Hausdorff Dimension Of Kuperberg Minimal Sets, 2017 University of Illinois at Chicago

#### Hausdorff Dimension Of Kuperberg Minimal Sets, Daniel Ingbretson

*Summer Conference on Topology and Its Applications*

The Seifert conjecture was answered negatively in 1994 by Kuperberg who constructed a smooth aperiodic flow on a three-manifold. This construction was later found to contain a minimal set with a complicated topology. The minimal set is embedded as a lamination by surfaces with a Cantor transversal of Lebesgue measure zero. In this talk we will discuss the pseudogroup dynamics on the transversal, the induced symbolic dynamics, and the Hausdorff dimension of the Cantor set.

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 …

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 …

Entropy Of Induced Continuum Dendrite Homeomorphisms, 2017 Universidade Federal do Rio de Janeiro

#### Entropy Of Induced Continuum Dendrite Homeomorphisms, Jennyffer Bohorquez, Alexander Arbieto

*Summer Conference on Topology and Its Applications*

Let f: D → D be a dendrite homeomorphism. Let C(D) denote the hyperspace of all nonempty connected compact subsets of D endowed with the Hausdorff metric. Let C(f):C(D) → C(D) be the induced continuum homeomorphism. In this talk we sketch the proof of the following result: If there exists a nonrecurrent branch point then the topological entropy of C(f) is ∞.

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 a …

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 …

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.

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 …

Totally Geodesic Surfaces In Arithmetic Hyperbolic 3-Manifolds, 2017 Oberlin College

#### Totally Geodesic Surfaces In Arithmetic Hyperbolic 3-Manifolds, Benjamin Linowitz, Jeffrey S. Meyer

*Summer Conference on Topology and Its Applications*

In this talk we will discuss some recent work on the problem of determining the extent to which the geometry of an arithmetic hyperbolic 3-manifold M is determined by the geometric genus spectrum of M (i.e., the set of isometry classes of finite area, properly immersed, totally geodesic surfaces of M, considered up to free homotopy). In particular, we will give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial geometric genus spectrum and analyze the growth of the genera of minimal surfaces across commensurability classes. These results have applications to the study of …

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. …

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.

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.

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.

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.

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 …

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.

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(Ω) =⋂_Ω π F= …

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 …

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; etc.