In this talk we will introduce the main notions of the theory of vector lattices and see how they manifest in the case of the lattice C(X) of all continuous real-valued functions on a (Tychonoff) topological space X. The latter special class is rich enough so that some of the results about it can be generalized to the case of an abstract Archimedean vector lattice using appropriate representation theorems.

After some preliminaries on the subject, I am going to present an order continuous orthogonally additive operator which is an $\varepsilon$-approximation of the principal lateral band projection $Q_e$ (the order discontinuous lattice homomorphism $Q_e: E \to E$ which assigns to any element $x\in E$ the maximal common fragment $Q_e(x)$ of $e$ and $x$). Using it, I will provide the first example of an order discontinuous orthogonally additive operator which is both uniformly-to-order continuous and horizontally-to-order continuous. Another result, which I am going to talk about, gives sufficient conditions on Riesz spaces $E$ and $F$ under which such an example does not exist. The talk is based on a part of a just finished joint paper with V. Mykhaylyuk.

I will talk about the theory of weak* sequential closures (weak* derived sets), which S. Banach and S. Mazurkiewicz started to develop in 1929-1932. I plan to describe the history of the topic and its applications. The main new result: for every nonreflexive Banach space X and every countable successor ordinal α, there exists a convex subset A in X* such that α is the least ordinal for which the weak* derived set of order α coincides with the weak* closure of A. This result extends the previously known results on weak* derived sets by Ostrovskii (2011) and Silber (2021).

For Banach spaces $A,X,Y$, a bilinear operator $A\times X\to Y$, $(a,x)\mapsto ax$, is defined to preserve unconditional convergence if for any bounded sequence $(a_n)_{n\in\omega}$ and any unconditionally convergent series $\sum_{n\in\omega}x_n$ in $X$ the series $\sum_{n\in\omega}a_nx_n$ converges unconditionally in $Y$.
We prove that for numbers $p,q,r\in[1,\infty]$, the coordinatewise multiplication $\ell_p\times\ell_q\to\ell_r$ preserves unconditional convergence if and only if one of the following conditions is satisfied: (i) $p\le 2$ and $q\le r$, (ii) $2\lt p\lt q\le r$, (iii) $2\lt p=q\lt r$, (iv) $r=\infty$, (v) $2\le q\lt p\le r$, (vi) $q\lt 2\lt p$ and $\frac1p+\frac1q\ge\frac1r+\frac12$.

The Bernstein polynomials were introduced in 1912 by S. N. Bernstein, who used them to provide an elegant proof of the Weierstrass Approximation Theorem. Subsequently, many remarkable properties and important applications of these polynomials were discovered, while a great number of generalizations and analogues were introduced. In this presentation, the generalized Bernstein polynomials based on the q-integers, otherwise known as q-Bernstein polynomials, will be discussed. These q-Bernstein polynomials were defined by G. M. Phillips in 1997 and studied by a number of researches from different angles during the last decades. While for q=1, the q-Bernstein polynomials coincide with the classical ones, for q ≠ 1, one obtains new polynomials with rather different properties. In this talk, starting with the definition of the q-Bernstein polynomials, an overview of the results in this subject will be presented. The aim of the presentation is to show what do we obtain by involving parameter q and, as such, to discuss why the q-Bernstein polynomials deserve investigation.

It is shown that universal algebras that are injective in their equational classes are characterized by internal property that can be called completeness. We define universal algebra $A$ as complete (closed to simple extensions) if for each its subalgebra $A'$ and each set of extension conditions for this subalgebra there is $a \in A$ that satisfies these conditions. We define a set of extension conditions for $A'$ as the difference between factorization kernels of free algebras for $A'$ and corresponding extension. It's proved that each injective universal algebra is complete and each complete universal algebra belonging to the class of algebras with CEP is injective. It's checked directly that complete (in the sense of ordering) boolean algebras and divisible Abelian groups are complete in the sense defined here. More details can be found in the preprint https://arxiv.org/abs/2109.09539

In the talk we shall discuss the interplay between pairs of Hahn and separately continuous functions, the topic, initiated by V.K. Maslyuchenko. A pair $(g,h)$ of functions on a topological space is called a pair of Hahn if $g\le h$, $g$ is an upper semicontinuous function and $h$ is a lower semicontinuous function. We say that a pair of Hahn $(g,h)$ is generated by a function $f$, which depends on two variables, if the infimum of $f$ and the supremum of $f$ with respect to the second variable equals to $g$ and $h$ respectively. We prove that for any perfectly normal space $X$ and non-pseudocompact space $Y$ every pair of Hahn on $X$ is generated by a continuous function on $X\times Y$. We also obtain that for any perfectly normal space $X$ and for any space $Y$ having non-scattered compactification any pair of Hahn on $X$ is generated by a separately continuous function on $X\times Y$.

