Conference Program, 2017 University of Dayton

#### Conference Program, University Of Dayton

*Summer Conference on Topology and Its Applications*

Document provides a list of the sessions, speakers, workshops, and committees of the 32nd Summer Conference on Topology and Its Applications.

Enriched Topology And Asymmetry, 2017 Youngstown State University

#### Enriched Topology And Asymmetry, Stephen Rodabaugh, Jeffrey T. Denniston, Austin Melton

*Summer Conference on Topology and Its Applications*

Mathematically modeling the question of how to satisfactorily compare, in many-valued ways, both bitstrings and the predicates which they might satisfy-a surprisingly intricate question when the conjunction of predicates need not be commutative-applies notions of enriched categories and enriched functors. Particularly relevant is the notion of a set enriched by a po-groupoid, which turns out to be a many-valued preordered set, along with enriched functors extended as to be "variable-basis". This positions us to model the above question by constructing the notion of topological systems enriched by many-valued preorders, systems whose associated extent spaces motivate the notion of topological spaces ...

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.

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.

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 ...

Uncountably Many Quasi-Isometry Classes Of Groups Of Type Fp, 2017 University of Oklahoma

#### Uncountably Many Quasi-Isometry Classes Of Groups Of Type Fp, Ignat Soroko, Robert Kropholler, Ian Leary

*Summer Conference on Topology and Its Applications*

An interplay between algebra and topology goes in many ways. Given a space X, we can study its homology and homotopy groups. In the other direction, given a group G, we can form its Eilenberg-Maclane space K(G, 1). It is natural to wish that it is `small' in some sense. If K(G, 1) space has n-skeleton with finitely many cells, then G is said to have type F_{n}. Such groups act naturally on the cellular chain complex of the universal cover for K(G, 1), which has finitely generated free modules in all dimensions up to n ...

On The Tightness And Long Directed Limits Of Free Topological Algebras, 2017 Technische Universitat Darmstadt

#### On The Tightness And Long Directed Limits Of Free Topological Algebras, Gábor Lukács, Rafael Dahmen

*Summer Conference on Topology and Its Applications*

For a limit ordinal λ, let (A_{α})_{α < λ} be a system of topological algebras (e.g., groups or vector spaces) with bonding maps that are embeddings of topological algebras, and put A = ∪_{α < λ} A_{α}. Let (A, *T*) and (A, *A*) denote the direct limit (colimit) of the system in the category of topological spaces and topological algebras, respectively. One always has *T* ⊇ *A*, but the inclusion may be strict; however, if the tightness of *A* is smaller than the cofinality of λ, then *A*=*T*.

In 1988, Tkachenko proved that the free topological group F(X) is sequential ...

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 The Axiomatic Systems Of Steenrod Homology Theory Of Compact Spaces, 2017 Georgian Technical University

#### On The Axiomatic Systems Of Steenrod Homology Theory Of Compact Spaces, Leonard Mdzinarishvili, Anzor Beridze

*Summer Conference on Topology and Its Applications*

The Steenrod homology theory on the category of compact metric pairs was axiomatically described by J.Milnor. In Milnor, the uniqueness theorem is proved using the Eilenberg-Steenrod axioms and as well as relative homeomorphism and clusres axioms. J. Milnor constructed the homology theory on the category Top^{2}_{C} of compact Hausdorff pairs and proved that on the given category it satisfies nine axioms - the Eilenberg-Steenrod, relative homeomorphis and cluster axioms (see theorem 5 in Milnor). Besides, he proved that constructed homology theory satisfies partial continuity property on the subcategory Top^{2}_{CM} (see theorem 4 in Milnor) and the ...

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 ...

On Di-Injective T0-Quasi-Metric Spaces, 2017 North-West University (South Africa)

#### On Di-Injective T0-Quasi-Metric Spaces, Collins Amburo Agyingi

*Summer Conference on Topology and Its Applications*

We prove that every q-hyperconvex T0-quasi-metric space (X, d) is di-injective without appealing to Zorn’s lemma. We also demonstrate that QX as constructed by Kemajou et al. and Q(X) (the space of all Katˇetov function pairs on X) are di-injective. Moreover we prove that di-injective T0-quasi-metric spaces do not contain proper essential extensions. Among other results, we state a number of ways in which the the di-injective envelope of a T0-quasi-metric space can be characterized.

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.

A Compact Minimal Space Whose Cartesian Square Is Not Minimal, 2017 AGH University of Science and Technology, Krakow

#### A Compact Minimal Space Whose Cartesian Square Is Not Minimal, Jan P. Boronski, Alex Clark, Piotr Oprocha

*Summer Conference on Topology and Its Applications*

A compact metric space X is called *minimal* if it admits a minimal homeomorphism; i.e. a homeomorphism h:X→ X such that the forward orbit {h^{n}(x):n=1, 2, ...} is dense in X, for every x ∈ X. In my talk I shall outline a construction of a family of 1-dimensional minimal spaces from "A compact minimal space Y such that its square YxY is not minimal" whose existence answer the following long standing problem in the negative.

**Problem.** Is minimality preserved under Cartesian product in the class of compact spaces?

Note that for the fixed point property ...

Topology And Experimental Distinguishability, 2017 University of Michigan - Ann Arbor

#### Topology And Experimental Distinguishability, Gabriele Carcassi, Christine A. Aidala, David J. Baker, Mark J. Greenfield

*Summer Conference on Topology and Its Applications*

In this talk we are going to formalize the relationship between topological spaces and the ability to distinguish objects experimentally, providing understanding and justification as to why topological spaces and continuous functions are pervasive tools in the physical sciences. The aim is to use these ideas as a stepping stone to give a more rigorous physical foundation to dynamical systems and, in particular, Hamiltonian dynamics.

We will first define an experimental observation as a statement that can be verified using an experimental procedure. We will show that observations are not closed under negation and countable conjunction, but are closed under ...

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.

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.

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 ...}

Extension Theorems For Large Scale Spaces Via Neighbourhood Operators, 2017 University of Tennessee, Knoxville

#### Extension Theorems For Large Scale Spaces Via Neighbourhood Operators, Thomas Weighill, Jerzy Dydak

*Summer Conference on Topology and Its Applications*

Coarse geometry is the study of the large scale behaviour of spaces. The motivation for studying such behaviour comes mainly from index theory and geometric group theory. In this talk we introduce the notion of (hybrid) large scale normality for large scale spaces and prove analogues of Urysohn’s Lemma and the Tietze Extension Theorem for spaces with this property, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from ...

Entropy In Topological Groups, Part 2, 2017 University of Udine

#### Entropy In Topological Groups, Part 2, 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 ...

A New Class Of Dendrites Having Unique Second Symmetric Product, 2017 Universidad Autonoma del Estado de Mexico

#### A New Class Of Dendrites Having Unique Second Symmetric Product, David Maya, José G. Anaya, Fernando Orozco Zitli

*Summer Conference on Topology and Its Applications*

The second symmetric product of a continuum X, F_{2}(X), is the hyperspace consisting of all nonempty subsets of X having at most two points. A continuum X has unique hyperspace F_{2}(X) provided that each continuum Y satisfying that F_{2}(X) and F_{2}(Y) are homeomorphic must be homeomorphic to X. In this talk, a new class of dendrites having unique F_{2}(X) will be presented.