Scientific Seminar
at Geometry and Topology Department of
Ivan Franko National University of Lviv
Archive for (2010/11) academic year



Topology
&
Applications

Date: Speaker:       Title (click on the title to see the Abstract):
08.06.11 Ф.Сохацький
(Вінниця)
  • Про деякі алгебри топологічних квазігрупових операцій
    • Багатомісна операція називається оборотною або квазігруповою, якщо вона оборотна по кожній своїй змінній. Квазігрупова операція називається топологічною, якщо неперервна як сама операція, так і всі її ділення. Множина всіх топологічних квазігрупових операцій деякого топологічного простору замкнена відносно безповторної композиції операцій, тобто композиції, в якій предметні змінні не повторюються. Над такими алгебрами багатомісних операцій знаходяться розв'язки функційних рівнянь асоціативності, медіальності, тощо. Наводяться наслідки для топологічної прямої.
30.05.11 T.Radul
  • Non-isomorphness of monads Ek of k-Lipschitz functionals.
    • For a real number k≥1 and a compact Hausdorff space X by Ek(X) we denote the subspace of constant-preserving k-Lipschitz functionals φ:C(X)→R. It is known that the functor Ek can be completed to a monad in the category Comp of compact Hausdorff spaces. We prove that for distinct numbers k,n the monads Ek and En are not isomorphic.
23.05.11 V.Shepelskaya
(Kharkiv)
  • Slicely countably determined Banach spaces.
    • We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the Radon-Nikodym property and all spaces without copies of l1. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1.
11.04.11 T.Banakh
  • On Borel σ-ideals on Rn and their cardinal characteristics.
    • We shall prove that the ideal μ of meager subsets in Rn is the largest topologically invariant σ-ideal on Rn. This fact will be used for calculating cardinal characteristics of an arbitrary topologically invariant σ-ideal with Borel base.
11.04.11 T.Radul
  • An involution map for the functional monad.
    • We define an involution map for the functional monad which generalise the conjugation map for capacities and the transversality map for inclusion hyperspaces.
28.03.11 T.Banakh
  • Isometric homogeneity versus topological and coarse homogeneity
    • We shall discuss the relation between various sorts of homogeneity of metric spaces: isometric, Lipschitz, topological, uniform and coarse homogeneities.
21.03.11 O.Chervak
  • Characterization of the real line in the coarse category
    • We shall prove that a metric space X is coarsely equivalent to the real line if and only if X is coarsely homogeneous, has positive asymptotic dimension and each small subset of X has asymptotic dimension zero.
      This is a coarse analog of the following (folklore) topological result: a connected compact metrizable space X is homeomorphic to the circle if and only if X is topologically homogeneous and each nowhere dense subset of X has topological dimension zero.
14.03.11 L.Karchevska
  • On the Chigogidze Extension of Weakly Normal Functors in the Category Comp onto Tych.
    • In the present talk we show that preserving imbeddings by the Chigogidze extension of a weakly normal functor implies preserving 1-preimages by this functor. We also try to determine what properties from the definition of weak normality are really necessary.
28.02.11 N.Kolos
  • Operations on some classes of discontinous functions
    • We shall talk about the composition, Cartesian and diagonal product of some discontinuous functions.
21.02.11 All
  • Divertisement
    • The active participants of the topological seminar posed some open problems:
      Problem (T.Radul) Suppose X is a barycentrically opened convex compact space. Is the hyperspace cc(X) barycentrically open? The answer is not known even for X=P(Y).
      Problem (I.Guran) Does the group FS(X) of finitely supported permutations of a set X admit a locally compact group topology?
      Problem (O.Gutik) Is it true that each group topology on the permutation group FS(X) extends to a semigroup topology on the inverse semigroup IFS(X) of finitely supported partial bijections of X?
      Problem (I.Guran) Is there a universal second-countable (zero-dimensional) locally compact topological group?
14.02.11 All
  • Divertisement
    • The active participants of the topological seminar were discussing new (winter) results and posed some open problems:
      Problem (M.Zarichnyi) Study the topological properties of the hyperspace of rosettes of constant width.
      Problem (T.Banakh) Let X be a coarsely homogeneous metric space of asymptotic dimension 2 such that each small subset of X has asymptotic dimension ≤ 1. Is X coarsely equivalent to the Euclidean or hyperbolic plane?
