.Притула, Т.Банах, М.Зарічний, О.Гутік (Львів) 
Топологія у Львівському університеті (LvivTopoFest)
У програмі заходу відбудуться лекції:
Я. Притула: Історія Львівської топології
Т. Банах: Зигмунт Янішевський (18881920)
М. Зарічний: Іван Миколайович Песін (19301993)
О. Гутік: Про Львівський топологічний семінар та його засновника  Ігоря Йосиповича Гурана
Початок о 15:05!

17Feb2020 
Pavlo Khylynskyi (Lviv) 
The minimal semigroup topologies on an infinite semigroup of matrix units
We discuss on structure of the semilattice of topologies on an infinite semigroup of matrix units. Also, minimal semigroup topologies on such semigroups will be partially described.

10Feb2020 
All (Lviv) 
Divertissement
The active participants of the seminar will discuss open problems and
recent results obtained during Winter Holydays.

25Nov2019 
Taras Banakh (Lviv) 
JosefsonNissenzweig Property for locally convex spaces
We define a locally convex space E to have the
JosefsonNissenzweig property (JNP)
if the identity map (E',σ(E',E))→(E',β^{*}(E',E))
is not sequentially continuous.
By the classical JosefsonNissenzweig theorem,
every infinitedimensional Banach space has the JNP.
We show that for a Tychonoff space $X$, the function space
Cp(X) has the JNP iff there is a *weak nullsequence
of finitely supported signmeasures on X
with unit norm. However, for every Tychonoff space X,
neither the space B_{1}(X) of Baire1 functions on X
nor the free locally convex space L(X) over X has the JNP.
We also define two modifications of the JNP, called the universal JNP
and the JNP everywhere (briefly, the uJNP and eJNP),
and thoroughly study them in the classes of locally convex spaces,
Banach spaces and function spaces.
We provide a characterization of the JNP in terms of operators into
locally convex spaces with the uJNP or eJNP and
give numerous examples clarifying relationships between
the considered notions.
This is joint work with Saak Gabriyelyan.

18Nov2019 
Olena Hryniv (LNU) 
A parallel metrization theorem
Two nonempty sets A, B of a metric space (X , d) are called parallel if d(a, B) = d(A, B) = d(A, b) for any points a of A and b of B. Answering a question posed on mathoverflow.net, we prove that for a cover C of a metrizable space X by compact subsets, the following conditions are equivalent: (i) the topology of X is generated by a metric d such that any two sets A, B of C are parallel; (ii) the cover C is disjoint, lower semicontinuous and upper semicontinuous.

11Nov2019 
Taras Banakh (Lviv) 
A solution of the Tingley problem for C^{3}smooth 2dimensional Banach spaces
We shall prove that any isometry between the unit spheres of $C^3$smooth 2dimensional Banach spaces extends
to a linear isometry of the Banach spaces, which resolves the famous Tingley's
problem in the class of $C^3$smooth 2dimensional Banach spaces.

4Nov2019 
Taras Banakh (Lviv) 
Tingley Problem: isometries of spheres of Banach spaces
The famous Tingley problem (posed in 1987) asks whether every isometry between real
Banach spaces extends to a linear isometry of the whole spaces. Surprisingly,
but this problem still remains open even for 2dimensional Banach spaces.
In the talk we shall discuss an attept of a solution of Tingley problem for sufficiently
smooth 2dimensional Banach spaces by methods of the classical theory of curves.

28Oct2019 
Oleg Gutik (ЛНУ) 
Структурні властивості топологічних напівгруп, близьких до компактних
За матеріалами докторської дисертації (продовження)

21Oct2019 
Oleg Gutik (ЛНУ) 
Структурні властивості топологічних напівгруп, близьких до компактних
За матеріалами докторської дисертації

7Oct2019 
Mykhailo Zarichnyi (Lviv) 
On C*embeddings in sublinear and subpower coronas
J. Keesling proved that that the closure of a σcompact subset
in the Higson corona of a proper etric space is homeomorphic to its
StoneCech compactification.
The aim of the talk is to demonstrate that this is not the case
in the sublinear and subpower coronas.

30Sep2019 
Taras Banakh (Lviv) 
Some problems in Asymptology
We discuss some open problems related to Asymptology and Topology.

23Sep2019 
Taras Banakh (Lviv) 
Geometry Multiverse
We describe various geometry categories and functors between them.

16Sep2019 
Mykhailo Zarichnyi (Lviv) 
Presentation of the book net/лінне
Презентація нової прозової збірки М.М. Зарічного net/лінне
(в Музей етнографії та художнього промислу).

2Sep2019 
Taras Banakh (Lviv) 
An example of a Hausdorff semitopological semilattice, which is not $\bar G_\omega$Lawson
A topologized semilattice $X$ is called $\bar G_\kappa$Lawson for a cardinal $\kappa$
if for any distinct points $x,y\in X$ there exists a family $\mathcal U$ of closed neighborhoods of
$x$ such that $\mathcal U\le\kappa$ and $\bigcap\mathcal U$ is a
subsemilattice that does not contain the point $y$. We prove that each Hausdorff
topological semilattice is $\bar G_\omega$Lawson. On the other hand,
for every cardinal $\kappa$ we construct an example of a Hausdorff semitopological semilattice,
which is not $\bar G_\kappa$Lawson.

22Jul2019 
Iryna Kuz (University of Florida) 
Introduction to Topological Data Analysis
We will discuss what is topological data analysis (TDA) and why it is a
powerful tool for studying and understanding data.
We will also talk about the workflow of TDA and briefly study each
of its components.
Finally, we will discuss some topological properties of the space of
persistence diagrams with the 1Wasserstein distance.

10Jun2019 
Oleg Gutik (Lviv) 
Compact semitopological semigroups and their compact topological I^{n}_{λ}extensions
In the talk we shall present proofs of some results announced 5 June 2019 on
the seminar "Topological Algebra and its Applications".

3Jun2019 
Alex Ravsky (Lviv) 
On structure of minimal feasible lists of swaps representing chaotic attractors, II
This is continuation of the talk, which was started on 27 May 2019.

