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 ...

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.

Cohen Reals And The Sequential Order Of Groups, 2017 Tennessee Technological University

#### Cohen Reals And The Sequential Order Of Groups, Alexander Shibakov

*Summer Conference on Topology and Its Applications*

We show that adding uncountably many Cohen reals to a model of diamond results in a model with no countable sequential group with an intermediate sequential order. The same model has an uncountable group of sequential order 2. We also discuss related questions.

Sequential Order Of Compact Scattered Spaces, 2017 University of North Carolina at Charlotte

#### Sequential Order Of Compact Scattered Spaces, Alan Dow

*Summer Conference on Topology and Its Applications*

A space is sequential if the closure of set can be obtained by iteratively adding limits of converging sequences. The sequential order of a space is a measure of how many iterations are required. A space is scattered if every non-empty set has a relative isolated point. It is not known if it is consistent that there is a countable (or finite) upper bound on the sequential order of a compact sequential space. We consider the properties of compact scattered spaces with infinite sequential order.

Spaces With No S Or L Subspaces, 2017 University of Wisconsin-Oshkosh

#### Spaces With No S Or L Subspaces, Joan Hart, Kenneth Kunen

*Summer Conference on Topology and Its Applications*

We show it consistent for spaces X and Y to be both HS and HL even though their product X ×Y contains an S-space. Recall that an S-space is a T_{3} space that is HS but not HL.

More generally, consider spaces that contain neither an S-space nor an L-space. We say a space is ESLC iff each of its subspaces is either both HS and HL or neither HS nor HL. The "C" in "ESLC" refers to HC; a space is HC iff each of its subspaces has the ccc (countable chain condition) (iff the space has no ...

Disjoint Infinity Borel Functions, 2017 University of Denver

#### Disjoint Infinity Borel Functions, Daniel Hathaway

*Summer Conference on Topology and Its Applications*

Consider the statement that every uncountable set of reals can be surjected onto R by a Borel function. This is implied by the statement that every uncountable set of reals has a perfect subset. It is also implied by a new statement D which we will discuss: for each real a there is a Borel function f_{a} : RtoR and for each function g : RtoR there is a countable set G(g) of reals such that the following is true: for each a in R and for each function g : R to R, if f_{a} is disjoint from g ...

On The Chogoshvili Homology Theory Of Continuous Maps Of Compact Spaces, 2017 Batumi Shota Rustaveli State University

#### On The Chogoshvili Homology Theory Of Continuous Maps Of Compact Spaces, Anzor Beridze, Vladimer Baladze

*Summer Conference on Topology and Its Applications*

In this paper an exact homology functor from the category **Mor**_{C} of continuous maps of compact Hausdorff spaces to the category **LES** of long exact sequences of abelian groups is defined (cf. [2], [3], [5]). This functor is an extension of the Hu homology theory, which is uniquely defined on the category of continuous maps of finite CW complexes and is constructed without the relative homology groups [9]. To define the given homology functor we use the Chogoshvili construction of projective homology theory [7], [8]. For each continuous map f:X → Y of compact spaces, using the notion of ...

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.

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 ...

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 ...

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.

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.

Some Applications Of The Point-Open Subbase Game, 2017 Universidad Autonoma Metropolitana - Iztapalapa

#### Some Applications Of The Point-Open Subbase Game, David Guerrero Sanchez

*Summer Conference on Topology and Its Applications*

Given a subbase S of a space X, the game PO(S,X) is defined for two players P and O who respectively pick, at the n-th move, a point xn 2 X and a set Un 2 S such that xn 2 Un . The game stops after the moves {xn, Un : n 2 !} have been made and the player P wins if the union of the Un’s equals X; otherwise O is the winner. Since PO(S,X) is an evident modification of the well-known point-open game PO(X), the primary line of research is to describe the ...

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 ∞.

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.

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.

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 ...

Properties Of Weak Domain Representable Spaces, 2017 University of Dayton

#### Properties Of Weak Domain Representable Spaces, Joe Mashburn

*Summer Conference on Topology and Its Applications*

We will explore some of the basic properties of weak domain representable (wdr) spaces, including hereditary properties and properties of products. In particular, we will construct a Baire space that is not wdr, show that products of wdr spaces are wdr, and demonstrate that the factors of a product that is wdr need not themselves be wdr. We will also show that if X is a wdr space and Y ⊆ X such that |Y|=|X| then Y is wdr. We can declare a subset of a wdr space X to be open or to consist of isolated points without losing ...