A continuation of the preceding talk. We shall apply Hadamard matrices in order to show that the coordinatewise multiplication $\ell_p\times\ell_q\to\ell_r$ does not preserve unconditional convergence if $r\lt q$ or $\frac1r+\frac12\gt\frac1p+\frac1{\min\{2,q\}}$.

By a Banach action we understand any continuous bilinear function $A\times X\to Y$, where $A,X,Y$ are Banach spaces. We say that a Banach action $A\times X\to Y$ preserves unconditional convergence if for any bounded sequence $(a_i)_{i\in \omega}$ in $A$ and any unconditionally convergent series $\sum_{i\in\omega}x_i$ in $X$ the series $\sum_{i\in\omega}a_ix_i$ is unconditionally convergent in $Y$. Applying the famous Grothendieck inequality, we prove that a Banach action $\ell_2\times X\to Y$ preserves unconditional convergence if there exists an orthonormal basis $(e_i)_{i\in\omega}$ in $\ell_2$ such that for any $x\in X$ the series $\sum_{i\in\omega}e_ix$ is weakly unconditionally Cauchy. Applying this result to the coordinatewise multiplication $\ell_p\times\ell_q\to\ell_r$ where $p,q,r\in[1,\infty]$ and $\frac1r\le \frac1p+\frac1q$, we prove that this multiplication preserves unconditional convergence if either $r=\infty$ or $p\in[1,\infty]$ and $q\le r$. On the other hand, applying famous Hadamard matrices, we prove that this multiplication does not preseve unconditional convergence if $r\lt q$ or $\frac12+\frac1r\gt\frac1p+\frac1{\min\{2,q\}}$.

This is a continuation of the preceding talk.

We discuss finite lattices which can be represented as a poset of free open filters on a given Hausdorff space. For each positive integer n we construct a space admitting exactly n free open filters which form a chain. This way we solved a few problems of Mooney. Also, assuming the existence of enough many measurable cardinals we construct spaces with different finite nonlinear lattices of free open filters. On the other hand, we showed that there exists a finite (even 5-element) lattice which cannot be represented as a lattice of free open filters over some Hausdorff space.

A topological ring R is defined to be Hirsch, if for any unconditionally convergent series ∑_i x_i in R and any neighborhood U of the additive identity 0 of R there exists a neighborhood V⊂ R of 0 such that ∑_i∈If_ix_i∈ U for any finite subset I of ω and elements {f_i}_i∈I⊂ V. We investigate a question of recognizing Hirsch rings in certain known classes of topological rings. We prove that a topological ring R is Hirsch provided R is locally compact or R is sequentially complete and has a base at the zero consisting of open ideals, or R is a closed subring of the Banach ring C(K) where K is a compact Hausdorff space. This implies that the Banach ring l_∞ and its subrings c₀ and c are Hirsch. Grothendieck's Theorem on absolutely summing operators implies that the Banach ring l₁ is Hirsch. On the other hand, the Banach space l₁ admits a structure of a unital commutative Banach ring, which is not Hirsch. Also for any p∈(1,∞), the Banach ring L(l_p) of continuous endomorphisms of the Banach ring l_p is not Hirsch. We do not know whether the Banach ring l₂ is Hirsch.

We construct a functor assigning to each poset X a Hausdorff zero-dimensional topological pospace FX containing X as a dense discrete subspace such that X=FX iff X is well-founded. We show that this functor admits a lifting to the category of ordered universal algebras of arbitary signature. As an application of this result, we prove that an inverse semigroup X if H-closed in the class if Hausdorff zero-dimensional topological inverse semigroups if and only if the maximal semilattice of X is chain-finite.

Let C be a class of topological semigroups. A semigroup X is C-closed if X is closed in each topological semigroup Y∈C that contains X as a discrete subsemigroup. We discuss this notion for two basic classes of semigroups -- groups and semilattices -- and prove the following characterizations.

Theorem. Every group is C-closed in the classes T₁IS of T₁ topological inverse semigroups and T₁CS of T₁ topological Clifford semigroups.