27May2019 
Alex Ravsky (Lviv) 
On structure of minimal feasible lists of swaps representing chaotic attractors
Our research is based on a recent paper "[Visualizing the Template of a Chaotic Attractor](http://arxiv.org/abs/1807.11853)" of Olszewski et al.~ who use \emph{tangles} (which they call \emph{templates}) to visualize chaotic attractors, which occur in chaotic dynamic systems. Such systems are considered in physics, celestial mechanics, electronics, fractals theory, chemistry, biology, genetics, and population dynamics. In the framework of Olszewski et al., one is given a set of wires that hang off a horizontal line in a fixed order, and a list of swaps between the wires; a tangle then is a visualization of these swaps, i.e., an order in which the swaps are performed, where only adjacent wires can be swapped and disjoint swaps can be done simultaneously.
More formally, we study the following problem. Given a set of $n$ ymonotone \emph{wires}, a \emph{tangle} determines the order of the wires on a number of horizontal \emph{layers} such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a list ~$L$ of \emph{swaps} (that is, quantities of unordered pairs of numbers between~1 and~$n$) and an initial order of the wires, a tangle \emph{realizes}~$L$ if each pair of wires changes its order exactly as many times as specified by~$L$. The aim is to find a tangle that realizes~$L$ using the smallest number of layers.
We showed that this problem is NPcomplete, and we provided an algorithm that computes an optimal tangle for $n$ wires and a given list~$L$ of swaps in $O((2L/n^2+1)^{n^2/2}\varphi^n n)$ time, where $\varphi \approx 1.618$ is the golden ratio and $L$ is the total quantity of swaps in the list $L$. We can treat lists where every swap occurs at most once in $O(n!\varphi^n)$ time. For very long lists we expect an algorithm of complexity $e^O(n^7 log n)\log L$.
But our present talk is devoted to a problem to determine whether a given list of swaps is \emph{feasible}, that is it can be realized by a tangle starting from the identity permutation. We shall consider structure of minimal feasible lists, some results and conjectures about it.

20May2019 
Taras Banakh (Lviv) 
Baire category properties of some function spaces
We shall discuss the problem of characterization of topological spaces X
for which the space B_{1}(X) of functions of the
first Baire class is Baire. In particular,
we prove that this function space is Baire (resp. meager) if
X is a λspace (resp. X is Ukrainian).

6May2019 
Taras Banakh (Lviv) 
Selection properties of the split interval and the Continuum Hypothesis
Let $S$ be the split interval (= the Aleksandrov double arrow space) and
$p:S\to I$ be the natural projection onto the unit interval $I=[0,1]$.
We observe that any selection of the map $p^{1}:I\to S$ is
$F_\sigma$measurable. On the other hand, for the projection
$P:S^2\to I^2$ of the square of $S$ to the unit square,
the inverse multimap $P^{1}$ has a Borel (F_\sigma$measurable)
selection if and only if the Continuum Hypothesis holds.
This CHcounterexample shows that some known selection theorems
for usco multimaps with values in fragmentable compact spaces
do not extend to the class of Rosenthal compacta.

22Apr2019 
Oleg Pikhurko (University of Warwick) 
The minimum number of triangles in graphs of given order and size
In 1941 Rademacher asked for the minimum number of triangles
in a graph of given order and size. This problem has attracted much
attention. It was solved asymptotically by Razborov in 2008. I will
discuss the history of this problem, the methods that were introduced
for attacking it as well as the recent exact results obtained in our
joint work with Hong Liu and Katherine Staden.

15Apr2019 
Taras Banakh (Lviv) 
Площа поверхні сфери та об'єм кулі
Як обчислити площу поверхні сфери та об'єм кулі елементарними засобами без інтегрального та диференціального числення.

8Apr2019 
Mykhailo Zarichnyi ((Lviv)) 
Topology of the space of persistence diagrams endowed with the bottleneck distance
The persistence diagrams are widely used in Topological Data Analysis (TDA),
in particular, for visualization of the persistence homology.
The set of persistence diagrams is usually endowed with different metrics,
in particular, with the bottleneck metric.
The main result of the talk is to describe the topology of this space.

1Apr2019 
Taras Banakh (Lviv) 
σcontinuous functions and related small uncountable cardinals
A function f:X\to Y between topological spaces is called σcontinuous (resp. $\bar\sigma$continuous)
if there exists a countable (closed) cover {X_{n}}_{{n\in\omega}} of X such that for every $n\in\omega$
the restriction $fX_n$ is continuous. Let σ (resp. $\bar\sigma$) be the smallest
cardinality of a subset $X\subset\mathbb R$ such that every function $f:X\to \mathbb R$ is σcontinuous
(resp. $\bar\sigma$continuous). We prove that
$\mathfrak p\le\bar\sigma\le\sigma\le\min\{non(\mathcal M),non(\mathcal N)\}$,
which implies that under Martin's Axiom the cardinals
$\bar\sigma$ and σ are equal to the continuum.

18Mar2019 
Taras Banakh (Lviv) 
A metrizable semitopological semilattice with nonclosed partial order
We construct a metrizable semitopological semilattice $X$ whose partial order $P=\{(x,y)\in X\times X:xy=x\}$ is a nonclosed dense subset of $X\times X$. As a byproduct we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergent sequences.
More details can be found in this paper, written jointly with S.Bardyla and A.Ravsky.

11Mar2019 
Taras Radul (Lviv) 
Functional representation of a capacity monad
Functional representations of the capacity monad based on the max and min operations were considered by Radul and Nykyforchyn. Nykyforchyn considered some alternative monad structure for the possibility capacity functor based on the max and usual multiplication operations.
We show that such capacity monad (which we call the capacity multiplication monad) has a functional representation, i.e. the space of capacities on a compactum can be naturally embedded (with preserving of the monad structure) in some space of functionals on C(X,I). We also describe this space of functionals in terms of properties of functionals.

4Mar2019 
Alex Ravsky (Lviv) 
Two new problems from MathOverflow
The [first](https://mathoverflow.net/questions/323713/adiscontinuousconstruction) was posed by [James Baxter](https://mathoverflow.net/users/132446/jamesbaxter) and concerns set and measure theoretical properties of the unit segment.
The [second](https://mathoverflow.net/questions/311325/vertexcoloringinheritedfromperfectmatchingsmotivatedbyquantumphysics) is “a purely graphtheoretic question motivated by quantum mechanics” posed by [Mario Krenn](http://mariokrenn.wordpress.com/). It is a special case of the questions asked in only a half of a month old arXiv paper "[Questions on the Structure of Perfect Matchings inspired by Quantum Physics](https://arxiv.org/abs/1902.06023)” by Mario Krenn, Xuemei Gu and Daniel Soltész). According to it, "A bridge between quantum physics and graph theory has been uncovered recently [1, 2, 3]. [These are fresh papers, among others, of the first two authors and [Anton Zeilinger](https://en.wikipedia.org/wiki/Anton_Zeilinger), a famous specialist in quantum physics. AR.] It allows to translate questions from quantum physics – in particular about photonic quantum physical experiments – into a purely graph theoretical language. The question can then be analysed using tools from graph theory and the results can be translated back and interpreted in terms of quantum physics. The purpose of this manuscript is to collect and formulate a large class of questions that concern the generation of pure quantum states with photons with modern technology. This will hopefully allow and motivate experts in the field to think about these issues. ... Every progress in any of these purely graph theoretical questions can be immediately translated to new understandings in quantum physics. Apart from the intrinsic beauty of answering purely mathematical questions, we hope that the link to natural science gives additional motivation for having a deeper look on the questions raised above".

25Feb2019 
Inna Hlushak (IvanoFrankivsk) 
Approximations of nonadditive measures
Доповідь за матеріалами кандидатської дисертації

18Feb2019 
Mykhailo Zarichnyi (Lviv) 
Spaces of persistence diagrams in Topological Data Analysis
The persistence diagrams naturally arise in Topological Data Analysis
and become an important tool for description of Big Data.
There are different metrics and topologies on the spaces of persistence
diagrams.
The aim of the talk is to discuss some properties of the obtained
spaces.

