
Date:

Speaker:

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



30.06.15

A.Katok (USA)

On some connections between topology and dynamics
We will discuss some connections between topology of a differentiable manifold and
properties of individual diffeomorphisms of that manifold, and, more generally,
group actions by diffeomorphisms. Classical results of that sort deal with periodic orbits,
Lefschetz fixed point formula and Nielsen theory being prime examples.
Development of hyperbolic dynamics during the last half century produced results
guaranteeing existence of complicated invariant sets with highly nontrivial (``chaotic’’)
dynamics. During the last ten years new striking phenomena were found related with
smooth actions of higher rank abelian groups.







02.06.15

O.Yampolski (Харків)

Geometry of submanifolds in fiber bundles
The tangent bundle π:TM→M of a Riemannian manifold (M,g) can be endowed with a natural
Riemannian metric, called the Sasaki metric. The main stream of the talk
is devoted to metric properties of the (unit) tangent bundle and geometry
of submanifolds in the (unit) tangent bundle with special emphasis on
totally geodesic properties of unit sections of the unit tangent bundle with the Sasaki metric.
The (maximal number of) topics include:
 Totally geodesic foliations on unit tangent bundle;
 Sectional curvature of unit tangent bundle over spaces of constant curvature;
 Geodesics on (unit) tangent bundle over space forms;
 Bergertype unit sphere bundle over Kahlerian manifold and its geodesics;
 Lift of submanifolds and foliations to the tangent bundle;
 Totally geodesic and minimal unit sections of unit tangent bundle;
 Totally geodesic property of the Hopf unit vector field on spheres;
 Stability of totally geodesic unit vector fields;
 The Sasakitype metric on vector bundle with metric connection over Riemannian manifolds:
definition, properties, geometry of sections, examples.
No strong backgrount is supposed. The number of topics to be covered is flexible.







25.05.15

L.Karchevska

Metrization of functors
We shall discuss the problem of metrization of functors with finite supports and functional functors.







18.05.15

О.Мироник (Чернівці)

Вичерпні, напіввичерпні простори та нарізно неперервні відображення
Буде дано огляд результатів досліджень множини C(f) точок сукупної
неперервності нарізно неперерервних відображень
f : X_{1}x ... x X_{n}→ Z зі значеннями у неметризовних топологічних
просторах Z. Зокрема, в ролі простору Z виступають
напіввичерпні, але не вичерпні простори C_{p}[0,1] і площина Бінґа
B, простір C_{k}(T) неперервних функцій
z:T→R з топологією компактної збіжності, який буде
вичерпним, коли T  польський простір, площина Сідра M
 вичерпний і неметризовний простір і пряма
Зорґенфрея L, яка не є напіввичерпним простором.







27.04.15

T.Banakh

Ascoli spaces and their properties
A topological space is called Ascoli if each compact subset in the function space C_{k}(X) is evenly continuous.
We shall discuss topological properties of Ascoli spaces, their relation to kspaces, and also detect Ascoli spaces among
topological groups and function spaces.







06.04.15

O.Ravsky

Characteristic subsets of locally compact abelian topological groups
A subset A of a locally compact abelian topological group G we shall call characteristic,
if there is no sequence {φ_{k}} of nontrivial characters of the group G
such that {φ_{k}(a)} converges to the unit for each element a of A.
Investigating the problem which subsets A are characteristic,
we obtain some beginning results and pose some problems and directions of investigation.







30.03.15

T.Banakh

Topological and Banach fractals
A topological space X is called a topological fractal if X=∪_{f∈F}f(X) for a finite family
F consisting of continuous selfmaps of X such that for every open cover of X there is a number n such that for every functions
f_{1},...,f_{n} in F the set f_{1}...f_{n}(X) is contained in some set of the cover.
If all maps f∈F have Lipschitz constant <1 with respect to some continuous metric on X, then X is called a Banach fractal.
It can be shown that each topological fractal is metrizable and compact. We shall discuss the open problem asking if every
Peano continuum X is a topological fractal. We shall show that the answer is affirmative if X contains an open subspace homeomorphic to R^{n} for some natural n.
Also we shall prove that a Peano continuum X containing an open subset homeomorphic to R^{n} is a
Banach fractal if and only if X has finite Holder dimension, which means that X is the image of the unit interval [0,1] under a continuous map
f:[0,1]→X which is Holder which respect to some metric generating the topology of X.







23.03.15

N.Pyrch

On generalized retracts related to topological groups
The subspace Y of topological space X is called Gretract if any continuous mapping from
Y to topological group H admits continuous extension onto X.
In the research we apply the theory of free topological groups for investigating Gretracts.
We obtain some new results concerned to the isomorphisms of free topological groups and retral spaces.







16.03.15

M.Zarichnyi 
Correspondences of measures with restricted marginals
We discuss the question of continuity of the correspondence that assigns to every couple of probability measures on the factors
the set of measures on the product with these marginals.







02.03.15

M.Zarichnyi,
O.Gutik,
A.Ravsky,
I.Guran

Igor Guran: scientific achievements and perspectives
This seminar is devoted to the 60th anniversary of Igor Guran, the founder of Lviv topological seminar.
M.Zarichnyi will discuss principal milestones of scientific biography of Igor Guran and survey his main
scientific achievements.
His former Ph.D. students Oleg Gutik and Alex Ravsky will analyse development of ideas
of I.Guran in Topological Algebra, namely in the theory of topological inverse semigroups
and the theory of paratopological groups.
Finally, I.Guran will discuss some open problems related to his latest scientific interests.