A semigroup X is 0-closed if it is closed in its 0-extension X⁰=X∪{0} endowed with any Hausdorff semigroup topology.

Theorem 2. A (countable) cancellative semigroup X is 0-closed if (and only if) X is closed in the class T₁S of T₁ topological semigroups if (and anly if) X is polybounded in the sense that X=∪_i≤n{x∈ X:f_i(x)=b_i} for some semigroup polynomials f₁,...,f_n:X→X and some elements b₁,...,b_n.

Theorem 3. If a semigroup X is T_zS-closed in the class T_zS of zero-dimensional Hausdorff topological semigroups, then the center Z(X)={z∈X:∀x∈X (xz=zx)} is chain-finite.

A semigroup X is chain-finite if each chain in X is finite. A subset C of X is a chain if xy∈{x,y} for any x,y∈C.

Corollary. A semilattice X is T₁S-closed if and only if X is T_zS-closed if and only if S is chain-finite.

Communication complexity is one of the strongest computational settings where we already know how to prove meaningful lower bounds (hardness). This can be used, in particular, to prove qualitative advantage of some quantum models over classical ones. We will see a two-party communication problem that:

• has an efficient solution in one of the weakest quantum models – namely, simultaneous message passing (SMP);
• is hard for one of the strongest feasible classical models – namely, two-way (interactive) protocols.

The talk is based on the paper of the same title https://users.math.cas.cz/~gavinsky/papers/QuSiClIR.pdf

Ми наведемо достатню умову для того, щоб сума скінченного числа доповнювальних підпросторів банахового простору була доповнювальна (в цьому банаховому просторі). Наведемо теорему про точність цієї умови. За допомогою цієї достатньої умови ми отримаємо достатню умову для того, щоб сума скінченного числа маргінальних підпросторів у просторі $L^p$ була доповнювальна в $L^p$.

We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear isometry of the Banach spaces. This resolves the famous Tingley's problem in the class of 2-dimensional Banach spaces.
More details can be found in this preprint: researchgate.net/publication/350075310

In this talk we consider motion of the random curve in the space. For description of such motion we use the stochastic differential equation with interaction. Such equation allows to take into account the action of external forces on the parts of the curve and interaction between them. The main question is the changing of the complexity of the curve with time. For nonsmooth random curves like a trajectiry of Brownian motion we consider possibility to contain an ordered polygonal line withthe long edges and big number of vertices. In these cases the visitation measure for certain images of the curve approximates discrete measures. This can serve as an explanation of the intermittensy of its density.

Various constructions of metric spaces have their counterparts in the category of fuzzy metric spaces. The talk is devoted to the notion of fuzzy Fréchet distance between curves in a fuzzy metric space.
This is a joint talk with Oleh Berezsky (WUNU).

In 1981, Hatzenbuhler and Mattson involved the axiom of choice to give a number of conditions that are necessary and sufficient for a locally compact non-compact Hausdorff space X to have all non-empty metrizable compact spaces as remainders of Hausdorff compactications. I am going to show several recent results of my joint work with Kyriakos Keremedis and Eleftherios Tachtsis concerning necessary and sufficient conditions for a space X to have all non-empty second countable compact Hausdorff spaces as remainders in the absence of the axiom of choice. I shall present some independence results on compactications. Without involving the axiom of choice, I shall also show direct constructions of Hausdorff compactications with a given second-countable compact remainder for every locally compact non-compact Hausdorff space which satisfies our modications of Hatzenbuhler-Mattson conditions.

Напівгрупа $S$ називається квадратною, якщо існує така її підмножина $A$, що відображення множення $A\times A\to S$ є бієктивним. Спершу ми зауважимо, що квадратні напівгрупи існують (серед прямокутних в'язок), потім побудуємо приклад квадратної нескінченної групи. Врешті решт, за допомогою групових кілець та матричних зображень, ми доведемо, що усі скінченні квадратні групи тривіальні. Аргументи тут належать експертам з (https://mathoverflow.net/q/384292/61536).
Оскільки скінченних нетривіальних квадратних груп не існує, то ми обсудимо також слабше поняття майже квадратної напівгрупи. Такою називається скінченна напівгрупа $S$, що містить підмножину $A$ таку, що відображення множення $A\times A\to S$ сюр'єктивне і $|S|=|A|^2-1$. На даний момент за допомогою GAP знайдено лише 5 майже квадратних груп: $C_3$, $D_8$, $S_4$, $GL(2,3)$ і $S_5$, які містять 3, 8, 24, 48 i 120 елементів, відповідно. На основі аналізу структури цих майже квадратних груп, ми висунемо певні гіпотези та відкриті проблеми щодо майже квадратних груп. Більше деталей (в динаміці) можна знайти на https://mathoverflow.net/q/384488/61536).