11Feb2019 
All (Lviv) 
Divertissement
Active participants of the seminar will discuss some open problems and
perspective directions of research.

4Feb2019 
Taras Banakh (Lviv) 
Metrization of functors with finite supports
In 20082009 in papers of Zarichnyi, Shukel, Radul a canonical
metrization of functors with finite supports was suggested.
We introduce a 1parametric family of such metrizations of functors and
analyse the obtained metrizations for some classical functors: the
functor of nth power, of hyperspace, of free Abelian (Boolean) group.
As partial cases, we obtain the Hausdorff distance on
for the functor of hyperspace and
the Graev metric on the free Boolean group.

28Jan2019 
Taras Banakh (Lviv) 
The spread of a topological groups containing an uncountable subspace of the Sorgenfrey line
We prove that a topological group containing a topological copy of the
Sorgenfrey line contains a discrete subspace of cardinality continuum.
More generally, if a topological group G contains an uncountable subspace X
of the Sorgenfrey line, then G has spead $s(G)\ge s(X\times X)$.
This implies that under OCA (the Open Coloring Axiom), a cometrizable topological
group is cosmic if and only if its has countable spread.
On the other hand,
under CH (the Continuum Hypothesis) we construct an example of a
cometrizable topological group G that contains
an uncountable subspace of the Sorgenfrey line and has countable spread
(more precisely, the countable power of G is hereditarily Lindelof).
This is a joint work with I.Guran and A.Ravsky.

21Jan2019 
Taras Banakh (Lviv) 
Analytic and banalytic spaces and their applications in Topological Algebra
A topological space is defined to be banalytic if it is a Borel image of a Polish space.
It is clear that each analytic space is analytic. The Sorgenfrey line is an example of a banalytic space which is not analytic.
We shall discuss properties of banalytic topological groups and shall prove that under PFA each
Baire banalytic topological group is Polish.

14Jan2019 
Taras Banakh (Lviv) 
Analytic and banalytic spaces

10Dec2018 
Taras Banakh (Lviv) 
On base zerodimensional spaces
A zerodimensional topological space $X$ is called base zerodimensional
if for any clopen base $B$ of the topology of $X$, any open cover of $X$ has a
disjoint refinement consisting of basic sets.
We prove that each countable space is base zerodimensional but the Cantor set is
not base zerodimensional. Also we shall try to prove that base
zerodimensional spaces coincide with the
spaces that have the Rothberger property. More details can be found
here.

19Nov2018 
Iryna Pastukhova (Lviv) 
Ditopological Inverse Semigroups
За матеріалами кандидатської дисертації. Основні результати:
 Означено нове поняття дiтопологiчної iнверсної напiвгрупи i доведено, що клас дiтопологiчних
iнверсних напiвгруп i включає всi топологiчнi групи, топологiчнi напiвгратки, компактнi
топологiчнi iнверснi напiвгрупи i є замкненим вiдносно взяття пiднапiвгруп, тихонiвських
добуткiв, напiвпрямих добуткiв та зведених добуткiв, зокрема конусiв та нульрозширень.
 Доведено теореми вкладення дiтопологiчних iнверсних напiвгруп в тихонiвськi добутки конусiв
та нульрозширень топологiчних груп. Цi теореми є некомпактними узагальненнями теорем вкладення
О. Гринiв, що об'єднують теореми вкладення Лоусона та ПетераВейля.
 Доведено теореми метризацiї дiтопологiчних клiфордових iнверсних напiвгруп (субiнварiантними
метриками).
 Доведено теореми про автоматичну неперервнiсть борелівських чи EHнеперервних гомоморфiзмiв мiж
топологiчними iнверсними напiвгрупами.
Список публікацій
 T. Banakh, I. Pastukhova, Topological and ditopological unosemigroups // Mat. Stud. 39:2 (2013) 119–133.
 T. Banakh, I. Pastukhova, On topological Clifford semigroups embeddable into products of cones over
topological groups // Semigroup Forum, 89:2 (2014) 367–382.
 T. Banakh, I. Pastukhova, Automatic continuity of homomorphisms between topological semigroups //
Semigroup Forum, 90:2 (2015) 280295.
 I. Pastukhova, On continuity of homomorphisms between
topological Clifford semigroups // Carpathian Math. Publ.
6:1 (2014) 123129.
 I. Pastukhova, Automatic continuity of homomorphisms
between topological inverse semigroups // Topological
Algebra and its Applications, 6:1 (2018) 6066.

12Nov2018 
Denys Onypa (Chernivtsi) 
Limit sets and oscillation of functions
За матеріалами кандидатської дисертації, виконаної під керівництвом проф. О.В.Маслюченка (Чернівецький національний університет імені Федьковича).

5Nov2018 
Taras Banakh (Lviv) 
The normality of products of balleans
We shall prove that if the product $X\times Y$ of two
unbounded balleans is normal, then the bornology of
$X\times Y$ has a linearly ordered base.
This resolves one open problem of I.V.Protasov.
More details can be found in this preprint.

29Oct2018 
Taras Banakh (LNU) 
The normality of ball structures on groups
We shall prove that a group $G$ is countable if and only if its finitary ballean is normal.
This solves one problem of I.Protasov. More details can be found in
this paper.

22Oct2018 
Taras Banakh (Lviv) 
The normality of finitary balleans on groups
We shall discuss the notion of normality of
balleans and coarse spaces, introduced by Protasov in 2003.
Our main result says that a group is at most countable if and only if its
finitary ballean is normal. This solves one problem of Protasov.
More information on normal balleans can be found in
this preprint.

8Oct2018 
Agnieszka Bojarska  Sokołowska (Uniwersytet WarmińskoMazurski w Olsztynie. ) 
Uniwersytety dziecięce w Polsce i
interaktywne nauczanie matematyki.

24Sep2018 
Taras Radul (LNU) 
On Ibaricentrically soft compacta
We investigate softness of idempotent barycenter map.

12Sep2018 
Dmitry Gavinsky (Institute of Mathematics of Czech Academy of Sciences) 
Entangled simultaneity versus classical interactivity in communication complexity
In 1999 Raz demonstrated a partial function that had an efficient quantum
twoway communication protocol but no efficient classical twoway
protocol and asked, whether there existed a function with an efficient
quantum oneway protocol, but still no efficient classical twoway protocol.
In 2010 Klartag and Regev demonstrated such a function and asked,
whether there existed a function with an efficient quantum
simultaneousmessages protocol, but still no efficient classical twoway
protocol. In this work we answer the latter question affirmatively and
present a partial function, which can be computed by a protocol sending
entangled simultaneous messages of polylogarithmic size,
and whose classical twoway complexity is lower bounded by a
polynomial.

