
Date:

Speaker:

Title (click on the title to see the Abstract):



27.05.13

T.Banakh


Metrization of the functor of idempotent measures with applications to idempotent fractals

Answering a question of M.Zarichnyi we find a metrization of the functor of idempotent measures.
For this we construct an embedding of the space I(X) of idempotent measures on a compact Hausdorff space X
to the hyperspace exp(I× X) and endow I(X)
with the metric induced from the Hausdorff metric on I×X.
Using this metrization, we prove that any contracting maps f_{1},...,f_{n}: X→X on a metric compact space X and any nonnegative real
numbers p_{1},...,p_{n} with max{p_{1},...,p_{n}} = 1 the map
Φ = max_{1≤i≤n}p_{i}I(f_{i}): I(X)→I(X) has a unique fixed point, called the idempotent fractal,
which can be found as the limit of the sequence of iterations Φ^{n}(μ_{0}), n=1,2,3,...







20.05.13

O.Gutik


On Hclosed pospaces

We give a criterion for Hclosedness of a point separated pospace with compact maximal antichains.







13.05.13

O.Ravsky


Evaluating the cardinality of Kuratowski monoid of an ntopological space

We prove that the sequence K(n)=∑^{n}_{i,j=0}(C^{i}_{i+j})^{2} equal to the cardinality of the free Kuratowski of type (n,n)
has asymptotic growth (1+o(1))16^{n+1}/9πn as n→∞.







29.04.13

T.Banakh


Kuratowski operations in ntopological spaces

Given a set X endowed with n comparable topologies we calculate the number of sets that can be obtained from a given set A of X
by repeated application of the operations of closure and interior in these topologies.







22.04.13 15.04.13

T.Banakh A.Ravsky


Boundedness and separability in topological groups, II

We shall introduce several (more precisely, infinitely many) cardinal characteristics expressing boundedness and separability properties
of topological groups and shall present several distinguishing examples. Also we shall calculate these cardinal characteristics for
the symmetric group S(X) and its dense normal subgroup S_{ω}(X) consisting of finitely supported permutations of an infinite set X.







08.04.13

O.Chervak


Limits of bounded degree graphs and hyperfiniteness.

This is an expository talk based on an article of G.Elek. We introduce notions of BenjaminiSchramm limit and graphing, give examples.
We define a notion of hyperfiniteness for graphings and graph sequences and prove that graph sequence is hyperfinite iff its limit graphing is hyperfinite.
We mention the Kaimanovich theorem and prove that weakly equivalent graphings are hyperfinite simultaneously.
We briefly discuss different notions of equivalent graphings (weakly, localglobal, strong) and mention that they coincide for hyperfinite ones.







1.04.13

T.Banakh


Boundedness and separability in topological groups

A topological group G is called
 ωbounded if for each neighborhood U of the unit 1_{G} in G there is a countable set C in G such that CU=G=UC;
 separable if there is a countable set C in G such that CU=G=UC for each neighborhood U of the unit 1_{G} in G;
 ωduobounded if for each neighborhood U of the unit 1_{G} in G there is a countable set C in G such that G=CUC;
 duoseparable if there exists a countable set C in G such that G=CUC for each neighborhood U of the unit 1_{G} in G.
It is clear that each separable topological group is ωbounded and duoseparable. On the other hand, a topological group is
ωdoubounded if it is duoseparable or ωbounded.
We show that for any uncountable set X the group FSym(X) of finitely supported permutations of X endowed with the topology of pointwise
convergence is ωduobounded but not ωbounded and not duoseparable. Also we prove that each topological group embeds into a duoseparable
(and hence ωduobounded) topological group. This answers a question of I.Guran.







25.03.13

T.Banakh


On invariant partitions of groups

We prove that for each partition G=A_{1}∪...∪A_{n} of an infinite group G
there is a cell A_{i} of the partition and a subset F⊂G of cardinality F≤n such that
G=F (A_{i}A_{i}^{1})^{G} where A^{G}=∪_{x∈G} xAx^{1}
for a subset A⊂G. This implies that for any partition G=A_{1}∪...∪A_{n} of G
into innerinvariant sets A_{i}=A_{i}^{G} there is a cell A_{i}
of the partition such that G=FA_{i}A_{i}^{1} for some finite subset F⊂G of cardinality F≤n.
This is a joint work with I.V.Protasov and S.Slobodianiuk.







18.03.13

M.Zarichnyi


Hyperspaces of maxplus convex subsets in the function spaces

The notion of maxplus convex set can be naturally defined for the spaces of continuous functions.
The main result is the following one: the hyperspace of compact maxplus convex sets in the Banach lattice C(X) of continuous functions
on an infinite compact space X is homeomorphic to the separable Hilbert space.
This is a joint work with Dusan Repovs.







11.03.13

T.Radul


Absolute retracts and equiconnected monads

We introduce a general notion of equiconnected functor and show that each such functor has similar topological properties
as probability measure functor and idempotent measure functor, which have some natural equiconnectedness structures.







04.03.13

T.Kudryk


Nonstandard Methods in Topology and Number Theory

Classification of Mahler of complex numbers and local distances (by P.Philippon).
Diophantian approximation groups and nonstandard methods (by T.Gendron).







25.02.13

T.Banakh


On Slobodianiuk's approach to solution of a Protasov's problem
about partitions of Gspaces

We shall discuss recent progress in resolving a Protasovs' problem on
partitions of Gspaces by Slobodianiuk's method.







18.02.13

All


Divertissement

Participants of the seminar present some problems they have recently investigated.







