Schedule

Henry Adams (Colorado State University)

Introduction to applied topology

This talk is an introduction to applied and computational topology. I will introduce Cech (nerve) simplicial complexes, Vietoris-Rips (clique) simplicial complexes, persistent homology, the stability theorem, and how these techniques can be used to approximate the shape of a dataset when given only a finite sample. A motivating example is the conformation space of the cyclo-octane molecule, which is a Klein bottle glued to a 2-sphere along two circles of singularity.

Henry Adams (Colorado State University)

Vietoris-Rips complexes and thickenings

Given a metric space \(X\) and a scale parameter \(r\), the Vietoris-Rips simplicial complex \(VR(X;r)\) has \(X\) as its vertex set, and contains a finite subset as a simplex if its diameter is at most \(r\). If a point cloud is sampled from a manifold, then as more samples are drawn, the Vietoris-Rips complex of the point cloud converges to the Vietoris-Rips complex of the manifold. But what are the homotopy types of Vietoris-Rips complexes of manifolds? The Vietoris-Rips complexes of the circle, as the scale \(r\) increases, obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until they are finally contractible. Only a little is understood about the homotopy types of the Vietoris-Rips thickenings of the \(n\)-sphere. I will explain connections to optimal transport, and why a new Morse theory is needed.

Henry Adams (Colorado State University)

Bridging applied and quantitative topology

The Gromov-Hausdorff distance between two metric spaces is an important tool in geometry, but it is difficult to compute. I will show how to provide new lower bounds on the Gromov-Hausdorff distance between unit spheres of different dimensions by combining Vietoris-Rips complexes with Borsuk-Ulam theorems. This is joint work with my 15 co-authors at https://arxiv.org/abs/2301.00246. Many questions remain open!

Hugo Adrian Maldonado Garcia (Universidad Nacional Autónoma de México)

Complexity of \(n\)-od-like continua

W. Lewis asked in "Indecomposable Continua. Open problems in topology II" whether there exists, for every \(n\geq 2\), an atriodic simple \((n+1)\)-od-like continuum which is not simple \(n\)-od-like and, if such continuum exists, whether it has a variety of properties such as being planar or being an arc-continuum, among others. Some partial results have been obtained by W.T. Ingram, P. Minc, C.T. Kennaugh and L. Hoehn. In each case, the most substantial challenge is in proving that a continuum is not \(T\)-like, for a given tree \(T\). We present the notion of a combinatorial \(n\)-od cover of a graph, a tool which may enable one to prove that certain examples of continua are not \(n\)-od-like. Also, we suggest the construction of an atriodic simple \((n+1)\)-od-like continuum which is not simple \(n\)-od-like which is a planar, arc-continuum and has span zero (This is a work in progress).

Mariam Alhawaj (University of Toronto)

Generalized pseudo-Anosov maps and Hubbard trees

The Nielsen-Thurston classification of the mapping classes proved that every orientation preserving homeomorphism of a closed surface, up to isotopy is either periodic, reducible, or pseudo-Anosov. Pseudo-Anosov maps have particularly nice structure because they expand along one foliation by a factor of \(\lambda >1\) and contract along a transversal foliation by a factor of \(\frac{1}{\lambda}\). The number \(\lambda\) is called the dilatation of the pseudo-Anosov. Thurston showed that every dilatation \(\lambda\) of a pseudo-Anosov map is an algebraic unit, and conjectured that every algebraic unit \(\lambda\) whose Galois conjugates lie in the annulus \(A_\lambda = \{z: \frac{1}{\lambda} < |z| < \lambda\}\) is a dilatation of some pseudo-Anosov on some surface \(S\).

Pseudo-Anosov maps have a huge role in Teichmuller theory and geometric topology. The relation between these and complex dynamics has been well studied inspired by Thurston.