Це спільна доповідь з В.Гаврилківим з Прикарпатського національного університету.

We study some variants of the notion of an equi-Borel 1 family. We prove that a pointwise convergent sequence of continuous functions between Polish spaces forms an equi-Borel 1 family. We characterize equi-Borel 1 countable families of characteristic functions. This leads us to an example showing that the exact counterpart of the Arzela-Ascoli theorem for families of real-valued Borel 1 functions is false. Also, the weak counterpart of this theorem dealing with restrictions to nonmeager sets turns out false. On the other hand, we obtain a simple proof of the Arzela-Ascoli type theorem for sequences of equi-Borel 1 functions in the case of pointwise convergence.

Згiдно з теоремою Серпiнського довiльна дійнозначна нарiзно неперервна функцiя двох дійсних змінних однозначно визначається своїми значеннями на всюди щiльнiй підмножинi площини. Ми розглянемо загальнi варiанти цiєї теореми для ширших класiв просторiв i вiдображень, що задовольняють умови слабшi, нiж нарiзна неперервнiсть.

We consider a recent The Vee’s fair soup division problem, provide its partial solution, and pose a related open problem.

We consider some algebras of analytic functions that are generated by a countable family of polynomials on complex Banach spaces. Analytic and algebraic structures of their spectra will be discussed. Also, we will propose some special examples.

We characterize compact subspace of the space S(X x Y) of separately continuous real-valued 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.

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) 68-108 arXiv:2008.11991, doi: 10.15673/tmgc.v13i2.1781
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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??)
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The talk has been mainly prepared for the International Conference dedicated to 70-th anniversary of Professor Oleh Lopushanski, September 16-19, 2019, Ivano-Frankivsk (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 infinite-dimensional subspace of L₁ is isomorphic to either L₁ or l₁. 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.
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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 переставно-інваріантним 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.

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.

We construct a few (consistent) examples of such spaces and resolve a few problems of Tzannes, Banakh, Ravsky and the speaker.
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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.
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Participants of newly created West-Ukrainian Mathematical Seminar will discuss their recent results, open problems and plans for future.
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The seminar talk is devoted to recent activities in studies on fractals and fractal sets, which is divided to two main parts: polytope-based fractals, and multi- and hypercomplex fractal sets. In the first part the presenter will discuss his early studies on determination of polyhedra and higher-dimensional 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 4-polytopes. 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 z²+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 to-date within the topic of the talk together with a definition of the open problems.

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 G⁰ 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.

(за матеріалами кандидатської дисертації)

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.

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.

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26.05.2020
(10:50--11:30)
T. Banakh, Ya Stelmakh
A universal coregular countable second-countable 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 second-countable, countable, semiregular and superconnetced. More details can be found in this preprint.

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26.05.2020
(14:00--14:30)
M.Zaricnhyi
Functors and fuzzy metric spaces
This is a talk at the online conference Geometry in Odesa - 2020.

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26.05.2020
(14:30--14:50)
M.Romanskyi
Конус, надбудова та джойн в асимптотичних категорiях. Лiпшицева та груба еквiвалентностi деяких функторiальних конструкцiй
Це доповідь (14:30--14:50) на онлайн конференції Геометрія в Одесі - 2020.

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28.05.2020
(15:50--16:10)
M.Khylynskyi
Minimal topologies on semigroup of matrix units
Доповідь на онлайн конференції Геометрія в Одесі - 2020.

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This is a talk at the online conference Geometry in Odesa - 2020 on the platform Zoom.

Це доповідь (14:30--14:50) на онлайн конференції Геометрія в Одесі - 2020, яка відбудеться у режимі відеоконференії ZOOM.

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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.
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We shall discuss the structure of betweenness, which is ternary relation satisfying certain natural axioms.

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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.
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Доповідь відбудеться online на базі відеоконференції Uber https://www.uberconference.com/room/mishampopov Початок в 15:00.

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.

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

We are going to discuss some algebraic properties of the Minkowski-Rådström-Hörmander construction of linear space of closed convex bounded subsets as well as some its variations.

We consider recent results about existence of metric spaces with given transfinite dimension.

