Schedule

Henry Adams (University of Florida)

The connectivity of Vietoris-Rips complexes of spheres

For \(X\) a metric space and \(r>0\), the Vietoris-Rips simplicial complex \(VR(X;r)\) contains \(X\) as its vertex set, and a finite subset of \(X\) as a simplex if its diameter is less than \(r\). Though these complexes are frequently built in applied topology to approximate the "shape" of a dataset, their theoretical properties are poorly understood. Interestingly, the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, …, as the scale parameter increases. But little is known about Vietoris-Rips complexes of the \(n\)-sphere \(S^n\) for \(n\ge 2\).

We show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projective spaces. For \(\delta \ge 0\), suppose that the first nontrivial homotopy group of \(VR(S^n;\pi-\delta)\) occurs in dimension \(k\), i.e., suppose that the connectivity is \(k-1\). Then there exist \(2k+2\) balls of radius \(\delta\) that cover \(S^n\), and no set of \(k\) balls of radius \(\delta/2\) cover the projective space \(\mathbb{R}P^n\).

Joint work with Johnathan Bush and Žiga Virk.

John C. Mayer (University of Alabama at Birmingham)

Complex Dynamics: Polynomials, Julia Sets, Parameter Spaces, and Laminations

Laminations are a combinatorial and topological way to study connected Julia sets of polynomials. While each locally connected Julia set has a corresponding lamination, laminations also give information about the structure of parameter space of degree \(d\ge 2\) polynomials with connected Julia sets. A \(d\)-invariant lamination of the unit disc consists of a closed collection of chords, called leaves, which meet at most at their endpoints, and which is forward and backward invariant under the angle-\(d\)-tupling map on the unit circle. Of particular interest are leaves in a lamination which are periodic, return for the first time by the identity, and whose endpoints are in different orbits. Such leaves play an important and understood role in the parameter space of quadratic polynomials and in the parameter space of unicritical higher degree polynomials, but more study is needed in the more general case of multiple criticality. Here we focus on the first case where there are open questions about the laminations: the angle tripling map corresponding to degree \(3\) polynomials with connected Julia set.

Coauthors: Brittany E. Burdette and Thomas C. Sirna

Forrest Hilton (University of Alabama at Birmingham)

Structure of the Pullback Tree

A lamination is a closed set of chords of the unit disk such that no two chords intersect in the open disk. Laminations are often used as a model of the Julia set of a degree \(d\) polynomial. There are well-defined classes of laminations that model polynomials. Essentially those laminations are \(d\) to \(1\) invariant under the degree \(d\) covering map \(\sigma_d: S \to S\). The chords, also called leaves, of the lamination are mapped by the covering map according to how their end points move.

A polynomial will have an infinite lamination, but since a Lamination is just a set of chords of a disc, it is possible to contemplate the set of polynomials which share a finite lamination as a subset of their lamination. For this purpose, we define a class of laminations which we call Finite Dynamic Laminations or FDL, and we arrange these laminations into a tree called the pullback tree. Any polynomial matching an FDL must match at least one of its children in a pullback tree.

In this talk, we will discuss the structure and meaning of the pullback tree. We attempt to discuss the limits of different kinds of sequences of FDL. We welcome audience questions to help guide our summer research project.

Jernej Činč (University of Maribor & ICTP Trieste)

Interval maps with dense set of periodic points

This talk is based on a recent study of the class of interval maps with dense set of periodic points CP and its closure Cl(CP) equipped with the metric of uniform convergence where we proved the following results:

1.) Cl(CP) is dynamically characterized as the set of interval maps for which every point is chain-recurrent.

2.) Topological exactness (or leo property) is attained on the open dense set of maps in CP.

3.) Every second category set in both CP and Cl(CP) contains uncountably many conjugacy classes.

