Date 
Speaker 
Title 
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) 
Íclosed 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.