У програмі заходу відбудуться лекції:

Я. Притула: Історія Львівської топології

Т. Банах: Зигмунт Янішевський (1888-1920)

М. Зарічний: Іван Миколайович Песін (1930-1993)

О. Гутік: Про Львівський топологічний семінар та його засновника - Ігоря Йосиповича Гурана

Початок о 15:05!

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.

The active participants of the seminar will discuss open problems and recent results obtained during Winter Holydays.

We define a locally convex space E to have the Josefson--Nissenzweig property (JNP) if the identity map (E',σ(E',E))→(E',β^*(E',E)) is not sequentially continuous. By the classical Josefson--Nissenzweig theorem, every infinite-dimensional 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 null-sequence of finitely supported sign-measures on X with unit norm. However, for every Tychonoff space X, neither the space B₁(X) of Baire-1 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.

Two non-empty sets A, B of a metric space (X , d) are called parallel if d(a, B) = d(A, B) = d(A, b) for any points a 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.

We shall prove that any isometry between the unit spheres of $C^3$-smooth 2-dimensional 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 2-dimensional 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 2-dimensional Banach spaces. In the talk we shall discuss an attept of a solution of Tingley problem for sufficiently smooth 2-dimensional Banach spaces by methods of the classical theory of curves.

За матеріалами докторської дисертації (продовження)

За матеріалами докторської дисертації

J. Keesling proved that that the closure of a σ-compact subset in the Higson corona of a proper etric space is homeomorphic to its Stone-Cech compactification. The aim of the talk is to demonstrate that this is not the case in the sublinear and subpower coronas.

We discuss some open problems related to Asymptology and Topology.

We describe various geometry categories and functors between them.

Презентація нової прозової збірки М.М. Зарічного net/лінне (в Музей етнографії та художнього промислу).

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.

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 1-Wasserstein distance.

In the talk we shall present proofs of some results announced 5 June 2019 on the seminar "Topological Algebra and its Applications".

This is continuation of the talk, which was started on 27 May 2019.

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 $n$ y-monotone \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 NP-complete, and we provided an algorithm that computes an optimal tangle for $n$ wires and a given list~$L$ of swaps in $O((2|L|/n^2+1)^{n^2/2}\varphi^n n)$ 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.

We shall discuss the problem of characterization of topological spaces X for which the space B₁(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 λ-space (resp. X is Ukrainian).

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 CH-counterexample shows that some known selection theorems for usco multimaps with values in fragmentable compact spaces do not extend to the class of Rosenthal compacta.

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.

Як обчислити площу поверхні сфери та об'єм кулі елементарними засобами без інтегрального та диференціального числення.

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.

A function f:X\to Y between topological spaces is called σ-continuous (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 $f|X_n$ 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 σ-continuous (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.

We construct a metrizable semitopological semilattice $X$ whose partial order $P=\{(x,y)\in X\times X:xy=x\}$ is a non-closed dense subset of $X\times X$. As a by-product 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.

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.

The [first](https://mathoverflow.net/questions/323713/a-discontinuous-construction) was posed by [James Baxter](https://mathoverflow.net/users/132446/james-baxter) and concerns set- and measure- theoretical properties of the unit segment. The [second](https://mathoverflow.net/questions/311325/vertex-coloring-inherited-from-perfect-matchings-motivated-by-quantum-physics) is “a purely graph-theoretic 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".

Доповідь за матеріалами кандидатської дисертації

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.

Active participants of the seminar will discuss some open problems and perspective directions of research.

In 2008-2009 in papers of Zarichnyi, Shukel, Radul a canonical metrization of functors with finite supports was suggested. We introduce a 1-parametric family of such metrizations of functors and analyse the obtained metrizations for some classical functors: the functor of n-th 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.

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.

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.

A zero-dimensional topological space $X$ is called base zero-dimensional 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 zero-dimensional but the Cantor set is not base zero-dimensional. Also we shall try to prove that base zero-dimensional spaces coincide with the spaces that have the Rothberger property. More details can be found here.

За матеріалами кандидатської дисертації. Основні результати:

1. 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 нуль-розширень.
2. Доведено теореми вкладення дiтопологiчних iнверсних напiвгруп в тихонiвськi добутки конусiв та нуль-розширень топологiчних груп. Цi теореми є некомпактними узагальненнями теорем вкладення О. Гринiв, що об'єднують теореми вкладення Лоусона та Петера-Вейля.
3. Доведено теореми метризацiї дiтопологiчних клiфордових iнверсних напiвгруп (субiнварiантними метриками).
4. Доведено теореми про автоматичну неперервнiсть борелівських чи EH-неперервних гомоморфiзмiв мiж топологiчними iнверсними напiвгрупами.

