Locally Compact Groups: Traditions And Trends, 2017 Technische Universitat Darmstadt

#### Locally Compact Groups: Traditions And Trends, Karl Heinrich Hofmann, Wolfgang Herfort, Francesco G. Russo

*Summer Conference on Topology and Its Applications*

For a lecture in the Topology+Algebra and Analysis section, the subject of locally compact groups appears particularly fitting: Historically and currently as well, the structure and representation theory of locally compact groups draws its methods from each of theses three fields of mathematics. Nowadays one might justifiably add combinatorics and number theory as sources. The example of a study of a class of locally compact groups called “near abelian,” undertaken by W. Herfort, K. H. Hofmann, and F. G. Russo, may be used to illustrate the liaison of topological group theory with this different areas of interest. Concepts like ...

Entropy In Topological Groups, Part 1, 2017 University of Udine

#### Entropy In Topological Groups, Part 1, Dikran Dikranjan

*Summer Conference on Topology and Its Applications*

Entropy was introduced first in thermodynamics and statistical mechanics, as well as information theory. In the last sixty years entropy made its way also in topology, ergodic theory, as well as other branches of mathematics as algebra, geometry and number theory where dynamical systems appear in one way or another.

Roughly speaking, entropy is a non-negative real number or infinity assigned to a "selfmap" T of a "space" X, where the "space" X can be a topological or uniform space, a measure space, an abstract or topological group (or vector space) or just a set. The "selfmap" T can be ...

Uncountable Discrete Sets And Forcing, 2017 University of South Carolina - Beaufort

#### Uncountable Discrete Sets And Forcing, Akira Iwasa

*Summer Conference on Topology and Its Applications*

Suppose that a space X has no uncountable discrete subspace. We will discuss if forcing can create an uncountable discrete subspace of X.

Dense Subsets Of Function Spaces With No Non-Trivial Convergent Sequences, 2017 Universidad Autonoma Metropolitana - Iztapalapa

#### Dense Subsets Of Function Spaces With No Non-Trivial Convergent Sequences, Vladimir V. Tkachuk

*Summer Conference on Topology and Its Applications*

We will show that a monolithic compact space X is not scattered if and only if C_{p}(X) has a dense subset without non-trivial convergent sequences. Besides, for any cardinal κ ≥ *c*, the space **R**^{κ} has a dense subspace without non-trivial convergent sequences. If X is an uncountable σ-compact space of countable weight, then any dense set Y ⊂ C_{p}(X) has a dense subspace without non-trivial convergent sequences. We also prove that for any countably compact sequential space X, if C_{p}(X) has a dense k-subspace, then X is scattered.

Lifting Homeomorphisms Of Cyclic Branched Covers Of The Sphere, 2017 University of Wisconsin - Milwaukee

#### Lifting Homeomorphisms Of Cyclic Branched Covers Of The Sphere, Rebecca R. Winarski, Tyrone Ghaswala

*Summer Conference on Topology and Its Applications*

Birman and Hilden ask: given finite branched cover X over the 2-sphere, does every homeomorphism of the sphere lift to a homeomorphism of X? For covers of degree 2, the answer is yes, but the answer is sometimes yes and sometimes no for higher degree covers. In joint work with Ghaswala, we completely answer the question for cyclic branched covers. When the answer is yes, there is an embedding of the mapping class group of the sphere into a finite quotient of the mapping class group of X. In a family where the answer is no, we find a presentation ...

Compactness Via Adherence Dominators, 2017 Morgan State University

#### Compactness Via Adherence Dominators, Bhamini M. P. Nayar, Terrence A. Edwards, James E. Joseph, Myung H. Kwack

*Summer Conference on Topology and Its Applications*

This talk is based on a joint work by T. A. Edwards, J. E. Joseph, M. H. Kwack and B. M. P. Nayar that apperared in the *Journal of Advanced studies in Topology,* Vol. 5 (4), 2014), 8 - 15. B

An adherence dominator on a topological space X is a function π from the collection of filterbases on X to the family of closed subsets of X satisfying A(Ω) ⊆ π(Ω) where A(Ω) is the adherence of Ω. The notations π(Ω) and A(Ω) are used for the values of the functions π and A and π ...

Some New Completeness Properties In Topological Spaces, 2017 Gazi University

#### Some New Completeness Properties In Topological Spaces, Cetin Vural, Süleyman Önal

*Summer Conference on Topology and Its Applications*

One of the most widely known completeness property is the completeness of metric spaces and the other one being of a topological space in the sense of Cech. It is well known that a metrizable space X is completely metrizable if and only if X is Cech-complete. One of the generalisations of completeness of metric spaces is subcompactness. It has been established that, for metrizable spaces, subcompactness is equivalent to Cech-completeness. Also the concept of domain representability can be considered as a completeness property. In [1], Bennett and Lutzer proved that Cech-complete spaces are domain representable. They also proved, in ...

