Scientific Seminar at Geometry and Topology Department of Ivan Franko National University of Lviv Archive for (2011/12) academic year Topology & Applications
 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 L-semimodules of Scott continuous functions from domains to a completely distributive quantale. Approximations of Scott continuous functions are also discussed. 07.05.12 M.MytrofanovO.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: R-factorizable 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 H-closed topological semilattice with non-H-closed quotient We construct an example of an H-closed topological semilattice admitting a surjective homomorphism on a discrete semilattice, which is not H-closed. 23.04.12 M.PankovOlsztyn Jonson graphs and Grassman graphs The Jonson graph J(k,n) is a graph whose vertices are k-element subsets of an n-element set, and two vetrices A,B are connected by an edge if the intersection A∩B has cardinality k-1. A Grassman graph G(k,V) over a vector space V is the graph whose vertices are linear k-dimensional subspaces of V and two vertices A,B are linked by an edge if their intersection A∩ B is a linear subspae of dimension k-1. 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) R-scattered spaces; 3) normality of the space SCp(X); 4) σ-convex subspaces of the space SCp(X). 26.03.12 M.Popov Narrow Operators Âóçüê³ îïåðàòîðè áóëè ââåäåí³ ³ ñèñòåìàòè÷íî äîñë³äæåí³ àâòîðîì ó ñóì³ñí³é ðîáîò³ ç À.Ì.Ïë³÷êîì ó 1990 ð., ÿê óçàãàëüíåííÿ êîìïàêòíèõ îïåðàòîð³â, çàäàíèõ íà ïðîñòîðàõ ôóíêö³é. Îêðåì³ ðåçóëüòàòè ïðî âóçüê³ îïåðàòîðè áóëè â³äîì³ ùå äî öüîãî. Â³ä òîãî ÷àñó çà 20 ðîê³â òåîð³ÿ âóçüêèõ îïåðàòîð³â çàçíàëà ³ñòîòíîãî ðîçâèòêó. Äîïîâ³äü ïðèñâÿ÷åíà êîðîòêîìó åêñêóðñó â öþ òåîð³þ. 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 non-limit ordinal if X=Φ(X) for some contracting finite-valued 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 FK(X) of X can be identified with a closed subalgebra of the free universal algebra FK(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 Sω(X) ) of (finitely supported) bijections of a set X. In particular, confirming a Dikranjan's conjecture, we prove that the topology Tp of pointwise convergence on each subgroup G of S(X) that contains Sω(X) is the coarsest Hausdorff group topology on G (more generally, the coarsest T1-topology which turns G into a [semi]-topological group), and Tp 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 Sω(X), the centralizer topology coincides with the topology Tp if and only of G= Sω(X). Also we prove that the group Sω(X) is σ-discrete in each Hausdorff shift-invariant topology. This is a joint work with I.Guran and I.Protasov. 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 t-notm $\frac{ab}{max(a,b,\alpha)}$. 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 ω-primitives We describe some methods of constructing functions with given oscilation. 05.12.11 T.Banakh Fractals generated by multi-valued functions on topological spaces Given a multi-valued 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 multi-functions (called micro- and macro-fractals) 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 1-dimensional Peano continuum which is not homeomorphic to a deterministic fractal. Answering an old (1985) question of M.Hata, we construct an example of a 1-dimensional Peano continuum which is not homeomorphic to a deterministic fractal. A metric compact space X is called a deterministic fractal if X=f1(X)∪...∪fn(X) for some contracting self-maps f1,...,fn: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. This is a joint work with M.Nowak 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 2-colorings of Euclidean spaces Generalizing a classical result of I.V. Protasov, we prove that for every 2-coloring of Euclidean space Rn and for every affinely independent subset A with |A|=n+1 there exists a monochromatic subset S of Rn of asymptotic dimension ≥ n-1, 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 non-trivial 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 Borsuk-Ulam type theorems We explain a homological method of proof of some theorems of combinatorial topology, including Borsuk-Ulam Theorem and topological Tverberg Theorem. 3.10.11 R.Cauty Approximation of multi-valued maps by chain morphisms We introduce a general functorial approach to the theory of exteding and averaging operators which unify linear and non-linear theories. 19.09.1112.09.11 T.Radul Extenders and averaging operators generated by functors We define a notion of approximation of a multi-valued 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.