Schedule

Henry Adams (University of Florida)

Hausdorff vs Gromov-Hausdorff distances

Hausdorff and Gromov-Hausdorff distances measure the distance between metric spaces. Though Hausdorff distances are easy to compute, Gromov-Hausdorff distances are not. When \(X\) is a sufficiently dense subset of a closed Riemannian manifold \(M\), we lower bound the Gromov-Hausdorff distance between \(X\) and \(M\) by 1/2 the Hausdorff distance between them. The constant 1/2 can be improved depending on the dimension and curvature of the manifold, and obtains the optimal value 1 (meaning the Hausdorff and Gromov-Hausdorff distances coincide) in the case of the circle. Our proofs convert discontinuous functions between metric spaces into simplicial maps between Vietoris-Rips and Čech complexes, and obstruct the existence of such maps using the nerve lemma and the fundamental class of the manifold. Joint with Florian Frick, Sushovan Majhi, Nicholas McBride, available at https://arxiv.org/abs/2309.16648.

Lori Alvin (Furman University)

Minimal Bounded Speedups of Toeplitz Flows

Given a minimal Cantor system \((X,T)\), a bounded topological speedup of \((X,T)\) is a dynamical system \((X,S)\) where \(S\) is a homeomorphism such that \(S(x) = T^{p(x)}(x)\) for some function \(p:X\to \mathbb{N}\). We assume the function \(p\) is continuous (and thus bounded) and the resulting system \((X,S)\) is minimal. One can ask what properties of the underlying initial system \((X,T)\) are preserved under minimal bounded speedups. We investigate the class of Toeplitz flows, which are minimal symbolic almost one-to-one extensions of odometers. Although the minimal bounded speedup of an odometer is always a conjugate odometer, we demonstrate that the minimal bounded speedup of a Toeplitz flow need not be Toeplitz. We then provide sufficient conditions to guarantee that the minimal bounded speedup will be a Toeplitz flow; in this case, it is never conjugate to the original Toeplitz flow but has the same underlying odometer. This is joint work with Silvia Radinger (University of Vienna).

Andrea Ammerlaan (Nipissing University)

A study on a tree-like continuum which does not have the fixed-point property

A continuum is tree-like if it can be expressed as an inverse limit on trees. In 1980, David Bellamy constructed the first example of a tree-like continuum which does not have the fixed-point property. Several others have been constructed since, most recently in 2018 by Rodrigo Hernández-Gutiérrez and Logan Hoehn. In this talk, I give an overview of the construction by Hernández-Gutiérrez and Hoehn and discuss some analysis of the mechanisms exhibited there. This is joint work with Logan Hoehn.

Alexander Blokh (University of Alabama, Birmingham)

Degree \(d\) Siegel-dendritic topological polynomials and their space

A topological Siegel-dendritic (“Side”) polynomial can be visualized in two steps. First, consider a \(d\)-to-\(1\) branched-covering map of a planar dendrite with finitely many periodic critical points of infinite (necessarily countable) order, no points of infinite order outside their grand orbits, and no wandering continua. Then replace periodic points of infinite order by disks subdivided by a circle with an irrational rotation together with a countable concatenation of its unfolding preimages. We suggest a way of parameterizing the family of all such maps. The talk is joint with Lex Oversteegen (Birmingham, AL) and Vladlen Timorin (Moscow).

Kazuhiro Kawamura (University of Tsukuba)

Some topologically transitive linear operators on infinite-dimensional spaces

No linear maps on finite-dimensional vector spaces are topologically transitive, while there are many topologically transitive bounded linear operators on infinite-dimensional Banach spaces. A two-sided weighted backward shift operator, with an appropriate weight, on the sequence space \(\ell^p(Z) (1\leq p < \infty)\) is an example of such operators.