Список публікацій

1. T. Banakh, I. Pastukhova, Topological and ditopological unosemigroups // Mat. Stud. 39:2 (2013) 119–133.
2. T. Banakh, I. Pastukhova, On topological Clifford semigroups embeddable into products of cones over topological groups // Semigroup Forum, 89:2 (2014) 367–382.
3. T. Banakh, I. Pastukhova, Automatic continuity of homomorphisms between topological semigroups // Semigroup Forum, 90:2 (2015) 280-295.
4. I. Pastukhova, On continuity of homomorphisms between topological Clifford semigroups // Carpathian Math. Publ. 6:1 (2014) 123-129.
5. I. Pastukhova, Automatic continuity of homomorphisms between topological inverse semigroups // Topological Algebra and its Applications, 6:1 (2018) 60-66.

За матеріалами кандидатської дисертації, виконаної під керівництвом проф. О.В.Маслюченка (Чернівецький національний університет імені Федьковича).

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.

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.

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.

We investigate softness of idempotent barycenter map.

In 1999 Raz demonstrated a partial function that had an efficient quantum two-way communication protocol but no efficient classical two-way protocol and asked, whether there existed a function with an efficient quantum one-way protocol, but still no efficient classical two-way protocol. In 2010 Klartag and Regev demonstrated such a function and asked, whether there existed a function with an efficient quantum simultaneous-messages protocol, but still no efficient classical two-way 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 poly-logarithmic size, and whose classical two-way complexity is lower bounded by a polynomial.

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

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

We shall discuss a recent progress concerning topologizations of graph inverse semigroups.

We prove that a real-valued 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.

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.

We shall discuss a recent brakethrough result of Aubrey de Grey in direction of solution of the famous Hadwiger-Nelson 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 50-ies 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 Hadwiger-Nelson problem depends on Axioms of Set Theory.

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 non-empty θ-open regular subspace. Also we observe that each locally regular space is Bokalo regular. Using the non-regular space called Gutik's hedgehog, we construct a locally regular topological space without points of regularity.

We give an answer to a question of Zarichnyi about openness of the idempotent barycenter map.

We shall discuss topological properties, preserved by scatteredly continuous homeomorphisms.

We characterize 0-bisimple inverse semigroups which semilattice of idempotents is isomorphic to the λ-ary tree whith adjoint zero.

We shall prove that an Abelian topological group is compact if and only if it is complete in each weaker group topology.

We shall discuss the recent progress in the problem of detecting topological groups which are (absolutely or injectively) H-closed in some classes of topological semigroups. More details can be found in this preprint.

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

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|>2×10¹⁰ has difference weight Δ[G]²<4|G|/3.

We describe the Green relations and the congruence lattice of the semigroups of partial injective transformations of some partially ordered sets.

We discuss principal results of the Ph.D. Thesis (written under supervision of O.Gutik), which are related to:

• topologization of the α-bicyclic monoid,
• embeddings of polycyclic monoids into compact-like topological semigroups, and
• H-closedness of topological semilattices.

The active participants of the seminar will discuss some open problems and new results obtained during Winter Holydays.

We shall discuss some new results and open problems related to the weak continuity of functions.

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).

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).

We discuss Separations Axioms in quasitopological groups and construct an example of a regular quasi-topological groups, which is not functionally Hausdorff.

A topological space X is called strongly σ-metrizable 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 σ-metrizable. Answering a question of V.K.Maslyuchenko and O.I.Filipchuk, we prove that a topological space is strongly σ-metrizable if and only if it is super σ-metrizable.

Two questions from [V.Bykov, On Baire class one functions on a product space, Topol. Appl. 199 (2016) 55-62] 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.

We discuss interplay between (generically) Haar-meager and (generically) Haar-null sets in Polish groups.

The active participants of the seminar will discuss some open problems and possible ways of their solutions.

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.

A notion of Ramsey trees will be introduced. They will offer a useful framework for various Ramsey-type problems including the Ramsey multiplicity problem and the Erdős clique-independent set problem. The connection to Conlon's bound for Ramsey multiplicity constant will be discussed as well.

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 long-standing 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.