10Sep2018 
Dmitry Gavinsky (Institute of Mathematics in Prague) 
Якісні розділення між квантовою і класичною комунікаційною складністю
Модель комунікаційної складності, на відміну від складності обчислювальної, дозволяє демонструвати "безумовні" нижні границі, користуючись сучасними математичними інструментами. Це, зокрема, уможливлює застосовування надполіноміальних розділень між квантовою і класичною комунікаційною складністю спеціально підібраних проблем для демонстрації безумовної якісної переваги квантових протоколів (отже і комп'ютерів) над класичними. Ми розглянемо декілька прикладів таких розділень.
Dmitry Gavinsky

4Sep2018 
All 
Divertissement
The active participants of the seminar will discuss open problem and new results obtains during Summer holidays.

21May2018 
Serhiy Bardyla (Lviv) 
On graph inverse semigroups
We shall discuss a recent progress concerning topologizations of graph inverse semigroups.

14May2018 
Taras Banakh (Lviv) 
A real function is continuous if and only if it has closed graph and is peripherally bounded
We prove that a realvalued function f on a locally contractible space X is continuous if and only if it has closd graph and is
peripherally bounded at each point x of X. The peripheral boundedness of f at a point x of X means that for some bounded subset B of the real line and any neighborhood U of x there exists a neighborhood V⊂ U of x whose boundary &partial;V has image f(&partial V)⊂B.
This characterization resolves an open problem posed in Lviv Scottish Book by Julia Wodka from Lodz.

7May2018 
Taras Banakh (Lviv) 
A real function is continuous if and only if it has closed graph and a weak Darboux property
We prove that a real function is continuous if and only if
it has closed graph and a weak Darboux property.
This answers a problem written to Lviv Scottish Book by
Julia Wodka from Lodz.

23Apr2018 
Alex Ravsky (IPPMM, Lviv) 
The chromatic number of the plane is at least 5
We shall discuss a recent brakethrough result of
Aubrey de Grey in direction of solution of the famous
HadwigerNelson problem of calculation of the chromatic number of the
Euclidean plane. It is defined as the smallest number of colors which is sufficient for
coloring the Euclidean plane so that any points on distance 1 have different colors.
Since 50ies it was known that the chromatic number of the plane lies
in the interval [4,7]. Aubrey de Grey improved the lower bound to 5.
So, now we know that the chromatic number of the plane is 5,6, or 7.
It may happen that the answer to HadwigerNelson problem depends on
Axioms of Set Theory.

16Apr2018 
Taras Banakh 
Applying Gutik's hedgehods to recognizing Bokalo regular spaces
A topological space $Y$ is called a Bokalo regular if
each scatteredly continuous function into Y is weakly discontinuous.
By an old result of Bokalo, each regular topological space is Bokalo regular.
We prove that a topological space is Bokalo regular if each subspace of Y contains
a nonempty θopen regular subspace.
Also we observe that each locally regular space is Bokalo regular.
Using the nonregular space called Gutik's hedgehog,
we construct a locally regular topological space
without points of regularity.

2Apr2018 
Taras Radul (LNU) 
An answer to a question of Zarichnyi
We give an answer to a question of Zarichnyi about openness of the idempotent barycenter map.

26Mar2018 
Bogdan Bokalo 
Scattered continuity and the hereditary Lindelof number
We shall discuss topological properties, preserved by scatteredly continuous homeomorphisms.

9Oct2017 
Serhiy Bardyla (LNU) 
On 0bisimple inverse semigroups
We characterize 0bisimple inverse semigroups which semilattice of idempotents is isomorphic to the λary tree whith adjoint zero.

19Jun2017 
Taras Banakh (LNU) 
Generalizing ProdanovStoyanov Theorem on minimal topological groups
We shall prove that an Abelian topological group is compact if and only if it is complete in each weaker group topology.

12Jun2017 
Taras Banakh (LNU) 
Detecting Hclosed topological groups
We shall discuss the recent progress in the problem of detecting topological groups which are (absolutely or injectively) Hclosed in some classes of topological semigroups.
More details can be found in this preprint.

29May2017 
Serhiy Bardyla (LNU) 
Detecting Hclosed semigroups
A semigroup S is defined to be Hclosed if it is closed in each Hausdorff topological semigroup containing S as a discrete subsemigroup.
We shall try to characterize Hclosed semigroups in some classes of semigroups (completely simple, completely 0simple, completely regular, etc).

20Mar2017 
Taras Banakh (LNU) 
On rulers and difference bases in finite groups
We discuss the recent progress on the (open) problem of determining the difference weight Δ[G] of a finite group G,
which is defined as the smallest cardinality of a subset B in G such that BB^{1}=G. It is clear that Δ[G]^{2}>G.
Using known information on rulers we shall prove that each cyclic group G of cardinality
G>2×10^{10} has difference weight Δ[G]^{2}<4G/3.

13Mar2017 
Inna Pozdniakova (LNU) 
On semigroups of partial injective transformations of some partially ordered sets
We describe the Green relations and the congruence lattice of the semigroups of
partial injective transformations of some partially ordered sets.

27Feb2017 
Serhiy Bardyla (LNU) 
Нclosed topological semigroups and semilattices
We discuss principal results of the Ph.D. Thesis (written under supervision of O.Gutik), which are related to:
 topologization of the αbicyclic monoid,
 embeddings of polycyclic monoids into compactlike topological semigroups, and
 Hclosedness of topological semilattices.

20Feb2017 
All (LNU) 
Divertisement
The active participants of the seminar will discuss some open problems and new results obtained during Winter Holydays.

26Dec2016 
Bohdan Bokalo (LNU) 
Some open problems related to the weak continuity
We shall discuss some new results and open problems related to the weak continuity of functions.

12Dec2016 
Mykhailo Zarichnyi (LNU) 
Selfsimilar idempotent measures II
In the idempotent mathematics, the notion of idempotent measure
(Maslov measure) is a counterpart of the notion of probability measure. The aim
of the talk is to discuss the existence of an invariant idempotent measure for an
iterated function system on a complete metric space.
(This is a joint talk with N. Mazurenko).

5Dec2016 
Mykhailo Zarichnyi (LNU) 
Selfsimilar idempotent measures I
In the idempotent mathematics, the notion of idempotent measure
(Maslov measure) is a counterpart of the notion of probability measure. The aim
of the talk is to discuss the existence of an invariant idempotent measure for an
iterated function system on a complete metric space.
(This is a joint talk with N. Mazurenko).

28Nov2016 
Taras Banakh (LNU) 
Separation Axioms in Quasitopological groups
We discuss Separations Axioms in quasitopological groups and construct an example of a regular quasitopological groups,
which is not functionally Hausdorff.

21Nov2016 
Alex Ravsky (IAPMM) 
Strongly σmetrizable spaces are super σmetrizable
A topological space X is called strongly σmetrizable if X is the union of an increasing sequence
(X_{n})_{n∈ω}, of closed metrizable subspaces such that every convergence sequence in
X is contained in some X_{n}. If, in addition, every compact subset of X is contained in some
X_{n}, n∈ω, then X is called super σmetrizable.
Answering a question of V.K.Maslyuchenko and O.I.Filipchuk, we prove that a topological space is strongly σmetrizable
if and only if it is super σmetrizable.