In this project, I develop a new connection between the dynamics of quadratic polynomials on the complex plane and the dynamics of homeomorphisms of surfaces. In particular, given a quadratic polynomial, we show that one can construct an extension of it which is generalized pseudo-Anosov homeomorphism. Generalized pseudo-Anosov means the foliations have infinite singularities that accumulate on finitely many points. We determine for which quadratic polynomials such an extension exists. My construction is related to the dynamics on the Hubbard tree which is a forward invariant subset of the filled Julia set that contains the critical orbit.

Andrea Ammerlaan (Nipissing University)

Accessibility of points of chainable continua

This talk will discuss the Nadler-Quinn problem. Posed in 1972, the problem asks if, given any chainable continuum \(X\) and any point \(x\) in \(X\), we can embed \(X\) in the plane with \(x\) accessible. In 2001, Minc put forward a candidate for a counterexample to the problem - Minc's Continuum. However, Anušić proved in 2018 that it was not, in fact, a counterexample. I will give an overview of this proof and hint at our more recent approach to the problem.

Ana Anušić (Nipissing University)

Accessibility of points of chainable continua, continued

Continuing on Andrea Ammerlaan's talk, we will discuss recent progress on the Nadler-Quinn problem. We restrict our attention to chainable continua which can be represented as inverse limits with simplicial bonding maps on the interval. With some technical restrictions, we show that for every point \(x\) of such a chainable continuum \(X\), there exists a planar embedding of \(X\) in which \(x\) is accessible. This is a joint work with Andrea Ammerlaan and Logan Hoehn (Nipissing University).

Alexander Blokh (University of Alabama at Birmingham)

On the Sharkovsky Theorem and some of its developments and interpretations

Dedicated to the memory of Oleksandr Mykolayovych Sharkovs'ky.

We will discuss the coexistence of periods of cycles for interval maps discovered by Sharkovs'ky and consider two ways to view it: one based on over-rotation numbers and the other one related to the structural properties of interval cycles. The speaker will also make some historical remarks.

Brittany Burdette (University of Alabama at Birmingham)

The global and local correspondence between unicritical rotational polygons and maximally multicritical rotational polygons

The purpose of this research is to establish the correspondence between locally unicritical rotational polygons and locally maximally rotational multicritical polygons in a lamination. Laminations are a topological and combinatorial way of modeling complex polynomials. Locally unicritical polygons have an all critical polygon associated with them that is strictly less than the global degree. The correspondence in the global case has been previously shown in [1]. The Lavaur’s Algorithm which is an integral part of the correspondence will also be discussed.

[1] Brittany Burdette, Caleb Falcione, Cameron Hale, John Mayer. 2022. Unicritical and Maximally Critical Laminations. Under review: Contemporary Mathematics Proceedings of AMS

Jacob Cleveland (Colorado State University)

Tropical routing and topological star tracking

In this talk, we describe two important applications of pure mathematics to NASA engineering problems. The first applies the framework of tropical geometry to routing in time-varying networks. This approach extends classical Dijkstra's algorithm on a static graph by doing computations in a polynomial semiring. Semirings have been shown to have the flexibility necessary to properly model time-varying networks. The second applies a topological localization version of the Whitney Embedding theorem to star tracking, the process of determining pointing direction based on a view of stars. This is possible because bounds can be put on the number of observations necessary to determine one's location in a manifold.

Mitch Haslehurst (University of Victoria)

Constructing examples of classifiable \(C^\ast\)-algebras through topological dynamics

Since the mid 20th century, \(C^\ast\)-algebras have exploded into a huge area of research that have found their way into many areas of mathematics. One of these areas is topological dynamical systems, and it is an interesting question to ask, given a \(C^\ast\)-algebra, whether or not it can be constructed from an dynamical algebraic object known as a groupoid, and K-theory is a very useful tool when attempting to find such a groupoid. I will begin this presentation with a short introduction to \(C^\ast\)-algebras, K-theory, groupoids, and Bratteli diagrams. I will then describe how to construct a certain quotient space of the path space of a Bratteli diagram to create a factor groupoid whose \(C^\ast\)-algebras has predetermined K-theory data, and whose unit spaces tend to exhibit self-similar behaviour.