      Problem (T.Banakh) Is there an injective F-convergent sequence in βω for an analytic filter F?
All
  • Winter Holidays
27.12.10 I.Guran
  • Bogdan Bokalo: a sketch of scientific biography
    • We shall survey the principal achievements of Bogdan Bokalo for the years of his mathematical activity:
      - tangential topologies and scattered properties,
      - scatteredly continuous functions and corresponding function spaces;
      - the structure of cancellative sequentially compact semigroups;
      - Bokalo's Problem of the metrizability of compact topological inverse semigroups.
20.12.10 T.Banakh
  • On functors preserving skeletal maps and skeletally generated spaces
    • A map f: X→Y between topological spaces is skeletal if the preimage f –1(A) of each nowhere dense subset A of Y is nowhere dense in X. We prove that a normal functor F: CompComp is skeletal (which means that F preserves skeletal epimorphisms) if and only if for any open surjective map f: X→Y between metrizable zero-dimensional compacta the map Ff: FX→FY is skeletal. This characterization implies that each open normal functor is skeletal. Also we prove that each normal functor F: CompComp preserves the class of skeletally generated compacta.
      This is a joint work with A.Kucharski and M.Klymenko.
13.12.10 T.Bosenko
(Kharkiv)
  • The Daugavet property and Daugavet centers
    • A nonzero operator G:X→Y between Banach spaces is called a Daugavet center if the equation ||G+T||=||G||+||T|| holds for every rank 1 operator T:X→Y. We consider examples of Daugavet centers, study properties of spaces X and Y admitting a Daugavet center, and show for every Daugavet center G:X→Y the class of operators T:X→Y satisfying ||G+T||=||G||+||T|| is much wider than the class of rank 1 operators.
29.11.10 T.Banakh
  • Means on scattered compact spaces
    • We prove that a separable scattered compact space containing a copy of the ordinal segment [0,ω1) admits no separately continuous commutative idempotent operation.
22.11.10 I.Getman
  • The topological structure of hyperspaces of closed convex sets in a Banach space
    • We describe the topological structure of the hyperspace of closed convex sets in a Banach space, endowed with the Hausdorff metric.
15.11.10 V.Korzhyk
(Chernivtsi)
  • Minimal embeddings of complete graphs and 1-immersions of graphs in two-dimensional surfaces
    • A graph is defined to be embedded (resp. 1-immersed) in a two-dimensional surface, if the graph is drawn on the surface so that its edges do not cross (resp. each edge is crossed by at most one other edge). By a minimal embedding of a graph we mean an embedding of the graph in a surface of minimal genus. We give a survey of author's results on minimal embeddings of complete graphs and 1-immersions of graphs in two-dimensional surfaces.
08.11.10 L.Karchevska,
T.Radul
  • On Extension of weakly normal functors in the Category Comp onto Tych
    • In the present talk we shall discuss questions concerning extension of functors from Comp onto Tych, as well as study some properties of the Chigogidze extension of certain functors. It is well known that in the case of a normal functor F, its Chigogidze extension Fβ is normal. We investigate Fβ for weakly normal functors F. In particular, we study the properties of F that guarantee nice properties of Fβ such as preserving weight and embeddings. We say that a functor F in the category Comp preserves 1-preimages, if for any spaces X,Y, any closed subset A of Y and any mapping f : XY between them such that f | f -1(A) is a homeomorphism we have that (Ff)-1(FA) = F(f -1 (A)). It is easy to see that in case a weakly normal functor F preserves 1-preimages, Fβ preserves weight and embeddings. It is likely that the statement inverse to the previous one holds. At least, we can prove that if Fβ preserves embeddings, then F preserves 1-preimages in the class of 0-dimensional spaces and mappings between them.
01.11.10 R.Vorobel'
  • Construction of algebras of logarithmic type
    • Розглянуто підхід до конструювання універсальних алгебр двох змінних з двома бінарними операціями. Особливістю підходу є побудова алгебр, арифметичні операції яких відображають властивості наперед заданих нелінійних функцій логарифмічного типу. Представлено спосіб конструювання таких алгебр. Доведено теореми, які є основою отримання явного виразу операцій додавання та множення на скаляр. Базовим елементом конструювання розглянутих алгебр є задана аналітично функція-генератор. Наведено приклади нових алгебр логарифмічного типу. Ці алгебри є основою побудови нових методів опрацювання цифрових зображень і відображають психофізичні властивості сприйняття світла людиною.

