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 ...

Liouville Numbers And One-Sided Ergodic Hilbert Transformations, 2017 Wesleyan University

#### Liouville Numbers And One-Sided Ergodic Hilbert Transformations, David Constantine, Joanna Furno

*Summer Conference on Topology and Its Applications*

We examine one-sided ergodic Hilbert transforms for irrational circle rotations and some mean-zero functions. Our approach uses continued fraction expansions to specify rotations by Liouville numbers for which the transformation has everywhere convergence or divergence.

Normal Images Of A Product And Countably Paracompact Condensation, 2017 University of Kansas

#### Normal Images Of A Product And Countably Paracompact Condensation, Jila Niknejad

*Summer Conference on Topology and Its Applications*

In 1997, Buzjakova proved that for a pseudocompact Tychonoff space X and λ = | βX|^{+}, X condenses onto a compact space if and only if X×(λ+1) condenses onto a normal space. This is a condensation form of Tamano's theorem. An interesting problem is to determine how much of Buzjakova's result will hold if "pseudocompact" is removed from the hypothesis.

In this talk, I am going to show for a Tychonoff space X, there is a cardinal λ such that if X×(λ+1) condenses onto a normal space, then X condenses onto a countably paracompact space.

Balanced And Functionally Balanced P-Groups, 2017 University of Udine

#### Balanced And Functionally Balanced P-Groups, Menachem Shlossberg

*Summer Conference on Topology and Its Applications*

In relation to Itzkowitz's problem, we show that a c-bounded P-group is balanced if and only if it is functionally balanced. We prove that for an arbitrary P-group, being functionally balanced is equivalent to being strongly functionally balanced. A special focus is given to the uniform free topological group defined over a uniform P-space. In particular, we show that this group is (functionally) balanced precisely when its subsets B_{n}, consisting of words of length at most n, are all (resp., functionally) balanced.

Domains And Probability Measures: A Topological Retrospective, 2017 Tulane University

#### Domains And Probability Measures: A Topological Retrospective, Michael Mislove

*Summer Conference on Topology and Its Applications*

Domain theory has seen success as a semantic model for high-level programming languages, having devised a range of constructs to support various effects that arise in programming. One of the most interesting - and problematic - is probabilistic choice, which traditionally has been modeled using a domain-theoretic rendering of sub-probability measures as valuations. In this talk, I will place the domain-theoretic approach in context, by showing how it relates to the more traditional approaches such as functional analysis and set theory. In particular, we show how the topologies that arise in the classic approaches relate to the domain-theoretic rendering. We also describe ...

On Continua With Regular Non-Abelian Self Covers, 2017 Bradley University

#### On Continua With Regular Non-Abelian Self Covers, Mathew Timm

*Summer Conference on Topology and Its Applications*

We look at a planar 2-dimensional continuum X which satisfy the following:

Given any finite group G there is an |G|-fold regular self cover f:X → X with G as its group of deck transformations.

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.

Generic Approximation And Interpolation By Entire Functions Via Restriction Of The Values Of The Derivatives, 2017 University of Prince Edward Island

#### Generic Approximation And Interpolation By Entire Functions Via Restriction Of The Values Of The Derivatives, Maxim R. Burke

*Summer Conference on Topology and Its Applications*

A theorem of Hoischen states that given a positive continuous function ε:**R**^{n}→**R**, an unbounded sequence 0 ≤ c_{1} ≤ c_{2} ≤ ... and a closed discrete set T ⊆ **R**^{n}, any C^{∞} function g:**R**^{n}→**R** can be approximated by an entire function f so that for k=0, 1, 2, ..., for all x ∈ **R**^{n} such that |x| ≥ c_{k}, and for each multi-index α such that |α| ≤ k,

- (a) |(D α f)(x)-(D α g)(x)| < ε(x);

- (b) (D α f)(x)=(D α g)(x) if x ∈ T.

We show that if C ⊆ **R**^{n ...}

Shift Maps And Their Variants On Inverse Limits With Set-Valued Functions, 2017 Lamar University

#### Shift Maps And Their Variants On Inverse Limits With Set-Valued Functions, Judy Kennedy, Kazuhiro Kawamura, Van Nall, Goran Erceg

*Summer Conference on Topology and Its Applications*

We study inverse limits with set-valued functions using a pull-back construction and representing the space as an ordinary inverse limit space, which allows us to prove some known results and their extensions in a unified scheme. We also present a scheme to construct shift dynamics on the limit space and give some examples using the construction.