Forrest Hilton (University of Alabama at Birmingham)

Counting sibling portraits

We count the sibling portraits of a rotational polygon with \(n\) sides in degree \(d\). A polygon in this context is a set of \(n\) points on the unit circle that are connected with chords. Sibling portraits are created by duplicating the original points across the unit circle, and then joining them into polygons that correspond one-to-one in circular order with the original. Degree describes the number of times that each point of the original is duplicated. The problem is to count the number of ways to join those points such that the chords do not cross.

We conjectured that the unicritical case is counted by the Fuse-Catalan numbers, and eventually proved this by describing a bijection with \(n\)-ary trees with \(d-1\) internal vertices. We rediscovered a recurrence for the Fuse-Catalan numbers that is documented for the application of \(n\)-ary trees, and this recurrence relation describes how to count non-unicritical cases.

Computational methods were essential in forming the original conjecture, and many of the findings are calling out to be animated. We not only reimplemented the existing computational infrastructure for laminations of the unit disk, but added animation capabilities. This presentation will demonstrate my visualization program, which will be made public.

The original problem arose from the use of laminations of the unit disk as a combinatorial and topological model of the (connected) Julia sets of polynomials of degree \(d\). The rotational polygons correspond to some of the fixed points of the corresponding polynomial.

Krystyna Kuperberg (Auburn University)

On a construction of a class of continua

I will present an interesting, fairly easy to describe class of 1-dimensional, non-locally connected continua exhibiting a certain kind of symmetry. Among the properties to investigate are:

1. Indecomposability

2. The Mittag-Leffler condition

3. Movability in the sense of Borsuk

4. Homogeneity properties

5. Mappings onto graphs

It is known that every arc component in these continua is dense. S. Hurder and A. Rechtman have shown that similar 2-dimensional continua are Mittag-Leffler, although not shape stable. It is not clear if they are movable. For comparison, solenoids are not movable, not Mittag-Leffler. The Case-Chamberlin continuum is not movable, but it is Mittag-Leffler. The Denjoy continua are movable and thus Mittag-Leffler.

John C. Mayer (University of Alabama at Birmingham)

Pullback laminations for Julia sets of complex polynomials

The Julia set of a complex polynomial of degree \(d \ge 2\) is a fully invariant subset of the complex plane under iteration of the polynomial. It is the fractal set of points on which the polynomial is "chaotic." Laminations of the unit disc consist of non-crossing chords, called leaves, and are a topological, geometric, and combinatorial way to study the connected Julia sets of polynomials. Our focus will be on polynomials and laminations of degree \(> 2\). We will discuss how to determine its corresponding lamination given a polynomial Julia set, via a version of the Riemann Mapping Theorem and external rays. More to the point, we will show how to find interesting laminations, show that a polynomial exists whose Julia set has that lamination, and explore ways to determine specific polynomials that have that lamination. The latter route, from lamination to polynomial, begins with defining a "pullback lamination" given initial data about the behavior of periodic leaves and criticality. Brittany Burdette, Thomas Sirna, and Md. Abdul Aziz will talk in this context about their related work.

John C. Mayer (University of Alabama at Birmingham)

joint with Md. Abdul Aziz (University of Alabama at Birmingham)

Fixed point portraits for laminations of the unit disc

Laminations are a combinatorial and topological model for studying the Julia sets of complex polynomials. Every complex polynomial of degree \(d\) has \(d\) fixed points. From the point of view of laminations at most \(d-1\) of these fixed points are peripheral (approachable by "external rays" from outside the Julia set of the polynomial). Hence, at least one of the \(d\) fixed points is "hidden" from the laminational point of view. The purpose of this study is to identify, classify and count the possible fixed point portraits for a lamination of degree \(d\). We will identify the "simplest" lamination for a given fixed point portrait and will show that there are polynomials that have these canonical simplest laminations. This work is connected to that of Brittany Burdette on "locally unicritical laminations."

