Scientific Seminar
at Geometry and Topology Department of
Ivan Franko National University of Lviv
Archive for (2008/09) academic year


08.06.09 O.Pikhurko
  • Maximizing the number of colorings
    • An old problem of Linial and Wilf asks for f(n,m,l), the maximum number of l-colorings that a graph with n vertices and m edges can have. We solve this problem for every fixed l when C≤ m≤ cn2 for some C,c>0 depending on l only. Moreover, for l=3, we establish the structure of optimal graphs for all large m≤ n2/4 confirming (in a stronger form) a conjecture of Lazebnik from 1989.
      This is joint work with Po-Shen Loh and Benny Sudakov.
25.05.09 R.Cauty
  • Cohomological properties of subspaces of symmetric powers
    • We investigate cohomological properties of subspaces of the n-th symmetric power X[n] of k-dimensional space X. The obtained results imply that the n-dimensional sphere Sn does not embed into the n-th symmetric power X[n] of a 1-dimensional compact space X. This resolves a problem of Illanes and Nadler.
  • Constructing economical connected metric spaces
    • A metric space (X,d) is economic if for each infinite subspace A of X the set d(A×A) has cardinality |d(A×A)|≤dens(A). It is clear that each economic metric space X of density dens(X)<c is zero-dimensional.
      We show that each (connected) sequential topological space X is the image of a (connected) economic complete metric space Eco(X) under a quotient map Eco(X)→X. The construction the space Eco(X) determines a functor Eco:TopMetr from the category Top of topological spaces and their continuous maps to the category Metr of metric spaces and their non-expanding maps.
  • Embeddings and closedness of inverse semigroups of finite partial bijections
    • We shall prove that for every infinite cardinal λ the inverse semigroup Iλn of partial bijections defined on subsets of cardinality ≤ n of λ does not embed into a countably compact topological semigroup. Also we shall describe certain H-closed topologies on the inverse semigroup Iλn.
N.Kolos, B.Bokalo
  • When is SC(X)=RX?
    • In the talk we discuss properties of topological spaces X on which every function f:X→R is scatteredly continuous. A map f:X→ Y is called scatteredly continuous if for each non-empty subspace A of X the restriction f|A has a continuity point.
  • The packing completeness of translation invariant ideals on groups
    • An invariant ideal I on a group G is defined to be packing complete if each subset A of G with infinite I-packing index I-pack(A) belongs to I. Here
      I-pack(A)=sup{|B|:B ⊂ G   {bA}b∈ B is I-disjoint}
      where a family F of subsets of G is called I-disjoint if the intersection A∩B of any two distinct sets A,B from F belongs to the ideal I. We prove that several well-known ideals on groups are packing complete. In particular, so are the ideal of universally null subsets of an amenable group and the ideal of small subsets of an abelian group. We prove that each invariant ideal I on an amenable group G has the packing completion pack(I), (which is the smallest packing complete ideal containing I) and describe the inner structure of the packing completion.
16.03.09 Z.Blaszczyk
  • Free actions of finite groups on spheres
    • By Wolf, a complete description list of all finite subgroups G of SO(n+1) acting freely on the sphere Sn and thus the quotient spaces Sn/G (spherical forms) are known.
02.03.09 L.Karchevska
  • On functors OH, OS and absolute retracts
    • We shall consider functors of semiadditive OS and positively homogeneous OH functionals. We detect compact spaces X for which OS(X) and OH(X) are absolute retracts and discuss the connection between property of being an AR and the property of binarity of the monads generated by the latter functors.
23.02.09 T.Radul
  • A functional representation of the capacity monad
    • We construct an embedding of the capacity monad M into the universal monad V and show that the intersection of M with the monad O in V is equal to the inclusion hyperspace monad G.
16.02.09 O.Shukel', T.Radul
  • Functors of finite degree and asymptotic dimension
    • We prove that a finitary weakly normal functor of finite degree preserves the class of metric spaces of finite asymptotic dimension.
30.12.08 - 10.02.09
  • * * * Winter Holidays * * *
22.12.08 O.Pikhurko
  • Limits of sequences of finite graphs
    • We shall consider the notion of convergence of sequences of finite graphs introduced in 2003 by C. Borgs, J. T. Chayes, L. Lovasz, V. T. Sos, and K. Vesztergombi (see [BCLSV]) and shall discuss some recent results on this topic.
15.12.08 O.Verbitsky
  • Fermat's spiral and the line between Yin and Yang
    • Let D denote a disk of unit area. We call a set A of D perfect if it has measure 1/2 and, with respect to any axial symmetry of D, the maximal symmetric subset of A has measure 1/4. We call a curve β in D an yin-yang line if
      - β crosses each concentric circle of D twice,
      - β crosses each radius of D once,
      - β splits D into two congruent perfect sets.