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 ...

Rigidity And Nonrigidity Of Corona Algebras, 2017 Miami University - Oxford

#### Rigidity And Nonrigidity Of Corona Algebras, Paul Mckenney, Alessandro Vignati

*Summer Conference on Topology and Its Applications*

Shelah proved in the 1970s that there is a model of ZFC in which every homeomorphism of the Cech-Stone remainder of the natural numbers is induced by a function on the natural numbers. More recently, Farah proved that in essentially the same model, every automorphism of the Calkin algebra on a separable Hilbert space must be induced by a linear operator on the Hilbert space. I will discuss a common generalization of these rigidity results to a certain class of C*-algebras called corona algebras. No prerequisites in C*-algebra will be assumed.

On Cardinality Bounds Involving The Weak Lindelöf Degree And H-Closed Spaces, 2017 California Lutheran University

#### On Cardinality Bounds Involving The Weak Lindelöf Degree And H-Closed Spaces, Nathan Carlson, Angelo Bella, Jack Porter

*Summer Conference on Topology and Its Applications*

1. Bella and Carlson give several classes of spaces X for which |X| ≤ 2^{wL(X)χ(X)}. This includes locally compact spaces and, more recently, extremally disconnected spaces. Three proofs of the former lead to more general results. One such result is that any regular space X with a π-base consisting of elements with compact closure satisfies |X| ≤ 2^{wL(X)χ(X)}. It is also shown that if X is locally compact and power homogeneous that |X| ≤ 2^{wL(X)t(X)}, an extension of De la Vega's Theorem.

2. Porter and Carlson give a new cardinality ...

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 ...

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 ...}

Aperiodic Colorings And Dynamics, 2017 Universidade de Santiago de Compostela

#### Aperiodic Colorings And Dynamics, Ramon Barral Lijo, Jesús A. Álvarez López

*Summer Conference on Topology and Its Applications*

A graph coloring is strongly aperiodic if every colored graph in its hull has no automorphisms. The talk will describe a method to define strongly aperiodic colorings on graphs with bounded degree. This also provides an optimal bound for the strongly distinguishing number of a graph. Then some applications to the theory of foliated spaces and to tilings will be discussed.

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.

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 ...

On The Lindelöf Σ-Property And Some Related Conclusions, 2017 Universidad Nacional Autonoma de Mexico

#### On The Lindelöf Σ-Property And Some Related Conclusions, Reynaldo Rojas-Hernandez, Fidel Casarrubias-Segura, Salvador Garcia-Ferreira

*Summer Conference on Topology and Its Applications*

We will present some known and some new results about Lindelöf Σ-spaces. We extend some classical results about the Lindelöf and the Lindelöf Σ-property in spaces C_{p}(X) for compact X to the case when X is a Lindelöf Σ-space. We also present some results about the Lindelöf Σ-property in Σ_{s}-products. A result of Tkachenko is generalized by showing that the bound w(X) ≤ nw(X)^{Nag(X)} holds for regular (not necessarily Tychonoff) spaces. Finally we present the solution for two question posed by V. V. Tkachuk about Eberlein and Corson compact spaces.

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 ...

Quotients Of N-Fold Hyperspaces, 2017 Universidad Autonoma del Estado de Mexico

#### Quotients Of N-Fold Hyperspaces, Sergio Macías, Javier Camargo

*Summer Conference on Topology and Its Applications*

iven a continuum X and an integer n ≥ 2, let C_{n}(X) be the n-fold hyperspace of X consisting of all nonempty closed subsets of X with at most n components. We consider the quotient space C^{n}_{1}(X)=C_{n}(X)/C_{1}(X) with the quotient topology. We prove several properties. For example: C^{n}_{1}(X) is unicoherent; if X has the property of Kelley, C^{n}_{1}(X) is contractible; dim(C_{n}(X))=dim(C^{n}_{1}(X)); both C^{n}_{1}([0, 1]) and C^{n}_{1}(S^{1}) are Cantor manifolds ...