Classifying Matchbox Manifolds, 2017 University of Illinois at Chicago

#### Classifying Matchbox Manifolds, Olga Lukina

*Summer Conference on Topology and Its Applications*

A matchbox manifold is a compact connected foliated space, locally homeomorphic to the product of a Euclidean disk and a Cantor set. Strange attractors in dynamical systems, and exceptional minimal sets of smooth foliations present examples of matchbox manifolds. Many actions of profinite groups on trees can be suspended to obtain matchbox manifolds, and similar examples arise in other contexts and in other parts of mathematics.

Thus there is a natural problem of classifying matchbox manifolds. The most tractable class of matchbox manifolds is the class of weak solenoids which are the inverse limits of finite-to-one coverings of closed manifolds ...

A Trace Formula For Foliated Flows (Working Paper), 2017 Universidade de Santiago de Compostela

#### A Trace Formula For Foliated Flows (Working Paper), Jesús A. Álvarez López, Yuri A. Kordyukov, Eric Leichtnam

*Summer Conference on Topology and Its Applications*

The talk, based on work in progress, will be about our progress to show a trace formula for foliated flows on foliated spaces, which has been conjectured by V. Guillemin, and later by C. Deninger with more generality. It describes certain Leftchetz distribution of the foliated flow, acting on some version of the leafwise cohomology, in terms of local data at the closed orbits and fixed points.

Order, Distance, Closure And Convergence: Reconciling Competing Fundamental Topological Concepts, 2017 York University

#### Order, Distance, Closure And Convergence: Reconciling Competing Fundamental Topological Concepts, Walter Tholen

*Summer Conference on Topology and Its Applications*

Already in Hausdorff’s 1914 book, often considered the cradle of general topology, one finds traces of a discussion on the relative strengths of the concepts mentioned in the title of this talk. In fact, one may argue that Hausdorff anticipated the basic ideas of how to unify these concepts, which were developed only later on by many mathematicians over the course of a century, as propagated in Hofmann, Seal & Tholen. Indeed, Hausdorff thought of ordering points by assigning to every pair of them a (truth) value, just as a metric assigns to them a number. More importantly, he also ...

Virtual Seifert Surfaces And Slice Obstructions For Knots In Thickened Surfaces, 2017 Monmouth University

#### Virtual Seifert Surfaces And Slice Obstructions For Knots In Thickened Surfaces, Micah Chrisman, Hans U. Boden, Robin Gaudreau

*Summer Conference on Topology and Its Applications*

Here we introduce the notion of virtual Seifert surfaces. Virtual Seifert surfaces may be thought of as a generalization of Gauss diagrams of virtual knots to spanning surfaces of a knot. This device is then employed to extend the Tristram-Levine signature function to AC knots. Using the AC signature functions and Tuarev’s graded genus invariant, we determine the slice status of all 76 almost classical knots having at most six crossings. The slice obstructions for AC knots are then extended to all virtual knots via the parity projection map. This map, which is computable from a Gauss diagram, sends ...

Sequential Decreasing Strong Size Properties, 2017 Universidad Autonoma del Estado de Mexico

#### Sequential Decreasing Strong Size Properties, Miguel A. Lara, Fernando Orozco, Felix Capulín

*Summer Conference on Topology and Its Applications*

Let X be a continuum. A topological property *P* is said to be a sequential decreasing strong size property provided that if μ is a strong size map for C_{n}(X), {t_{n}} is a sequence in the interval (t, 1) such that limt_{n} = t and each fiber μ^{-1} (t_{n}) has the property *P*, then μ^{-1} (t) has the property *P*. We show that the following properties are sequential decreasing strong size properties: be a Kelley continuum, indecomposability, local connectedness, continuum chainability and unicoherence.

Pseudo-Contractibility, 2017 Universidad Autonoma del Estado de Mexico

#### Pseudo-Contractibility, Felix Capulín, Leonardo Juarez-Villa, Fernando Orozco

*Summer Conference on Topology and Its Applications*

Let X, Y be topological spaces and let f, g:X→ Y be mappings, we say that f is pseudo-homotopic to g if there exist a continuum C, points a, b ∈ C and a mapping H:X ×C → Y such that H(x, a)=f(x) and H(x, b)=g(x) for each x ∈ X. The mapping H is called a pseudo-homotopy between f and g. A topological space X is said to be pseudo-contractible if the identity mapping is pseudo-homotopic to a constant mapping in X. i.e., if there exist a continuum C, points a, b ∈ C ...

