
Date:

Speaker:

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







14.05.12

O.Nykyforchyn


Dualities for the lattices of Scott continuous functions

Lawson duality for domains and its restriction to continuous
semilattices is well known. We use it to construct analogues of
dual pairs and conjugate operators for continuous Lsemimodules
of Scott continuous functions from domains to a completely
distributive quantale. Approximations of Scott continuous functions
are also discussed.







07.05.12

M.Mytrofanov O.Ravsky


Separating polynomials and the approximation of continuous functions in linear topological spaces

Separating polynomials play an important role in the theory of approximation of continuous functions on Banach spaces. We shall consider general properties of separating polynomials (with examples) and its relation to the weak polynomial topology.
Also we shall consider draft generalizations of these notions and results for normed, Frechet and locally convex linear topological spaces and the problem of detecting spaces on which each continuous function with some properties is appoximated by analityc functions.
We shall try to use some topological tools and approachs such as: Rfactorizable topological groups and the dependence of functions defined on products on countably many coordinates, uniform analogue of the paracompactness and a locally finite analogue of the Souslin number.







30.04.12

O. Gutik S.Bardyla


An example of an Hclosed topological semilattice with nonHclosed quotient

We construct an example of an Hclosed topological semilattice admitting a surjective homomorphism on a discrete semilattice, which is not Hclosed.







23.04.12

M.Pankov Olsztyn


Jonson graphs and Grassman graphs

The Jonson graph J(k,n) is a graph whose vertices are kelement subsets of an nelement set,
and two vetrices A,B are connected by an edge if the intersection A∩B has cardinality k1.
A Grassman graph G(k,V) over a vector space V is the graph whose vertices are linear kdimensional subspaces of V and two vertices A,B are linked by an edge if their intersection A∩ B is a linear subspae of dimension k1. We discuss the problem of isometric embedding a Jonson graph J(k,n) into a Grassman graph G(k,V).







09.04.12

O. Gutik


On embedding of topological semigroup into a countably compact topological semigroup

We give a construction of embedding of topological semigroup into a countably compact topological semigroup.







02.04.12

N.Kolos


Scatteredly continuous maps and spaces of scatteredly continuous maps

We consider the main results of Ph.D. dissertation, which consists of 4 chapters:
1) operations on some classes of discontinuous maps;
2) Rscattered spaces;
3) normality of the space SC_{p}(X);
4) σconvex subspaces of the space SC_{p}(X).







26.03.12

M.Popov


Narrow Operators

Вузькі оператори були введені і систематично досліджені
автором у сумісній роботі з А.М.Плічком у 1990 р., як узагальнення
компактних операторів, заданих на просторах функцій. Окремі результати
про вузькі оператори були відомі ще до цього. Від того часу за 20
років теорія вузьких операторів зазнала істотного розвитку.
Доповідь присвячена короткому екскурсу в цю теорію.







12.03.12

T.Banakh


Hyperspaces of convex compacta usually are not barycentrically open

We prove that for any convex subset X of dimension dim(X)≥3 and
the hyperspace cc(X) of compact convex subsets of X the Minkowski mean μ: cc(X) × cc(X) → cc(X), μ(A,B)→(A+B)/2, is not an open map.
This implies that for any compact space X of cardinality X≥4 the space ccP(X) is not barycentrically open, which partially answers a question of T.Radul.







5.03.12

T.Banakh


On scattered spaces homeomorphic to fractals

We prove that the scattered height of a scattered compact metric space X is a nonlimit ordinal if X=Φ(X)
for some contracting finitevalued map
Φ:X→ X.







27.02.12

O.Hryniv


Embeddings of free universal algebras

Generalizing a theorem of Uspenski (on embeddings of free topological groups),
we prove that for any complete quasivariety K of topological algebras
of a given continuous signature and for each closed subspace X of a metrizable space Y
the free universal algebra F_{K}(X) of X can be identified
with a closed subalgebra of the free universal
algebra F_{K}(Y) of Y. For more details, see
here.







20.02.12

T.Banakh


Algebraically determined topologies on permutation groups

We shall answer several questions of Dikran Dikranjan about algebraically determined topologies
on the groups S(X) (and S_{ω}(X) ) of (finitely supported) bijections of a set X.
In particular, confirming a Dikranjan's conjecture, we prove that the topology T_{p}
of pointwise convergence on each subgroup G of S(X) that contains S_{ω}(X)
is the coarsest Hausdorff group topology on G
(more generally, the coarsest T_{1}topology which turns G into a [semi]topological group),
and T_{p} coincides with the Zariski and Markov topologies on G.
Answering another question of Dikranjan, we prove that the centralizer topology on the symmetric group G=S(X)
is discrete if and only if X≤c.
On the other hand, we prove that for each subgroup G of S(X) that contains S_{ω}(X),
the centralizer topology coincides with the topology T_{p} if and only of G= S_{ω}(X).
Also we prove that the group S_{ω}(X) is σdiscrete in each Hausdorff shiftinvariant topology.







