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.

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.

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 ...

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 ...

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 ...

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 ...

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 Quasi-Uniform Box Products, 2017 North-West University (South Africa)

#### On Quasi-Uniform Box Products, Hope Sabao, Olivier Olela Otafudu

*Summer Conference on Topology and Its Applications*

In this talk, we preset the quasi-uniform box product, a topology that is finer than the Tychonov product topology but coarser than the uniform box product.

We then present various notions of completeness of a quasi-uniform space that are preserved by their quasi-uniform box product using Cauchy filter pairs.

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 ...

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 ...

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 ...

On Product Stability Of Asymptotic Property C, 2017 University of North Carolina at Greensboro

#### On Product Stability Of Asymptotic Property C, Gregory C. Bell, Andrzej Nagórko

*Summer Conference on Topology and Its Applications*

Asymptotic property C is a dimension-like large-scale invariant of metric spaces that is of interest when applied to spaces with infinite asymptotic dimension. It was first described by Dranishnikov, who based it on Haver's topological property C. Topological property C fails to be preserved by products in very striking ways and so a natural question that remained open for some 10+ years is whether asymptotic property C is preserved by products. Using a technique inspired by Rohm we show that asymptotic property C is preserved by direct products of metric spaces.

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 ∞.

Topologically Homogeneous Continua, Isometrically Homogeneous Continua, And The Pseudo-Arc, 2017 California State University, Sacramento

#### Topologically Homogeneous Continua, Isometrically Homogeneous Continua, And The Pseudo-Arc, Janusz Prajs

*Summer Conference on Topology and Its Applications*

We use accumulated knowledge on topologically homogeneous continua, and, in particular, on the pseudo-arc, to investigate the properties of isometrically homogeneous continua.

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 ...

Relationships Between Topological Properties Of X And Algebraic Properties Of Intermediate Rings A(X), 2017 California State University, Long Beach

#### Relationships Between Topological Properties Of X And Algebraic Properties Of Intermediate Rings A(X), Joshua Sack

*Summer Conference on Topology and Its Applications*

A topological property is a property invariant under homeomorphism, and an algebraic property of a ring is a property invariant under ring isomorphism. Let C(X) be the ring of real-valued continuous functions on a Tychonoff space X, let C^{*}(X) ⊆ C(X) be the subring of those functions that are bounded, and call a ring A(X) an *intermediate ring* if C^{*}(X) ⊆ A(X) ⊆ C(X). For a class Q of intermediate rings, an algebraic property P *describes* a topological property T among Q if for all A(X), B(Y) ∈ Q if A(X) and B(Y ...

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 ...

On Roitman's Principle For Box Products, 2017 Universidad Nacional Autonoma de Mexico

#### On Roitman's Principle For Box Products, Hector Alonso Barriga-Acosta

*Summer Conference on Topology and Its Applications*

One of the oldest problems in box products is if the countable box product of the convergent sequence is normal. It is known that consistenly (e.g., b=d, d=c) the answer is affirmative. A recent progress is due to Judy Roitman that states a combinatorial principle which also implies the normality and holds in many models.

Although the countable box product of the convergent sequence is normal in some models of b < d < c, Roitman asked what happen with her principle in this models. We answer that Roitman's principle is true in some models of b < d < c.

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 ...

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.