7Nov2016 
PhDstudents (LNU) 
Reports

31Oct2016 
Olena Karlova (Chernivtsi) 
Baire one functions depending on finitely many coordinates
Two questions from [V.Bykov, On Baire class one functions on a product space, Topol. Appl. 199 (2016) 5562] will be discussed. In particular, we will prove that
every Baire one function on a subspace of a countable perfectly normal product is the pointwise limit of a sequence of continuous functions, each depending on finitely many coordinates.
It is proved also that a lower semicontinuous function on a subspace of a countable perfectly normal product is the pointwise limit of an increasing sequence of continuous functions,
each depending on finitely many coordinates, if and only if the function has a minorant which depends on finitely many coordinates.

24Oct2016 
Taras Banakh (LNU) 
On Haarnull and Haarmeager sets in Polish groups
We discuss interplay between (generically) Haarmeager and (generically) Haarnull sets in Polish groups.

10Oct2016 
All (LNU) 
Divertisement
The active participants of the seminar will discuss some open problems and possible ways of their solutions.

3Oct2016 
Ostap Chervak (LNU) 
Color guessing on graphs
A variant of a classic gnome and hats problem will be discussed. For an oriented graph let us consider the following guessing game.
A gnome is sitting on each vertex of an oriented graph and tries to guess its own hat color by looking on the colors of its neighbours.
By the colorguess number cg(G) denote the largest number of colors k such that gnomes have a strategy where at least one of them guesses its hat color correctly.
It is known that cg( K_{k} )=k. B.Bosek, J.Grytchuk and others asked if cg(G) is bounded if G is a bipartite graph or a simple directed graph. It will be proved that cg(K_{k,exp(k +3 log(k))})>k and there exist a simple directed graph on Cexp(k+3log(k)) vertices with cg(G)>k.

26Sep2016 
Ostap Chervak (LNU) 
On Ramsey trees
A notion of Ramsey trees will be introduced. They will offer a useful framework for various Ramseytype problems
including the Ramsey multiplicity problem and the Erdős cliqueindependent set problem. The connection to
Conlon's bound for Ramsey multiplicity constant will be discussed as well.

5Sep2016 
Oleg Pikhurko (University of Warwick) 
Measurable circle squaring
In 1990 Laczkovich proved that one can split a disk into finitely many
parts and move them to form a partition of a square, thus solving the
longstanding Tarski's circle squaring problem. I will discuss our
result with Andras Mathe and Lukasz Grabowski that, additionally,
one can require that all parts are Lebesgue measurable and have the
property of Baire.

5Oct2015 
Taras Banakh (LNU) 
Closed Steinhaus properties of σideals on Polish groups, II
For Polish locally compact groups G this results of Banach was generalized by Laczkovich (1998) who proved that each analytic subgroup of G is either open or belongs to the σideal E generated by closed Haar null subsets of G. These two theorems motivate the problem of detecting σideals on Polish groups G containing every nonempty analytic subgroup of G. Generalizing the results of Banach and Laczkovich we shall prove that any nonopen analytic subgroup H of a Polish group G belongs to every Fσsupported σideal with the closed nSteinhaus property for some n. An ideal I on a topological group G is defined to have the closed nSteinhaus property if for any Ipositive closed subsets A1,...,An of G the product A1··· An has nonempty interior in G. For every n we shall construct an σideal on R which has the closed (n+1)Steinhaus property but fails to have the closed nSteinhaus property. Also we shall discuss possible extensions of the Laczkovich Theorem to nonlocally compact Polish groups.

21Sep2015 
Taras Banakh (LNU) 
Closed Steinhaus properties of σideals on Polish groups
By a classical dichotomy of S.Banach (1931), any analytic subgroup H of a Polish group G is open or meager in G.
For Polish locally compact groups G this results of Banach was generalized by Laczkovich (1998) who proved that each analytic subgroup of G is either
open or belongs to the σideal E generated by closed Haar null subsets of G.
These two theorems motivate the problem of detecting σideals on Polish groups G containing every nonempty analytic subgroup of G.
Generalizing the results of Banach and Laczkovich we shall prove that any nonopen analytic subgroup H of a Polish group G belongs to every
F_{σ}supported σideal with the closed nSteinhaus property for some n. An ideal I on a topological group G
is defined to have the closed
nSteinhaus property if for any Ipositive closed subsets A_{1},...,A_{n} of G the product A_{1}···
A_{n} has nonempty interior in G.
For every n we shall construct an σideal on R which has the closed (n+1)Steinhaus property but fails to have the closed nSteinhaus property.
Also we shall discuss possible extensions of the Laczkovich Theorem to nonlocally compact Polish groups.

8Jun2009 
Oleg Pikhurko (University of Warwick) 
Maximizing the number of colorings
An old problem of Linial and Wilf asks for f(n,m,l), the maximum number
of lcolorings that a graph with n vertices and m edges can have. We
solve this problem for every fixed l when C≤ m≤ cn^{2} for some
C,c>0 depending on l only. Moreover, for l=3, we establish the
structure of optimal graphs for all large m≤ n^{2}/4 confirming (in a
stronger form) a conjecture of Lazebnik from 1989.
This is joint work with PoShen Loh and Benny Sudakov.

25May2009 
R.Cauty (Paris) 
Cohomological properties of subspaces of symmetric powers
We investigate cohomological properties of subspaces of the nth symmetric power X^{[n]} of kdimensional space X. The obtained results imply that the ndimensional sphere S^{n} does not embed into the nth symmetric power X^{[n]} of a 1dimensional compact space X. This resolves a problem of Illanes and Nadler.

27Apr2009 
Taras Banakh (LNU) 
Constructing economical connected metric spaces
A metric space (X,d) is economic if for each infinite subspace A of X the set d(A×A) has cardinality d(A×A)≤dens(A). It is clear that each economic metric space X of density dens(X)<c is zerodimensional.
We show that each (connected) sequential topological space X is the image of a (connected) economic complete metric space Eco(X) under a quotient map Eco(X)→X. The construction the space Eco(X) determines a functor Eco:Top→Metr from the category Top of topological spaces and their continuous maps to the category Metr of metric spaces and their nonexpanding maps.

13Apr2009 
Oleg Gutik (Lviv) 
Embeddings and closedness of inverse semigroups of finite partial bijections
We shall prove that for every infinite cardinal λ the inverse semigroup
I_{λ}^{n} of partial bijections defined on subsets of cardinality ≤ n of λ
does not embed
into a countably compact topological semigroup. Also we shall describe certain Hclosed topologies on the inverse
semigroup I_{λ}^{n}.

6Apr2009 
N. Kolos, B. Bokalo (Lviv) 
When is SC(X)=R^{X}?
In the talk we discuss properties of topological spaces X on which every function f:X→R is scatteredly continuous.
A map f:X→ Y is called scatteredly continuous if for each nonempty subspace A of X the restriction fA has a continuity point.