Fiber Strong Shape Theory For Topological Spaces, 2017 Batumi Shota Rustaveli State University

#### Fiber Strong Shape Theory For Topological Spaces, Ruslan Tsinaridze, Vladimer Baladze

*Summer Conference on Topology and Its Applications*

The purpose of this paper is the construction and investigation of fiber strong shape theory for compact metrizable spaces over a fixed base space B_{0} , using the fiber versions of cotelescop, fibrant space and SSDR-map. In the paper obtained results containing the characterizations of fiber strong shape equivalences, based on the notion of double mapping cylinder over a fixed space B_{0}. Besides, in the paper we construct and develop a fiber strong shape theory for arbitrary spaces over fixed metrizable space B_{0}. Our approach is based on the method of Mardešić-Lisica and instead of resolutions, introduced by ...

Compactly Supported Homeomorphisms As Long Direct Limits, 2017 Technische Universitat Darmstadt

#### Compactly Supported Homeomorphisms As Long Direct Limits, Rafael Dahmen, Gábor Lukács

*Summer Conference on Topology and Its Applications*

Let λ be a limit ordinal and consider a directed system of topological groups (G_{α})_{α < λ} with topological embeddings as bonding maps and its directed union G=∪_{α < λ}G_{α}. There are two natural topologies on G: one that makes G the direct limit (colimit) in the category of topological spaces and one which makes G the direct limit (colimit) in the category of topological groups.

For λ = ω it is known that these topologies almost never coincide (*Yamasaki's Theorem).*

In my talk last year, I introduced the *Long Direct Limit Conjecture*, stating that for λ = ω_{1 ...}

On A Construction Of Some Class Of Metric Spaces, 2017 Adam Mickiewicz University of Poznan

#### On A Construction Of Some Class Of Metric Spaces, Dariusz Bugajewski

*Summer Conference on Topology and Its Applications*

In this talk we are going to describe Sz´az’s construction of some class of metric spaces. Most of all we will analyze topological properties of metric spaces obtained by using Sz´az’s construction. In particular, we provide necessary and sufficient conditions for completeness of metric spaces obtained in this way. Moreover, we will discuss the relation between Sz´az’s construction and the “linking construction”. A particular attention will be drawn to the “floor” metric, the analysis of which provides some interesting observations.

The Specification Property And Infinite Entropy For Certain Classes Of Linear Operators, 2017 Christopher Newport University

#### The Specification Property And Infinite Entropy For Certain Classes Of Linear Operators, James Kelly, Will Brian, Tim Tennant

*Summer Conference on Topology and Its Applications*

We study the specification property and infinite topological entropy for two specific types of linear operators: translation operators on weighted Lebesgue function spaces and weighted backward shift operators on sequence F-spaces.

It is known from the work of Bartoll, Martinínez-Giménez, Murillo-Arcila (2014), and Peris, that for weighted backward shift operators, the existence of a single non-trivial periodic point is sufficient for specification. We show this also holds for translation operators on weighted Lebesgue function spaces. This implies, in particular, that for these operators, the specification property is equivalent to Devaney chaos. We also show that these forms of chaos imply ...

Revelation Of Nano Topology In Cech Rough Closure Spaces, 2017 Madurai Kamaraj University

#### Revelation Of Nano Topology In Cech Rough Closure Spaces, V. Antonysamy, Llellis Thivagar, Arockia Dasan

*Summer Conference on Topology and Its Applications*

The concept of Cech closure space was initiated and developed by E. Cech in 1966. Henceforth many more research scholars set their minds in this theory and developed it to a new height. Pawlak.Z derived and gave shape to Rough set theory in terms of approximation using equivalence relation known as indiscernibility relation. Further Lellis Thivagar enhanced rough set theory into a topology, called Nano Topology, which has at most five elements in it and he also extended this into multi granular nano topology. The purpose of this paper is to derive Nano topology in terms of Cech rough ...

The Isbell-Hull Of An Asymmetrically Normed Space, 2017 North-West University (South Africa)

#### The Isbell-Hull Of An Asymmetrically Normed Space, Olivier Olela Otafudu, Jurie Conradie, Hans-Peter Künzi

*Summer Conference on Topology and Its Applications*

In this talk, we discuss an explicit method to define the linear structure of the Isbell-hull of an asymmetrically normed space.

Topology And Order, 2017 Western Kentucky University

#### Topology And Order, Tom Richmond

*Summer Conference on Topology and Its Applications*

We will discuss topologies as orders, orders on sets of topologies, and topologies on ordered sets. More specifically, we will discuss Alexandroff topologies as quasiorders, the lattice of topologies on a finite set, and partially ordered topological spaces. Some topological properties of Alexandroff spaces are characterized in terms of their order. Complementation in the lattice of topologies on a set and in the lattice of convex topologies on a partially ordered set will be discussed.