On the basis of a theorem due to M. Pavone in 1992, we discuss topological transitivity of a weighted composition operator on the space \(L^{p}(\partial T)\) and \(C(\partial T)\), over the boundary \(\partial T\) of an infinite homogeneous tree \(T\) which is induced by a hyperbolic tree automorphism and a continuous weight. It turns out that the topological transitivity of such an operator is closely related to the values of the weight at the fixed points on the homeomorphism on \(\partial T\) associated with the automorphism. Some variants of topological transitivity are also discussed.

Ana Anušić (University of Zagreb)

Some results on entropy and commuting maps

I will present results from both previous and current work with Chris Mouron. We will begin by discussing how to obtain bounds on topological entropy of maps on inverse limits using their set-valued components, and when that can give an exact value of the entropy. This leads to interesting observations about the commutativity of (interval) maps, and raises several open questions which we will also explore.

Forrest Hilton (University of Alabama, Birmingham)

Modeling Bitransitive Hyperbolic Domains using the Quigs of \(\sigma_4^c\)

In this talk we discuss cubic polynomials and the circle maps \(\sigma_d(z)=z^d\). We provide some background on the combinatorial model of the boundary of the Cubic Principle Hyperbolic Domain, \(PHD_3\), the set of cubic polynomials with Jordan curve Julia sets. This model is centered around Quadratic Invariant Gaps, Quigs.

Many cubic hyperbolic domains are the same as \(PHD_3\). However, there are domains of cubic, Bitransitive polynomials. A Bitransitive polynomial is a cubic polynomial having an attracting periodic set whose first return map is a quartic polynomial; this quartic polynomial is the composition of two quadratic polynomials. The lamination of such a special quartic polynomial is said to be \(\sigma_4^c\)-invariant.

The focus of this talk is describing the Quigs of \(\sigma_4^c\).

Mikolaj Krupski (University of Warsaw)

Every Lindelöf scattered subspace of a \(\Sigma\)-product of real lines is \(\sigma\)-compact

We prove that every Lindelöf scattered subspace of a \(\Sigma\)-product of first-countable spaces is \(\sigma\)-compact. In particular, we obtain the result stated in the title. This answers some questions of Tkachuk from [Houston J. Math. 48 (2022), no. 1, 171–181].

John C. Mayer (University of Alabama, Birmingham)

Is the Combinatorial Structure of the Connectedness Locus for Cubic Polynomials a Tree?

Quadratic polynomials of the form \(z\mapsto z^2+c\) are represented by the complex plane \(\mathbb{C}\) for values of parameter \(c\in\mathbb{C}\), where \(c\) is the critical value of the quadratic polynomial. Every quadratic polynomial is represented up to conformal conjugacy in this parameter space. A subset of \(\mathbb{C}\), the quadratic connectedness locus, also called the Mandelbrot set, is the collection of all such polynomials with a connected Julia set. Equivalently, it is the set all polynomials whose critical point has bounded orbit, and consequently it is the intersection of a tower of closed disks in the plane. The analytic structure of the Mandelbrot set has been investigated by Mandelbrot and Douady/Hubbard since the 70's and others subsequently. The Mandelbrot set has often been described as the bifurcation locus for quadratic polynomials because of how it captures the change in Julia sets as the parameter is varied. The combinatorial structure of the quadratic bifurcation locus has been explicated by Thurston since the early 80's and others since. The combinatorial direction, the study of laminations of the unit disk, is an aid to increased understanding of the analytic direction. As Thurston showed, a model of the quadratic connectedness locus is the quadratic minor lamination, the QML. Consequently, the underlying combinatorial structure of the QML is a tree, the dual graph of the QML.