19May2008 
L.Karchevska (LNU) 
The functor of nonexpanding functionals is not open
The functor E of nonexpanding functionals was introduced by A.Stan'ko and J.Camargo.
It contains many known functors: hyperspace exp, space of probability measures P, superextension λ, space of inclusion hyperspaces G,
the functor of orderpreserving functionals O and many other as subfunctors.
All the functors mentioned above are open and hence bicommutative.
Theorem. The functor E is finitely open but fails to be open (moreover, E is not bicommutative).
A functor F is called (finitely) open if for any surjective open map f:X? Y between (finite) compact spaces the map Ff: FX? FY is open.

12May2008 
R.Voytsitskyy (LNU) 
Extensor and infinitedimensional properties of hyperspaces
Principal results of the Ph.D. Thesis of R.Voytsitskyy will be presented.

5May2008 
V.Mykhaylyuk (Chernivtsi) 
Some problems of the Baire and Lebesgue classification of separately continuous functions
The principal results of the Doctor Dissertation of V.Mykhaylyuk will be presented.

8Apr2008 
Mykhailo Zarichnyi (Lviv) 

31Mar2008 
Taras Banakh (LNU) 
Exdending vectorvalued functions
We prove that a Banach space Y is reflexive if and only if for each closed subspace A of a generalized ordered space X
there is a linear extender u:C_{∞}(A,Y)→C_{∞}(X,Y) if and only if such an extender exists
for the subset of rationals on the Michael line. For the proof we introduce a relative version of the strong Choquet
game created by G.Choquet for characterizing Polish spaces.
This is a joint work with I.Banakh and Kaori Yamazaki.

24Mar2008 
Taras Banakh (LNU) 
The coarse classification of countable groups
Two theorems are proved:
Theorem 1. Each countable group G of asymptotic dimension asdim(G)=0 is coarsely equivalent to the antiCantor set 2^{<ω}.
Theorem 2. A countable Abelian group G with asdim(G)=n is coarsely equivalent to:
 Z^{n} iff G is finitely generated or n=∞;
 Z^{n}×2^{<ω} iff G is infinitely generated.
This is a joint work with Jose Higes and Ihor Zarichnyi.

17Mar2008 
Taras Banakh (LNU) 
The coarse classification of homogeneous ultrametric spaces
We prove that two homogeneous ultrametric spaces are coarsely equivalent if and only
if they have the same sharp entropies.
This classification implies that each homogeneous proper ultrametric space is coarsely equivalent to the
antiCantor set. In particular, any two countable locally finite groups are coarsely equivalent.
For the proof of these results we develop a technique of towers which can have an independent interest.
This is a joint work with Ihor Zarichnyi.

3Mar2008 
Taras Banakh (LNU) 
The coarse classification of homogeneous ultrametric spaces
We prove that two homogeneous ultrametric spaces are coarsely equivalent if and only
if they have the same sharp entropies.
This classification implies that each homogeneous proper ultrametric space is coarsely equivalent to the
antiCantor set. In particular, any two countable locally finite groups are coarsely equivalent.
For the proof of these results we develop a technique of towers which can have an independent interest.
This is a joint work with Ihor Zarichnyi.

25Feb2008 
All (LNU) 
Divertissement
Some open problems have been posed.

18Feb2008 
All (LNU) 
Divertissement
Some open problems have been posed.

24Dec2007 
N.Lyaskovska (LNU) 
Packing index of subsets in Polish groups
For a subset A of a group G we study the packing index
ind_{P}(A)=sup{S:S⊂ G, {xA}_{x∈S} is disjoint} of A.
We show that the packing index of a σcompact subset of a Polish group cannot take an intermediate value between ℵ_{1}
and the contnuum c.
On the other hand, each nondiscrete Polish group contains a nowhere dense Haar null subset of arbitrary packing index κ≤c.

17Dec2007 
Taras Banakh (LNU) 
Algebra in superextensions of groups
We extend the binary operation from a group G to a righttopological semigroup operation on the superextension λ(G) of G and study the
properties of the obtained supercompact righttopological semigroup. We prove that all minimal left ideals of the superextension λ(Z) are metrizable topological semigroups, isomorphic to
minimal left ideals of the superextension λ(Z_{2}) of the compact group Z_{2} of integer 2adic numbers.

10Dec2007 
Oleg Gutik (Lviv) 
On finite partial bijections of bounded rank of a Hausdorff topological space.
We prove that the semigroup of finite partial bijections of bounded rank of a Hausdorff topological space is algebraically closed in the class of topological inverse semigroups.

3Dec2007 
Lyubomyr Zdomskyy (LNU) 
The strong Pytkeev property in function spaces C_{k}(X) and C_{p}(X)
We prove that for each Polish space X, the space C(X) of continuous
realvalued functions on X satisfies (a strong version of) the Pytkeev
property, if endowed with the compactopen topology.
We also consider the Pytkeev property in the case where
C(X) is endowed with the topology of pointwise convergence.
This is a joint work with Boaz Tsaban.

26Nov2007 
Lyubomyr Zdomskyy (LNU) 
The failure of the Lindelöf property in powers of Lspaces
We show that under PFA a hereditarily Lindelöf space X is hereditarily separable provided all finite powers of X are Lindelöf.
This is a joint work with Boaz Tsaban.

19Nov2007 
Lyubomyr Zdomskyy (LNU) 
Characterizing the Arkhangelski's α_{1}property in C_{p}spaces
We show that for a perfectly normal space X with Ind X=0 the following conditions are equivalent:
(i) the function space C_{p}(X) has the Arkhangelski's property α_{1};
(ii) for every metrizable space Y the space B_{p}(X,Y) of all Borel maps from X to Y, endowed with the topology of pointwise
convergence, has the property α_{1};
(iii) the function space C_{p}(X,{0,1}) has the property α_{1.5} (which is formally weaker than α_{1});
(iv) for each Borel map f : X→ ω^{ω} the image f(X) lies in a σcompact subset of ω^{ω};
(v) the space X satisfies the Selection Principle U_{fin}(B,Γ).
This is a joint work with Boaz Tsaban.

12Nov2007 
Taras Banakh (LNU) 
Righttopological semigroups of maximal linked systems on groups
We extend the binary operation from a group G to a righttopological semigroup operation on the superextension λ(G) of G and study the
properties of the obtained supercompact righttopological semigroup. We discuss possible applications of the semigroup λ(Z) to the
Problem of Owings who asked if for any partition of N into two pieces one of the pieces contains the sum set I+I of some infinite subset I.