Ali Emre Eysen (Nipissing University)

On Continuous Surjections Between Function Spaces

We consider uniformly continuous surjections between \(C_p(X)\) and \(C_p(Y)\) (respectively, \(C_p^*(X)\) and \(C_p^*(Y)\)) and show that if \(X\) has some dimensional-like properties, then so does \(Y\). In particular, we prove that if \(T:C_p(X)\to C_p(Y)\) is a continuous linear surjection, then \(\dim Y=0\) if \(\dim X=0\). This provides a positive answer to a question raised by Kawamura-Leiderman.

Coauthor: Vesko Valov

Ana Anušić (Nipissing University)

Generalization of the Nadler-Quinn problem and the Knaster continuum in the plane

A continuum is called arc-like if it is an inverse limit on arcs, or, equivalently, if it can be covered by an arbitrary small chain. If \(X\) is a continuum in the plane, then a point \(x\in X\) is called accessible if there is an arc \(A\subset\mathbb{R}^2\) such that \(A\cap X=\{x\}\). It is well-known that every arc-like continuum can be embedded in the plane, but, if a continuum is complicated enough, every such embedding necessarily leaves a lot of points inaccessible.

In 1972, Nadler and Quinn asked if for every arc-like continuum \(X\), and \(x\in X\), there exists a planar embedding of \(X\) in which \(x\) is accessible. The question was recently answered in positive (Ammerlaan, AA, Hoehn 2024). The Nadler-Quinn question can be naturally generalized as follows.

Given an arc-like continuum \(X\), \(n\in\mathbb{N}\), and \(x_1, \ldots, x_n\in X\), is there is a planar embedding of \(X\) in which all the points \(x_1, \ldots, x_n\) are accessible? In this talk, we will give some conditions when such an embedding exists, and then apply them to show that for every \(n\in\mathbb{N}\), the two-fold Knaster continuum has a planar embedding with (at least) \(n\) accessible composants.

Coauthor: Logan Hoehn

Silvia Radinger (University of Vienna)

Interval Translation Maps: Renormalization and Weak Mixing

In 2003 H. Bruin and S. Troubetzkoy studied a renormalization map for a two-parameter family of interval translation maps. For a non-typical subset of the parameter space the interval translation map has a Cantor attractor. The renormalization G is a procedure similar to the Rauzy induction. It acts as dynamics on the parameter space and can be used to find the attractor. In this talk we further study these systems focusing on weak mixing. We look at the symbolic representation of the interval translation map to define a S-adic subshift and use results about the eigenvalues of Bratteli-Vershik systems to determine whether the interval translation map is weakly mixing. Additionally we characterize the subset of linearly recurrent interval translation maps and their eigenvalues.

Coauthor: Henk Bruin

Judy Kennedy (Lamar University)

Chris Mouron (Rhodes College)

Dynamical properties of the Lelek fan

We have been investigating the dynamical properties of the Lelek fan for a couple of years now, and we continue this investigation. The Lelek fan admits surprisingly complex dynamics – it admits postive entropy maps, transitive maps, and mixing maps.

Coauthors: Iztok Banic, Goran Erceg, Chris Mouron, Van Nall

Lori Alvin (Furman University)

Countable Shifts on Compact Alphabets

We investigate dynamical properties of set-valued upper semicontinuous functions on compact countable sets. Each of these functions can be modeled as shift spaces over a countable alphabet, but unlike shifts of finite type or shifts of finite order over \(\mathbb{N}\), the shifts are not guaranteed to have shadowing. We provide necessary and sufficient conditions for such an usc function to have shadowing. We also investigate other dynamical properties including topological transitivity and topological mixing.

Paweł Krupski (Wrocław University of Science and Technology)

On hyperspaces of knots

New properties of Vietoris hyperspaces of simple closed curves in \(\mathbb{R}^2\) or in \(\mathbb{R}^3\) will be discussed. The hyperspaces are locally contractible. Moreover, the hyperspace of polygonal knots is a \(\sigma\)-compact, strongly countable-dimensional ANR and the hyperspace of tame knots is an absolute Borel, strongly infinite-dimensional Cantor manifold.