13.02.12

All


Divertissement

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







January

All


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







26.12.11

O.Savchenko


Metric and uniform properties of functors in topologic categories.

We consider functors in category of fuzzy (ultra)metric spaces and nonexpanding mappings.
We describe category of stationary fuzzy metric spaces for tnotm $\frac{ab}{max(a,b,\alpha)}$. We also consider spaces with fuzzy norm.







19.12.11

K.Koporkh


Spaces of factor objects in category of topological spaces

We consider various topologies on the space of quotient objects of a topological space.







12.12.11

O.V.Maslyuchenko


Construction of ωprimitives

We describe some methods of constructing functions with given oscilation.







05.12.11

T.Banakh


Fractals generated by multivalued functions on topological spaces

Given a multivalued function f on a topological space we define its fractal Fract(f) as the closure of the orbit of the set Fix(f) of fixed points of f. This definition agrees with the classical definition of a fractal if f is generated by a system of contracting maps on a complete metric space. We shall develop the theory of fractals of contracting and expanding multifunctions (called micro and macrofractals) and shall discuss algorithms of drawing such fractals.







28.11.11

R.Tymkiv
(Montreal)


Circulation, III. Pliroi in topology.

In topology we also have asymmetry and circulation. We organize it in a thing called plyros. We shall discuss pliroi and their typical applications in topology.







21.11.11

T.Banakh


A 1dimensional Peano continuum which is not homeomorphic to a deterministic fractal.

Answering an old (1985) question of M.Hata, we construct an example of a
1dimensional Peano continuum which is not homeomorphic to a deterministic fractal.
A metric compact space X is called a deterministic fractal if X=f_{1}(X)∪...∪f_{n}(X)
for some contracting selfmaps f_{1},...,f_{n}:X→X. Observe that a metric space X is a deterministic
fractal if it is an image of the interval [0,1] under a Lipschitz map f:[0,1]→X.







14.11.11

R.Tymkiv
(Montreal)


Circulation: new probability type category for applications, II

Many processes are not reversible in time. To study this phenomenon
we need an instrument which is similar to probability but is more adjusted to the study of assymetry.
It is called circullation. We shall discuss circulatory analysis and its typical applications in economics.







31.10.11 07.11.11

O.Chervak


Symmetrical sets in 2colorings of Euclidean spaces

Generalizing a classical result of I.V. Protasov, we prove that for every 2coloring of Euclidean space R^{n} and for every affinely independent subset A with A=n+1 there exists a monochromatic subset S of R^{n} of asymptotic dimension ≥ n1, symmetric with respect to some point of A. In the proof we shall use the technique of ultrahomology, a new method in Algebraic Asymptology.







24.10.11

R.Tymkiv
(Montreal)


Circulation: new probability type category for applications, I

Many processes are not reversible in time. To study this phenomenon
we need an instrument which is similar to probability but is more adjusted to the study of assymetry.
It is called circullation. We shall discuss circulatory analysis and its typical applications in economics.







17.10.11

I.Chuchman


On a semigroup of closed connected partial
homeomorphisms of the unit interval with a fixed point

We study the semigroup IC(I,[a])
(IO(I,[a])) of closed (open) connected partial
homeomorphisms of the unit interval I with a fixed point a from I.
We describe left and right ideals of IC(I,[0]) and the
Green's relations on IC(I,[0]). We show that the
semigroup IC(I,[0]) is bisimple and every nontrivial
congruence on IC(I,[0]) is a group congruence. Also we
prove that the semigroup IC(I,[0]) is isomorphic to the
semigroup IO(I,[0]) and describe the structure of a
semigroup II(I,[0]=IC(I,[0]) \sqcup IO(I,[0]). As a corollary we get structures of
semigroups IC(I,[a]) and IO(I,[a]) for an
interior point a.







10.10.11

I.Hetman


A "hidden" characterization of polyhedral convex sets

We prove that a convex subset C of a complete linear metric space X is polyhedral in its linear hull
if and only if C hides no infinite subset A of X\A in the sense that [a,b] meets C for any distinct points a,b of A.
This a joint work with T.Banakh.







6.10.11

R.Cauty


Simple homological proofs of BorsukUlam type theorems

We explain a homological method of proof of some theorems of combinatorial topology,
including BorsukUlam Theorem and topological Tverberg Theorem.







3.10.11

R.Cauty


Approximation of multivalued maps by chain morphisms

We introduce a general functorial approach to the theory of exteding and
averaging operators which unify linear and nonlinear theories.







19.09.11 12.09.11

T.Radul


Extenders and averaging operators generated by functors

We define a notion of approximation of a multivalued map by chain morphisms and use
it to prove some Fixed Point Theorems.







05.09.11

All


Divertissement

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