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 σ-ideal E generated by closed Haar null subsets of G. These two theorems motivate the problem of detecting σ-ideals on Polish groups G containing every non-empty analytic subgroup of G. Generalizing the results of Banach and Laczkovich we shall prove that any non-open analytic subgroup H of a Polish group G belongs to every Fσ-supported σ-ideal with the closed n-Steinhaus property for some n. An ideal I on a topological group G is defined to have the closed n-Steinhaus property if for any I-positive closed subsets A1,...,An of G the product A1··· An has non-empty interior in G. For every n we shall construct an σ-ideal on R which has the closed (n+1)-Steinhaus property but fails to have the closed n-Steinhaus property. Also we shall discuss possible extensions of the Laczkovich Theorem to non-locally compact 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 σ-ideal E generated by closed Haar null subsets of G. These two theorems motivate the problem of detecting σ-ideals on Polish groups G containing every non-empty analytic subgroup of G. Generalizing the results of Banach and Laczkovich we shall prove that any non-open analytic subgroup H of a Polish group G belongs to every F_σ-supported σ-ideal with the closed n-Steinhaus property for some n. An ideal I on a topological group G is defined to have the closed n-Steinhaus property if for any I-positive closed subsets A₁,...,A_n of G the product A₁··· A_n has non-empty interior in G. For every n we shall construct an σ-ideal on R which has the closed (n+1)-Steinhaus property but fails to have the closed n-Steinhaus property. Also we shall discuss possible extensions of the Laczkovich Theorem to non-locally compact Polish groups.

An old problem of Linial and Wilf asks for f(n,m,l), the maximum number of l-colorings that a graph with n vertices and m edges can have. We solve this problem for every fixed l when C≤ m≤ cn² for some C,c>0 depending on l only. Moreover, for l=3, we establish the structure of optimal graphs for all large m≤ n²/4 confirming (in a stronger form) a conjecture of Lazebnik from 1989. This is joint work with Po-Shen Loh and Benny Sudakov.

We investigate cohomological properties of subspaces of the n-th symmetric power X^[n] of k-dimensional space X. The obtained results imply that the n-dimensional sphere Sⁿ does not embed into the n-th symmetric power X^[n] of a 1-dimensional compact space X. This resolves a problem of Illanes and Nadler.

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 zero-dimensional.
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 non-expanding maps.

We shall prove that for every infinite cardinal λ the inverse semigroup I_λⁿ 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 H-closed topologies on the inverse semigroup I_λⁿ.

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 non-empty subspace A of X the restriction f|A has a continuity point.

The functor E of non-expanding 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 order-preserving 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.

Principal results of the Ph.D. Thesis of R.Voytsitskyy will be presented.

The principal results of the Doctor Dissertation of V.Mykhaylyuk will be presented.

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.

Two theorems are proved:
Theorem 1. Each countable group G of asymptotic dimension asdim(G)=0 is coarsely equivalent to the anti-Cantor set 2^<ω.
Theorem 2. A countable Abelian group G with asdim(G)=n is coarsely equivalent to:
- Zⁿ iff G is finitely generated or n=∞;
- Zⁿ×2^<ω iff G is infinitely generated.
This is a joint work with Jose Higes and Ihor Zarichnyi.

We prove that two homogeneous ultra-metric spaces are coarsely equivalent if and only if they have the same sharp entropies. This classification implies that each homogeneous proper ultra-metric space is coarsely equivalent to the anti-Cantor 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.

We prove that two homogeneous ultra-metric spaces are coarsely equivalent if and only if they have the same sharp entropies. This classification implies that each homogeneous proper ultra-metric space is coarsely equivalent to the anti-Cantor 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.

Some open problems have been posed.

Some open problems have been posed.

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 σ-compact subset of a Polish group cannot take an intermediate value between ℵ₁ and the contnuum c. On the other hand, each non-discrete Polish group contains a nowhere dense Haar null subset of arbitrary packing index κ≤c.

We extend the binary operation from a group G to a right-topological semigroup operation on the superextension λ(G) of G and study the properties of the obtained supercompact right-topological 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(Z₂) of the compact group Z₂ of integer 2-adic numbers.

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.

We prove that for each Polish space X, the space C(X) of continuous real-valued functions on X satisfies (a strong version of) the Pytkeev property, if endowed with the compact-open 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.

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.

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 α₁; (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 α₁; (iii) the function space C_p(X,{0,1}) has the property α_1.5 (which is formally weaker than α₁); (iv) for each Borel map f : X→ ω^ω the image f(X) lies in a σ-compact subset of ω^ω; (v) the space X satisfies the Selection Principle U_fin(B,Γ).
This is a joint work with Boaz Tsaban.