Coauthor: K. Omiljanowski

Žiga Virk (University of Ljubljana)

Uniform maps of spaces of persistence diagrams into the Hilbert space

The stability theorem of persistent homology for Rips and Cech complexes states that persistent homology is a 1-Lipschitz map from the collection of metric spaces into the space of persistence diagrams. In this context, the space of persistence diagrams equipped with the (non-Euclidean) bottleneck metric. In order to apply statistical tools or further data analytic techniques to collections of persistence diagrams, we thus need a map from the space of persistence diagrams into a Euclidean or Hilbert space. Dozens of such maps have been proposed in the past decade, including persistence landscape and persistence images. These maps are typically Lipschitz. However, none of them has explicit lower bounds on the distortion of distances and hence they provide no control on the loss of information. In this talk we will present Lipschitz maps from certain spaces of persistence diagrams into Hilbert and Euclidean spaces with explicit distortion functions. Some of these maps will be uniform embeddings. The maps are fairly geometric, consisting essentially of bottleneck distances to specific landmark diagrams, and are thus easily implementable. The idea for the construction comes from the quantification of classical constructions in dimension theory.

Joint work with Atish Mitra (Montana Tech, MT, USA).

Ihor Stasyuk (Thompson Rivers University)

Extension of functions and metrics with variable domains

The problem of extension of maps is fundamental in Mathematics. Extension theorems are extremely important in topology, functional analysis, differential equations and other disciplines. We outline some classical approaches for extending continuous real-valued functions on fixed domains and then discuss improvements of these techniques for simultaneous extensions. In the latter case maps (functions, or more specifically, metrics) to be extended are defined on variable subsets of metric spaces.

Thomas Sirna (University of Alabama at Birmingham)

No Wandering Triangles and No Identity Return Triangles in Symmetric Laminations

Laminations of the unit disc are, most basically, a closed collection of chords inside the unit disc which can be used to study the dynamics of complex polynomials. Depending on the specific object of interest – Julia sets, parameter spaces, etc. – more conditions are given to the laminations besides the basic requirement. In this talk we specialize to Symmetric Laminations of degree \(d\), which are laminations exhibiting a \(\frac{2\pi}{d-1}\) rotational symmetry. As one may guess, these laminations will help describe symmetric polynomials, which we define as polynomials whose locally connected Julia sets exhibit \(\frac{2\pi}{d-1}\) rotational symmetry.

For a lamination to give insight into Julia sets they must also satisfy a few invariant conditions under the map \(\sigma_d\) – the "angle \(d\)-tupling map". Our goal is to prove that for this map \(\sigma_d\), neither wandering triangles nor identity return triangles can exist in symmetric laminations, and we will interpret these results as the locally connected Julia set of a symmetric polynomial can neither have a wandering cutpoint of order 3 nor a periodic cutpoint of order 3 that returns without rotation.

We do this by first proving the Symmetric Critical Strip Lemma, which is analogous to Thurston's Central Strip Lemma for degree 2 laminations, before using it to prove the main results. "No Wandering Triangles" and "No Identity Return Triangles" are important results for general degree 2 laminations, so the fact that these results hold for symmetric laminations of all degrees makes their behavior very similar to laminations of degree 2. We will emphasize this throughout.

Andrea Ammerlaan (Nipissing University)

Embedding certain triod-like continua in the plane

In 1972, Nadler and Quinn asked if any arc-like continuum \(X\) can be embedded in the plane with some arbitrary point, \(x \in X\), accessible. The question was recently answered in the positive (AA, Anušić, Hoehn 2024). This talk will discuss a consequence of the result, namely that if \(Y\) is a continuum such that \(Y = R \cup X\), where \(X\) is an arc-like continuum, \(R\) is a ray, \(R \cap X = \emptyset\), and \(\overline{R} \setminus R \subseteq X\), then \(Y\) can be embedded in the plane.

Joint work with Logan Hoehn.