Date 
Speaker 
Title 
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 
M. 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 
M. 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.

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

21May2018 
S.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.