      A method for construction of logarithmic type algebras is presented. We obtain an explicit expression for the operations of addition and multiplication. As a basic element for construction of proposed algebras is a function-generator of analytic form. Examples for new logarithmic types algebras are presented.
25.10.10 T.Banakh
  • From Topology to Bornology: Five faces of Analysis Situs
    • We introduce a category of preuniform spaces PreU whose subcategories correspond to various branches of Analysis Situs: Topology, Theory of uniform spaces, Coarse Geometry etc. In each subcategory C of PreU we introduce the notions of isomorphism, equivalence, embedding, metrizability, homogeneity and dimension. We shall prove C-metrization theorems and shall classify C-homogeneous C-metrizable preuniform spaces of C-dimension zero up to the C-equivalence. Also we shall present characterizations of the object that appears in such classification.
18.10.10 O.Nykyforchyn
(Ivano-Frankivsk)
  • Non-additive measures on compact semilattices
    • A proper definition of non-additive measure on a compact Hausdorff Lawson semilattice, topologies on the set of measures, functors and monads in categories of compact semilattices, conjugate measures will be discussed.
04.10.10 R.Cauty
(Paris)
  • Sur une famille indenombrable de cubes de Hilbert
    • We shall discuss the topological structure of the uncountable systems {rQ: r in [0,1]} оf homothetic copies of the Hilbert cubes Q=[0,1]ω or Q=[-1,1]ω and shall try to prove that these two systems are homeomorphic.
27.09.10 T.Banakh
  • Pontryagin Duality for Topological Inverse Monoids
    • For a topological monoid S the dual inverse monoid is the topological monoid of all identity preserving homomorphisms from S to the circle with attached zero. A topological monoid S is defined to be reflexive if the canonical homomorphism from S to its second dual inverse monoid is a topological isomorphism. We prove that a (compact or discrete) topological inverse monoid S is reflexive (if and) only if S is abelian and the idempotent semilattice of S is zero-dimensional. For a discrete (resp. compact) topological monoid its dual inverse monoid is compact (resp. discrete). These results unify the Pontryagin-van Kampen Duality Theorem for abelian groups and the Hofmann-Mislove-Stralka Duality Theorem for zero-dimensional topological semilattices. For more details, look here.
20.09.10 I.Chuchman
  • On monoids of almost identity injective partial selfmaps
    • We study the semigroup $\mathscr{I}^{\infty}_\lambda$ of almost identity injective partial selfmaps of the set of cardinality $\lambda$. We describe the Green relations on $\mathscr{I}^{\infty}_\lambda$, all (two-sided) ideals and all congruences of the semigroup $\mathscr{I}^{\infty}_\lambda$. We prove that every Hausdorff hereditary Baire topology $\tau$ on $\mathscr{I}^{\infty}_\lambda$ such that $(\mathscr{I}^{\infty}_\lambda,\tau)$ is a semitopological semigroup is discrete and describe the closure of the discrete semigroup $\mathscr{I}^{\infty}_\lambda$ in a topological semigroup. Also we show that the discrete semigroup $\mathscr{I}^{\infty}_\lambda$ does not embed into a compact topological semigroup and construct two non-discrete Hausdorff topologies turning $\mathscr{I}^{\infty}_\lambda$ into a topological inverse semigroup.
13.09.10 O.Chervak
  • On asymptotic inductive dimension acInd
    • We introduce a new asymptotic inductive dimension acInd and prove that it is equal to Dranishnikov's asymptotic dimension asInd. As an application we prove that the asymptotic Dimensiongrad asDg(X) of any geodesic metric space X is equal to acInd(X) and asInd(X), which resolves an old problem in the Asymptotic Dimension Theory. Also the equality acInd=asInd gives a simple proof of the monotonicity theorem of T.Radul for asInd.
06.09.10 All
  • Divertisement
    • The active participants of the topological seminar will discuss new (summer) results and pose some open problems.