      We prove that Fermat's spiral is a unique yin-yang line in the class of smooth curves, algebraic in polar coordinates.
      This is a joint work with Taras Banakh and Yaroslav Vorobets.
8.12.08 T.Banakh
  • A connected complete metric space without connected separable subspaces
    • We construct a connected complete metric space whose separable subspaces are zero-dimensional. This resolves one problem of M.Morayne and M.R.Wojcik.
1.12.08 O.Ravsky
  • The continuity of inversion in paratopological groups
    • In the talk we concider a new approach to the question "When a paratopological group is a topological group?". Using the approach we show that some classes of paratopological groups are topological groups. We also formulate some questions about the contents of the classes. A group has two basic operations: the multiplication and the inversion. In a paratopological group the first operation is continuous but the second may be not continuous. But besides the multiplication, there are others continuous operations on a paratopological group, for instance, the n-th power. So, we can use the following approach. If we can find an open subset of a group such that the inversion on the set coincides with the other operation and the operation must be continuous then the inversion on the set is continuous too and so the group is topological.

  • Algebra in the superextensions of groups: representation theory
    • Given a group X we study the algebraic structure of the compact right-topological semigroup λ(X) consisting of maximal linked systems on X. This semigroup contains the semigroup β(X) of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup λ(X) in the semigroup of all self-maps of the power-set of X and using this representation describe the structure of the minimal ideal and minimal left ideals of λ(X). We show that for a finitely generated abelian group X the minimal left ideals of λ(X) are compact metrizable topological semigroups, topologically isomorphic to countable products of cyclic 2-groups and finite cardinals endowed with the left zero multiplication. We also prove that the superextension λ(Z) of the group of integers contains a topological copy of each second countable profinite semigroup. This contrasts with the famous Zelenuk's Theorem asserting that the semigroup of ultrafilters β(Z) contains no finite subgroup. More details can be found here.
27.10.08 D.Repovš
  • Topology of Busemann G-spaces
    • We present a survey of classical conjecture concerning the characterization of manifolds, the Busemann Conjecture which asserts that every n-dimensional G-space is a topological n-manifold.
      The key object in this conjecture is the so-called G-space, which is a generalization of a geodesic space. We look at the history, from the early beginnings in 1950's to the present day, concentrating on those geometric properties of these spaces which are particular for higher dimensions.
      In the second part of the talk we present the current state of the conjecture (the work of Busemann, Halverson - Repovš, Krakus, Thurston, and others). We also list open problems and related conjectures, see this survey article for more details.
20.10.08 T.Banakh
  • On functions having only local extrema
    • A real-valued continuous function f:X→R defined on a topological space X is called locally extremal if each point of X is a point of local minimum or local maximum for f.
      We resolve a problem of Michal Ryszard Wojcik constructing a non-constant locally extremal function f:X→R defined on a connected complete metric space X. The space X necessarily has density continuum becuase each locally extremal function f defined on a connected topological space T of network weight <c is constant.
13.10.08 M.Zarichnyi
  • Measures on ultrametric spaces.
    • We will consider the ultrametrizations of spaces of probablity measures and other measures on ultrametric spaces and also study the related functors of measures in the category of ultrametric spaces and their Lipschitz maps.
6.10.08 T.Banakh
  • Functor-semigroups
    • Each continuous monadic functor T on the category of compacta is lifted to the category of compact right-topological groups with dense topological center. Properties of the obtaines functor-semigroups are inverstigated.
29.09.08 T.Banakh
  • Embedding the bicyclic semigroup into a countably compact topological semigroup.
    • By a Hildebrant-Koch Theorem the bicyclic semigroup does not embed into a compact topological semigroup.
      We show that the same is true for each (Tychonov) topological semigroups with countably compact (pseudocompact) square.
      On the other hand, assuming the existence of a Tkachenko-Tomita group, we shall construct a Tychonov countably compact topological semigroup that does contain a copy of the bicyclic semigroup.
      By a Tkachenko-Tomita group we understand an abelian torsion-free countably compact topologcal group without non-trivial convergent sequences. The first example of a Tkachenko-Tomita group was constructed by M.Tkachenko in 1990 under CH, which was later weakened to some forms of Martin Axiom by A.Tomita at al.
      The details can be found in the paper.
15.09.08 T.Radul, O.Gutik
  • Divertissement.
    • Some open problems have been posed. In particular:
      T.Radul asked if the property of a monadic functor to preserve absolute retracts implies the binarity property of the monad.
      O.Gutik posed the problem of constructing an universal object for locally compact Lawson semilattices.
8.09.08 T.Banakh
  • Divertissement.
    • Recent results and open problems on topological semigroups close to being compact were discussed. For details see the "Mendeleev" Table.