Jan.2013

All


W i n t e r H o l y d a y s







24.12.12

O.Pikhurko


Pseudorandom permutations

A sequence of permutations is called pseudorandom if it contains each permutation of length k with frequency 1/k!+o(1).
Applying the analytic language of limits of permutations we shall show that it suffices to check the case k = 4.







17.12.12

O.Chervak


Small subset property of metric spaces.

In the spirit of MengerUrysohn theorem for meager subsets in manifolds we define a notion of a small subset property (ssp) for metric spaces.
Spaces with ssp are coarse analogues of topological manifolds. We prove that ndimensional hyperbolic an euclidean spaces have ssp. We discuss properties of groups with ssp.







10.12.12

M.Zarichnyi


Maxmin measures on ultrametric spaces

We construct a natural ultrametric on the set of maxmin measures on an ultrametric space
and establish some functorial properties of the corresponding constructions.







3.12.12

M.Gavrilovich


The language of category theoretic diagram chasing for set theory and topology

In the first part of the talk I shall describe the formalism
of a labelled category able to describe some aspects of basic notions
in set theory (equicardinality, subset, cofinality) and demonstrate
how some diagram chasing constructions correspond to set theory
arguments.
This formalism satisfies the axioms of a Quillen model category, a
formalism for homotopy theory; and thus indicates a connection between
set theory and homotopy theory.







24.11.12

T. Banakh


Subamenable groups and their partitions

A group G is called subamenable if it admits a leftinvariant monotone subadditive function μ
defined on the family of all subsets of G such that μ≠1 and for each subset A of G of measure μ(A)<1 and each ε>0 there is a left large subset L of X\A of measure μ(L)<ε.
We prove that the class of subamenable groups includes (i) all groups that has subamenable quotient group, (ii) all infinite groups admitting a totally bounded group topology, (iii) all groups containing an infinite amenable subgroup.
We prove that for every natural k a countable subamenable group G admits a partition G=A∪ B such that for every kelement subset K of G the sets KA and KB are not left thick. This resolves one problem of I.Protasov.
Reference:
T.Banakh, I.V.Protasov, S.Slobodianiuk, Subamenable groups and their partitions, preprint.







17.11.12

T. Banakh


Some examples of compact topological inverse semigroups

We present some canonical examples of compact topological inverse semigroups and discuss their role
in the structure theory of compact topological inverse semigroups.







12.11.12
05.11.12

T. Banakh


The Solecki submeasure on a group

The Solecki submeasure σ on a group G assigns to each subset A of G the real number σ(A)=inf_{F}max_{x,y}F∩ xAy/F
where the infimum is taken over all nonempty finite subsets of G.
We shall discuss properties of the Solecki submeasure and its interplay with the Haar measure λ on a compact topological group G.
In particular we show that σ(K)=λ(K) for each closed subset K of G.
1. T. Banakh, The Solecki submeasure on a group, preprint.







29.10.12

T. Radul


Binary convexities and monads

The notion of Lmonads was introduced in [1].
They have a functional representation, which preserve the monad structure.
In [2] it was introduced a convexity structure on each Falgebra for any Lmonad F in the category of compact Hausdorff spaces and continuous maps.
This general construction of convexities includes known convexities for probability measures, superextensions, inclusion hyperspaces, etc.
It was proved in [2] that each binary monad (i.e. a monad generating a binary convexity) has good topological properties. In particular, its functorial part transforms
openly generated compacta into Dugundji compacta. We will prove that the latter property characterizes binary monads.
1. T. Radul, On functional representations of Lawson monads, Applied Categorical Structures 9 (2001), 457–463.
2. T. Radul, Convexities generated by monads, Applied Categorical Structures, 19 (2011), 729–739.











22.10.12

O.Chervak


An introduction to forcing

We give a short introduction to forcing technics, briefly explain general ways of obtaining various independency results.
Also we build a model where Martin axiom holds and the Higson corona of the Cantor macro cube is homeomorphic to the StoneCech compactification of integers.











15.10.12

T. Radul


The HartmanMycielski functor cannot be completed to a monad

We prove that the HartmanMycielski functor cannot be completed to a (weak) monad in the category of compacta.







8.10.12

O. Chervak


Properties of Higson corona under different settheoretic assumptions

Higson corona is a natural object in coarse geometry, being a coarse analogue of StoneCech compactification. As a consequence to Parovicenko theorems, I. Protasov proved that under CH Higson coronas of all asymptotically zerodimensional metric spaces are homeomorphic and wondered if this result remains true in ZFC. We answer negatively on this question and prove that under some settheoretic assumptions Higson corona of antiCantor set is not homeomorphic to Higson corona of divergent sequence











1.10.12

R.Cauty (Paris)


Singular chains and fixed points of maps with a compact iterate

We shall explain some generic ideas underlying the use of singular chains for the proof of the following Fixed Point Theorem answering an old problem of Juliusz Shauder:
Theorem. A continuous map f:C→C defined on a convex subset C of a linear metric space has a fixed point if for some iterate f^{n} of f the image f^{n}(C) is contained is a compact subset of C.







24.09.12

R.Cauty (Paris)


A solution of West's Problem ANR 13

We shall prove that a metrizable σcompact space X is an ANR if and only if X is an ANE(Comp). The latter means that
each continuous map f:A→X defined on a closed subspace A
of a compact metrizable space B admits a continuous extension F:O(A)→X defined on a neighborhood O(A) of A in B. This resolves a
longstanding Problem ANR 13 from the West's list of open problems included to the book "Open Problems in Topology" (1990).







10.09.12

All


Divertissement

The active participants of the topological seminar
will discuss new (summer) results and pose some open problems.