We extend the binary operation from a group G to a right-topological semigroup operation on the superextension λ(G) of G and study the properties of the obtained supercompact right-topological 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.

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 Rⁿ 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 Rⁿ 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 Rⁿ:
Let K be a locally compact (possibly nonclosed) subset of Rⁿ. Then K is C^∞-homogeneous if and only if K is a C^∞-submanifold of Rⁿ, i.e. (i) if dim K = 0, then K is at most countable subset of isolated points in Rⁿ; (ii) if 0 < dim K < n, then K is at most countable collection of C^∞-submanifolds with pairwise disjoint neighborhoods in Rⁿ; and (iii) if dim K = n, then K is an open subset of Rⁿ.
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 Hilbert-Smith Conjecture, related work in chaos theory, etc.

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.

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.

We consider some properties of multiplication map of the monad of oder-preserving functionals and prove that it is soft for any compactum.

We survey existing results on the topological structure of the homeomorphism groups of compact and non-compact surfaces.

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.

Some open problems have been posed.

We shall present a variety of interesting applications of the classical example of a 1-dimensional connected non-Peano 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 2-dimensional noncontractible simply connected cell-like continua. We shall also present the solution of the Bestvina-Edwards problem: Does there exist a noncontractible cell-like compactum whose suspension is contractible? In conclusion we shall state some interesting open problems.

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.

A convex subset C of a linear topological space is called compactly convex if there is a continuous compact-valued 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 3-dimensional space R³ contains a convex subset which is not compactly convex.
This is a joint work with M.Mitrofanov and O.Ravsky.

A topological space X is called continuously homogeneous if there is a homeomorphism H of X³ 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^-1z). An example of a continuously homogeneous space homeomorphic to no topological group is the 7-dimensional sphere S⁷. On the other hand, the Hilbert cube fails to be continuously homogeneous.
This is joint work with Z.Kosztolowicz (Kilece).

A topological group G is called group-sequential if it carries the strongest group topology inducing the original topology on each convergent sequence. We give examples of such topological groups, characterize group-sequential free topological groups and pose some open problems.

In the report we shall discuss about some properties of H-closed topological semigroups and semilattices.

The notion of capacity was introduced into mathematics by G.Choquet. The family of all upper-semicontinuous 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.

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.

We discuss some open problems related to the transfinite separation dimension trt introduced by Arenas, Chatyrko and Puertas.

We investigate topological properties of Hartman-Mycielski functor on non -metrizable compacta.

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.

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 $f|A$ 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.

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.

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.

Kunzi and Schapiro (1997) found a continuous linear extension operator for all real-valued 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).

A compact space X is defined an absolute Z-space if for any embedding of X into the Hilbert cube Q the set Xx{x₀} is a Z-set in Qx[-1,1]. We discuss the relation of absolute Z-spaces to other dimension clases of compacta.

It is shown that the transfinite extension of the asymptotic counterpart of the large inductive dimension is not trivial.

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.

It is shown that the transfinite extension of the asymptotic counterpart of the large inductive dimension is not trivial.

We prove that
(1) Every compact metrizable space is weakly homotopy equivalent to a cell-like compactum and
(2) There exists a noncontractible cell-like compactum whose suspension is contractible (this gives an affirmative answer to the Bestvina-Edwards problem).

We shall discuss cardinal characteristics of the reals.

We shall discuss homologic algebra and generalization of Borsuk-Ulam theorem.

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 zero-dimensional spaces are considered. Also we consider the question of coarse embeddability of this spaces and their hypersymmetric powers.

We discuss P₁-metrics.

Answearing a question of Dikranjan and Protasov we prove that each infinite group containt a weakly P-small subset A which is not P-small. 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.

Some open problems are discussed.

We discuss the division and k-th Root Theorems for the Hilbert cube.

We present the answer to one of Maslyuchenko's questions.

We discuss operators extending partial ultrametrics.

We discuss subproper maps and k^*-spaces.

Answearing a question of Dikranjan and Protasov we prove that each infinite Abelian group containt a weakly P-small subset A which is not P-small. 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.

For the functors acting in the category of compact Hausdorff spaces, we introduce the so-called open multicommutativity property, which generalizes both bicommutativity and openness, and prove that this property is satisfied by the functor of probability measures.

We discuss Scheepers' diagram and a question of Zdomskyy.

We provide an answer to a question of Plicko about the cardinality of Hamel basis.