23.02.15

T.Banakh

An example of a nonseparable foreseparable space
A topological space X is called foreseparable if for each neighborhood assignment
(O_{x})_{x∈X} there exists a countable subset A of X which meets each neighborhood
O_{x}, x∈X. It is clear that each separable space is foreseparable and a first countable space is separable
if and only if it is foreseparable. We shall prove that each foreseparable space X of cardinality
X < ℵ_{ω} is separable. On the other hand, for every countable ordinal α
(with ℵ_{α} ≤ 2^{c}) we construct an example of a
nonseparable foreseparable space X of cardinality X = ℵ_{α}
(which is totally disconnected and hence functionally Hausdorff).
This is a joint work with Sasha Ravsky.







16.02.15

All

Divertissement
Active participants of the seminar will discuss recent results and open problems,
which can stimulate further investigations.







Winter

Holidays

* * *







1.12.14

T.Banakh

Linear separation axioms
Let L be a linearly ordered set, X be a topological space and x be a point of X.
An indexed family (U_{λ})_{λ∈L} of open neighborhoods of x
is called a regular Lneighborhood of x if cl_{X}(U_{λ})⊂U_{l}
for every λ < l in L.
We define a topological space X to be
• Lseparated if for any distinct points x,y in X there is a regular Lneighborhood (U_{λ})_{λ∈L} of x such that
y does not belong to ∪_{λ∈L}U_{λ};
• Lregular if for any point x∈X and a neighborhood O_{x} there is a regular Lneighborhood (U_{λ})_{λ∈L} of x such that
∪_{λ∈L}U_{λ} ⊂ O_{x};
• Lconnected if any regular Lneighborhood (U_{λ})_{λ∈L} of any point x in X has
∪_{λ∈L}U_{λ}=X.
It is clear that a topological space X is:
• a T_{1}space iff X is 1separated;
• Hausdorff iff X is 2separated;
• Urysohn iff X is 3separated;
• functionally Hausdorff iff X is Qseparated;
• regular iff X is 2regular;
• completely regular iff X is Qregular;
• connected iff X is αconnected for the ordinal α=c(X)^{+}.
We shall survey some known results related to Lseparated, Lregular or Lconnected spaces.
In particular, we show that the classical examples of countable connected Hausdorff spaces constructed by Bing (1953) and Golomb (1959) are 2separated and 3connected.
On the other hand, the example of a countable connected Urysohn space constructed by Ritter (1977) is 3separated and 4connected.
These example suggest the following:
Open Problem. For every natural number n construct an example of an nseparated and (n+1)connected space.
At the moment such examples are known only for n∈{1,2,3}.







24.11.14
8.12.14

T.Radul

Nash equilibria with Sugeno expected payoff
We consider a formal generalization of Nash equilibribrium
for games in capacities with Sugeno expected payoff.
We obtain an existence theorem using categorical and convex methods.







17.11.14

T.Banakh

Functionally stable closures of the class of separable metrizable spaces
We determine the smallest class of topological spaces, which contains all metrizable separable spaces and
is closed under the operations of taking subspace, homeomorphic image, countable topological sum, countable Tychonoff product, and
taking the function space C_{k}(X,Y) (resp. C_{p}(X,Y)).







20.10.14 27.10.14
10.11.14

T.Banakh, O.Ravsky

Continuous (quasipseudo) norms on (para)topological groups
We shall prove that for each closed neighborhood U of the unit e in a paratopological group G
there exists a continuous quasipseudonorm ·:G→[0,1] (i.e., a continuous function with e=0 and
x+y≤x+y for every x,y∈G) such that the unit ball B={x∈ G:x<1} is contained in U.
This implies that a topology of a regular paratopological group is generated by a family of
continuous quasipseudonorms and also by a family of leftinvariant right continuous quasipseudometrics. Also this implies
that the topology of a first countable regular paratopological group is generated by a continuous
quasinorm and also by a leftinvariant right continuous quasimetric. This resolves an old problem in the theory of paratopological groups.







13.10.14

O.Ravsky

(Quasi)metrization of (para)topological groups
We discuss different aspects of (quasi)(pseudo)metrization of (para)topological groups.
In particular, we are going to prove that the topology of each regular first countable
paratopological LSINgroup can be generated by a continuous quasinorm.







06.10.14

T.Banakh, O.Ravsky

Each regular paratopological group is Tychonoff
For each closed neighborhood U of the unit e in a paratopological group G we construct a continuous function
f:G→[0,1] such that f(e)=0 and f^{1}([0,1))⊂ U.
This implies that each regular paratopological group is Tychonoff and each Hausdorff paratopological group is functionally Hausdorff.
This resolves two (more than 50 years) old open problems in the theory of paratopological groups.
Reference:
T.Banakh, A.Ravsky,
A characterization of Tychonoff spaces with applications to paratopological groups, preprint (http://arxiv.org/abs/1410.1504)







15.09.14
22.09.14

All All

Divertissement
Active participants of the seminar will present and discuss interesting open problems
from various branches of topology and its applications.