The corresponding parameter space for cubic polynomials of the form \(z\mapsto z^3-3bz^2+c\) is \(\mathbb{C}^2\) for \((b,c)\in\mathbb{C}^2\), where \(c\) is the critical value of the critical point \(0\) and \(2b\) is the other critical point. Every cubic polynomial is represented up to conformal conjugacy in this parameter space (actually twice). The cubic connectedness locus is the set of parameter values for which the corresponding Julia set is connected. Equivalently, as shown by Douady/Hubbard, it is the set of cubic polynomials for which both critical orbits are bounded, and consequently is the intersection of a tower of closed real \(4\)-balls in \(\mathbb{C}^2\). A start on the combinatorial structure of the cubic connectedness locus has been made by Oversteegen, Blokh, and others using cubic laminations of the disk. Our intention in this talk is to provide evidence that the answer to the title question is “no.” We will do this by showing that there is more than one combinatorial bifurcation path to the same cubic lamination, and consequently the same hyperbolic polynomial and Julia set.

Atish Mitra (Montana Technological University)

Coarse Geometry of Graphs and a Coarse Menger type Theorem

Coarse geometry has natural connections with graph theory, and recent works have found new connections and suggested some promising directions. One of these directions is exploring the relations between the coarse geometry of metric graphs and their combinatorial structures such as exclusion of minors.

We introduce the notion of coarse bottlenecking and skeletons in graphs and show how bottlenecking guarantees that a skeleton resembles (up to quasi-isometry) the original graph. We show how these tools can be used to simplify the structure of graphs that have an excluded asymptotic minor up to quasi-isometry, reducing it to a skeleton of the original containing no 3-fat minor. This gives partial resolution to a conjecture of Georgakopoulos and Papasoglu. We then show how the notion of bottlenecking provides an approach to coarsening the notion of connectedness in a graph in the form of a Coarse Menger-type theorem.

We also discuss some variants of asymptotic dimension of graphs and show how minor exclusion affects these dimensions.

This is based on joint work with Michael Bruner and Heidi Steiger.

Lex Oversteegen (University of Alabama, Birmingham)

Uniformizations of planar domains

It is known that for every Domain \(U\) (simply connected open sets in the plane containing \(0\) with non-degenerate boundary), there exist a conformal map \(\varphi_U:\mathbb{D}\to U\) so that \(\varphi_U(0)=0\). Moreover, this map is unique if \(\varphi'_U(0)>0\).

One should think of this result of equipping every Domain with polar coordinates. It is known that if \(U\) and \(V\) are “close” then, on compact subsets of \(\mathbb{D}\), \(\varphi_U\) is close to \(\varphi_V\). In this talk we will introduce alternative maps, defined only through metric properties of the boundary of \(U\), with similar properties and raise the question if they have stronger properties (i.e., if \(U\) and \(V\) are close in an appropriate sense, then the maps are close on the entire disk).

Paul Szeptycki (York University)

Ramsey theoretic variations of compactness

We consider strengthenings of countable compactness and sequential compactness via notions of convergence and limit points of sequences enumerated by \([\omega]^{<\omega}\). The Ramsey and Nash-Williams theorems give rise to new classes of compact and sequentially compact spaces. Examples and applications will be presented.

Žiga Virk (University of Ljubljana)

From continuum theory to topological data analysis

We will discuss the development of topological approximations at multiple scales over the past hundred years. We will start with the Warsaw circle as one of the foundational continua, and conclude with modern filtrations in topological data analysis. Along the way we will show how similar ideas emerged in geometric group theory, wild topology, and persistent homology. If time permits, we will explain how related structures emerged in the covering space theory.

Yihan Zhu (University of Windsor)

Vector-valued almost periodic functions and almost periodic equidistribution

In 1893, P. Bohl introduced the concept of \(\epsilon\)-almost periods, which H. Bohr later adopted in 1923 to define what we now call almost periodic functions. In the first part of this talk, we discuss vector-valued almost periodic functions that take values in quasi-complete locally convex spaces. We extend the notion of \(\epsilon\)-almost periods to vector-valued functions defined on general topological groups and study those that satisfy the left and right Bohr conditions. Additionally, we show the existence of an invariant mean for these almost periodic functions. In the second part, we introduce a generalized notion of equidistribution for continuous functions. We define equidistribution with respect to the almost periodic mean and present a generalized version of Weyl’s criterion for this type of equidistribution.

(University)