Lex Oversteegen (University of Alabama at Birmingham)

On the parameter space of cubic symmetric polynomials

In this talk we will give a full combinatorial description of the boundary of the connectedness locus CSP of cubic symmetric polynomials of the form \(P(z)=Z^3+cz\). We will show that such polynomials can be associated to symmetric laminations of the unit disc. These laminations can be parametrized by a single lamination CCL (the cubic co-major lamination) in the unit disc \(D\). We show that there exists a monotone map \(\pi\) from the boundary of CSP to the quotient space \(D\)/CCL which is a homeomorphism if CSP is locally connected. Moreover, we provide an algorithm for constructing a dense set of leaves of CCL.

These results were inspired by Thurston's description of the quadratic connectedness locus and are joint work with A. Blokh, V. Timorin, N. Selinger and S. Vejandla.

Habibeh Pourmand (Jagiellonian University)

The mean orbital pseudo-metric in topological dynamics

We study properties and applications of the mean orbital pseudo-metric \(\overline{\rho}\) on a topological dynamical system \((X,T)\) defined by \[ \overline{\rho}(x,y) = \limsup_{n \to \infty} \min_{\sigma \in S_n} \frac{1}{n} \sum_{k=0}^{n-1} d \left( T^k(x), T^{\sigma(k)}(y) \right) ,\] where \(x,y \in X\), \(d\) is a metric for \(X\), and \(S_n\) is the permutation group of the set \(\{0,1,\ldots,n-1\}\). Writing \(\hat{\omega}(x)\) for the set of \(T\)-invariant measure generated by the orbit of a point \(x \in X\), we prove that the function \(x \mapsto \hat{\omega}(x)\) is \(\overline{\rho}\) uniformly continuous. This allows us to characterise equicontinuity with respect to the mean orbital pseudo-metric (\(\overline{\rho}\)-equicontinuity) and connect it to such notions as uniform or continuously pointwise ergodic systems studied recently by Downarowicz and Weiss. This is joint work with F. Cai, D. Kwietniak, and J. Li.

Thomas Sirna (University of Alabama at Birmingham)

A 2-to-1 correspondence between multi-critical-moment Identity Return Triangles and Base Leaves in a \(\sigma_3\)-invariant lamination.

Laminations of the unit disc were introduced by Thurston as a way to study the dynamics of polynomials with locally connected Julia set, with the degree \(d\) of the polynomial corresponding to the map \(\sigma_d\) on the unit circle \(\mathbb{S}\). As we increase the degree beyond 2 — and thus go beyond the standard Mandelbrot set — the degree of our parameter space increases. To compensate, a common method is to study only those polynomials whose laminations contain certain laminational phenomena. This talk shows 2-to-1 correspondence between multi-critical-moment Identity Return Triangles (MCM IRTs) and Base Leaves in invariant \(\sigma_3\) laminations. IRTs and Base Leaves will be linked by points called Co-Roots, which can be theoretically shown to always exist in our situation.

E. D. Tymchatyn (University of Saskatchewan)

From partitioning to length spaces and convex metrics

Every metric space which admits an equivalent convex metric is connected and locally arc connected. It is a longstanding open question whether the converse is true. Moise (1949) and Bing (1949) proved the converse for compact spaces and Tominaga and Tanaka (1956) did the same for locally compact spaces. Nikiel, Stasyuk, Tuncali and Tymchatyn (NSTT) in an unpublished manuscript (circa 2010) outlined a proof of the converse for spaces with property S. Building on the principal idea in (NSTT) we show that a connected, locally arc connected, metric space \(X\) which admits a defining sequence of simultaneous brick core partitions \(\{U_i \}\) of \(X\) and associated graphs \(\{ G_i \}\) is a length space. If these partitions are finite then \(X\) admits an equivalent convex metric. This together with a forthcoming paper of Stasyuk and Tymchatyn on partitioning gives a hopefully more accessible proof of the main result in (NSTT). This also answers a question in Mathoverflow (2013?) on whether Niemytzki's tangent disc half plane with all but countably many points on the \(x\)-axis removed admits a length metric or convex metric.