5Nov2007 
D.Repovs (Ljubljana) 
Characterization of smooth manifolds by smooth homogeneity
The talk will be devoted to a generalization of a problem originally
due to V. I. Arnol'd,
to arbitrary C^{∞}homogeneous compacta.
A locally compact subset
K of R^{n} is said to be C^{∞}homogeneous if for every
pair of points x,y in K there exist neighborhoods
O_{x}, O_{y} of x and y in R^{n}
and a C^{∞}diffeomorphism
h : (O_{x}, O_{x} ∩ K, x)→ (O_{y}, O_{y}∩ K, y). The following is a
characterization of
C^{∞}homogeneous subsets of the Euclidean space R^{n}:
Let K be a locally compact (possibly nonclosed) subset of R^{n}.
Then K is C^{∞}homogeneous if and only if K is a
C^{∞}submanifold of
R^{n}, i.e. (i) if dim K = 0, then K is at most countable subset
of
isolated points in R^{n}; (ii) if 0 < dim K < n, then K is at most
countable
collection of C^{∞}submanifolds
with pairwise disjoint neighborhoods in R^{n}; and
(iii) if dim K = n, then K is an open subset of R^{n}.
Our tools involve, among others, classical dimension theory (covering,
Hausdorff and cohomological dimension) and the Baire Category
Theorem. We shall also discuss further developments, e.g. the failure of such
a characterization for Lipschitz homogeneity, the connection with the
HilbertSmith Conjecture, related work in chaos theory, etc.

22Oct2007 
Taras Banakh (LNU) 
Each G_{δ}measurable map from a complete metric space into a regular space is F_{σ}measurable.
Answering a question of B.Bokalo, we prove that each G_{δ}measurable map from a complete metric space into a regular space
is F_{σ}measurable. A map f is called G_{δ}measurable (resp. F_{σ}measurable)
if the preimage of each open set is of type G_{δ} (resp. F_{σ}) in X. Details of the proof can be found in the paper
T.Banakh, B.Bokalo, On scatteredly continuous maps between topological spaces.

15Oct2007 
Taras Banakh (LNU) 
Each G_{δ}measurable map from a complete metric space into a regular space is F_{σ}measurable.
Answering a question of B.Bokalo, we prove that each G_{δ}measurable map from a complete metric space into a regular space
is F_{σ}measurable. A map f is called G_{δ}measurable (resp. F_{σ}measurable)
if the preimage of each open set is of type G_{δ} (resp. F_{σ}) in X. Details of the proof can be found in the paper
T.Banakh, B.Bokalo, On scatteredly continuous maps between topological spaces.

8Oct2007 
Taras Radul (LNU) 
Geometry of multiplication map of the orderpreserving monad.
We consider some properties of multiplication map of the monad of oderpreserving functionals and prove that it is soft for any compactum.

1Oct2007 
Taras Banakh (LNU) 
Homeomorphism groups of noncompact surfaces
We survey existing results on the topological structure of the homeomorphism groups of compact and noncompact surfaces.

17Sep2007 
All (LNU) 
Divertissement
Some open problems have been posed. In particular, B.Bokalo has asked if
each G_{δ}measurable map f from a Polish space X to a regular space Y is F_{σ}measurable.

10Sep2007 
All (LNU) 
Divertissement
Some open problems have been posed.

31Aug2007 
D.Repovs (Ljubljana) 
Interesting constructions based on the Topologist sine curve.
We shall present a variety of interesting applications of the classical example of a 1dimensional connected nonPeano planar continuum, the Topologist sine curve (and its derivatives, most notably the Warsaw circle) to diverse problems of geometric topology in dimensions 2 and 3. For example: an example showing that the classical van Kampen theorem fails without the openess condition, a counterexample to Molnar's theorem from 1950's, and a construction of a 2dimensional noncontractible simply connected celllike continua. We shall also present the solution of the BestvinaEdwards problem: Does there exist a noncontractible celllike compactum whose suspension is contractible? In conclusion we shall state some interesting open problems.

14May2007 
Taras Banakh (LNU) 
Symmetric subsets and partitions of the Lobachevsky plane.
We prove that for any partition of the Lobachevsky plane into finitely many Borel pieces one of the cells of the partition contains an unbounded centrally symmetric subset. This distinguishes the Lobachevsky plane from the Euclidean plane which can be divided into 3 Borel pieces containing no unbounded centrally symmetric subset.

7May2007 
Taras Radul (LNU) 
Hyperspace as intersection of idempotent measures and inclusion hyperspaces.

16Apr2007 
Taras Banakh (Lviv) 
Compactly convex sets.
A convex subset C of a linear topological
space is called
compactly convex if there is a continuous
compactvalued map
F assigning to each point x of C a compact subset F(x)
of C so that [x,y]
lies in the union of F(x) and F(y). We prove that each
convex subset of the plane is
compactly convex. On the other hand, the 3dimensional
space
R^{3} contains a convex subset which is
not compactly convex.
This is a joint work with M.Mitrofanov and O.Ravsky.

2Apr2007 
Taras Banakh (Lviv) 
Continuously homogeneous spaces.
A topological space X is called continuously homogeneous if there is a homeomorphism H of X^{3} such that H(x,y,z)=(x,y,h_{x,y}(z)) and H(x,y,x)=(x,y,y) for all points x,y,z in X.
It is clear that each topological group is continuously homogeneous with the homeomorphism H(x,y,z)=(x,y,yx^{1}z). An example of a continuously homogeneous space homeomorphic to no topological group is the 7dimensional sphere S^{7}. On the other hand, the Hilbert cube fails to be continuously homogeneous.
This is joint work with Z.Kosztolowicz (Kilece).

26Mar2007 
Taras Banakh (Lviv) 
Groupsequential topological groups.
A topological group G is called groupsequential if it carries the strongest group topology inducing the original topology on each convergent sequence. We give examples of such topological groups, characterize groupsequential free topological groups and pose some open problems.

12Mar2007 
Oleg Gutik (Lviv) 
Hclosed topological semigroups and semilattices.
In the report we shall discuss about some properties of Hclosed topological semigroups and semilattices.

5Mar2007 
K.Koporkh (IvanoFrankivs'k) 
On spaces of quotient maps.

26Feb2007 
Mykhailo Zarichnyi (Lviv) 
Spaces of uppersemicontinuous capacities.
The notion of capacity was introduced into mathematics by G.Choquet. The family of all uppersemicontinuous capacities is endowd with the weak* topology. The obtained construction is functorial in the category of compact Hausdorff spaces and the talk is devoted to properties of this functor. In particular, the open mapping theorem will be proved.

18Dec2006 
O.Shukel' (LNU) 
Functors of finite degree and asymptotic dimension zero.
For any finitary normal functor F in the category of compact Hausdorff spaces one can define its counterpart on the category of proper metric spaces and coarse maps. The aim of this talk is to show that the obtained functor preserves the class of proper metric spaces of asymptotic dimension zero in the sense of Gromov. The obtained result is a counterpart of the corresponding result of Basmanov in the category of compact Hausdorff spaces.

11Dec2006 
Taras Banakh (Lviv) 
Tranfinite separation dimension.

6Dec2006 
Taras Radul (LNU) 
Openess of HartmanMycielski functor.
We investigate topological properties of HartmanMycielski functor on non metrizable compacta.

6Dec2006 
Mykhailo Zarichnyi (Lviv) 
Idempotent probability measures.
The set of all idempotent probability measures (Maslov measures) on
a compact Hausdor. space endowed with the weak* topology determines is func
torial on the category Comp of compact Hausdor. spaces. We prove that the
obtained functor is normal in the sense of E. Shchepin. Also, this functor is the
functorial part of a monad on Comp. We prove that the idempotent probability
measure monad contains the hyperspace monad as its submonad. A counterpart
of the notion of Milyutin map is de.ned for the idempotent probability measures.
Using the fact of existence of Milyutin maps we prove that the functor of idem
potent probability measures preserves the class of open surjective maps. Unlikely
to the case of probability measures, the correspondence assigning to every pair of
idempotent probability measures on the factors the set of measures on the product
with these marginals, is not open.

20Nov2006 
Taras Banakh (LNU) 
On scatteredly continuous maps between topological spaces.
A bijective map $h:X\to Y$ between topological
spaces is a {\em scattered homeomorphism} if both $h$ and $h^{1}$
are scatteredly continuous. A map $f:X\to Y$ between topological
spaces is defined to be {\em scatteredly continuous} if for each
subspace $A\subset X$ the restriction $fA$ has a point of
continuity. We characterize scatteredly continuous maps in various terms and also study properties of topological spaces preserved by
scatteredly continuous maps and scattered homeomorphisms.

15Nov2006 
Szymon Głąb (Łódź) 
On Komjath property of sigmaideals.

13Nov2006 
Piotr BorodulinNadzieja (University of Wrocław) 
Minimally generated Boolean algebra

6Nov2006 
Alexander Balinsky (Cardiff University) 
Zero modes of Pauli operator

6Nov2006 
Taras Banakh (LNU) 
On scattered compactifications of scattered metrizable spaces.
We prove that each metrizable scattered space has a hereditarily paracompact scattered compactification. Also we show that the class of hereditarily paracompact scattered compact spaces coincides with the smallest class K of compacta closed with respect to taking one point compactification of discrete topological sums of compacta from K.
Consequently each compact of the class K is uniform Eberlein.

30Oct2006 
E. Tymchatyn (University of Saskatchewan) 
Convex metrics in the noncompact setting.
Bing (1949) showed that every Peano continuum X admits an
equivalent convex metric. To do this he showed
first that every Peano continuum can be partitioned into finitely
many small Peano continuous each pair of which meet only in their
common boundary.
We prove that if an arcwise connected and connected metric space
X admits a vanishing sequence of partitions U_{i} with U_{i+1} refining
U_{i}, then X admits an
equivalent convex metric. E.g., Plane with all points (x,y) where
x and y are both irrational removed.

16Oct2006 
E. Tymchatyn (University of Saskatchewan) 
Continuous extension operators for uniformly continuous functions and pseudometrics.
Kunzi and Schapiro (1997) found a continuous linear extension operator for all realvalued continuous functions with domains which are compact subsets of a metric space (X, d). We find analogous theorems for the space of uniformly continuous bounded functions with domains which are bounded subsets of (X, d).

9Oct2006 
Taras Banakh (LNU) 
Absolute Zspaces and their applications to Dimension Theory.
A compact space X is defined an absolute Zspace if for any embedding of X into the Hilbert cube Q the set Xx{x_{0}} is a Zset in Qx[1,1]. We discuss the relation of absolute Zspaces to other dimension clases of compacta.

2Oct2006 
Taras Radul (LNU) 
Asymptotic dimensions.
It is shown that the transfinite extension of the asymptotic counterpart of the large inductive dimension is not trivial.

11Sep2006 
E. Tymchatyn, A.Zagorodnyuk 
Free Banach spaces and extensions of Lipschitz maps.
We study the free Banach space B(X) over a metric space X, that is a predual space of the Banach space of all Lipschitz functions on X which preserve a marked point θ in X. Some applications to the extension theory of Lipschitz function are obtained.

6Sep2006 
Taras Radul (LNU) 
Asymptotic dimensions.
It is shown that the transfinite extension of the asymptotic counterpart of the large inductive dimension is not trivial.

2Sep2006 
D.Repovs (Ljubljana) 
Suspensions of celllike compactum.
We prove that
(1) Every compact metrizable space is weakly homotopy equivalent to a celllike compactum and
(2) There exists a noncontractible celllike compactum whose suspension is contractible (this
gives an affirmative answer to the BestvinaEdwards problem).

24May2006 
H.Mildenberger
(Vienna) 
On cardinal characteristics of the reals.
We shall discuss cardinal characteristics of the reals.

22May2006 
R.Cauty (Paris) 
Homologic algebra and generalization of BorsukUlam Theorem.
We shall discuss homologic algebra and generalization of BorsukUlam theorem.

6Mar2006 
O.Shukel' (LNU) 
Hypersymmetric powers and asymptotically zerodimesional spaces.
We study the properties of functor of hypersymmetric power in the
asymptotic category and category $R$ ( Roe category of proper
metric spaces and proper coarse embedding). Some examples of
asymptotically zerodimensional spaces are considered. Also we
consider the question of coarse embeddability of this spaces and
their hypersymmetric powers.

27Feb2006 
Taras Banakh (LNU) 
P_{1}metrics.
We discuss P_{1}metrics.

20Feb2006 
N.Lyaskovska (LNU) 
Weakly Psmall not Psmall subsets in a group.
Answearing a question of
Dikranjan and Protasov we prove that each infinite group
containt
a weakly Psmall subset A which is not Psmall. The latter means
that for every n there are pairwise disjoint translation copies of
A in group G but there is no infinitely many such disjoint copies.

13Feb2006 
All (LNU) 
Divertissement.
Some open problems are discussed.

28Nov2005 
Taras Banakh (LNU) 
Division and kth Root Theorems for the Hilbert cube.
We discuss the division and kth Root Theorems for the Hilbert cube.

14Nov2005 
Taras Banakh (LNU) 
An answer to Maslyuchenko's question.
We present the answer to one of Maslyuchenko's questions.

7Nov2005 
Ihor Stasyuk (LNU) 
On operators extending partial ultrametrics.
We discuss operators extending partial ultrametrics.

24Oct2005 
Taras Banakh (LNU) 
Subproper maps and k^{*}spaces.
We discuss subproper maps and k^{*}spaces.

5Oct2005 
N.Lyaskovska (LNU) 
Weakly Psmall not Psmall subsets in an Abelian group.
Answearing a question of Dikranjan and Protasov we prove that each infinite Abelian group containt a weakly Psmall subset A which is not Psmall. The latter means that for every n there are pairwise disjoint translation copies of A in group G but there is no infinitely many such disjoint copies.

26Sep2005 
R.Kozhan (University of Warwick) 
Openmulticommutativity of the probability measure functor.
For the functors acting in the category of compact
Hausdorff spaces, we introduce the socalled open multicommutativity property, which generalizes both
bicommutativity and openness, and prove that this property is satisfied by the functor of probability measures.

19Sep2005 
Taras Banakh (LNU) 
Scheepers' diagram and a question of Zdomskyy.
We discuss Scheepers' diagram and a question of Zdomskyy.

12Sep2005 
Taras Banakh (LNU) 
An answer to a question of Plicko about the cardinality of Hamel basis.
We provide an answer to a
question of Plicko about the cardinality of Hamel basis.
