Schedule

Henry Adams (University of Florida)

Evasion paths in mobile sensor networks

Suppose ball-shaped sensors wander in a bounded domain. A sensor doesn't know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. Relative homology or zigzag persistence give a necessary condition, depending only on the time-varying connectivity data of the sensors. However, no method with time-varying connectivity data (i.e. Cech complexes) as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, the existence of an evasion path depends on more than just the fibrewise homotopy type of the region covered by sensors. This talk will end with two open questions:

(1) In the setting of planar sensors that measure the alpha complex and weak rotation information, there is a necessary and sufficient condition for the existence of an evasion path. What about with Cech complexes?

(2) How can the Michael selection theorem (which the speaker learned about at Nipissing in 2024) be used to build evasion paths, especially in the context when the multivalued function has nonconvex values?

Alexander Blokh (University of Alabama, Birmingham)

joint with L. Oversteegen and V. Timorin

Limits of cubic laminations

Let \(\sigma_3:\mathbb{S}\to \mathbb{S}\) be the \(3\)-tupling map of the unit circle. For sequences \(\{\mathcal{L}_i\}\) of \(\sigma_3\)-invariant dendritic laminations we describe limits \((\overline{c}', \overline{d}')\) of their critical portraits assuming that one such limit \(\mathcal{P}=(\overline{c}, \overline{d})\) is given. It turns out that if the endpoints of \(\overline{c}\) and \(\overline{d}\) are non-periodic then there is a unique lamination \(\mathcal{L}\) with finite critical sets such that \(\overline{c}'\) and \(\overline{d}'\) can be any couple of critical chords compatible with \(\mathcal{L}\). As the extreme opposite case we consider \(\mathcal{P}=(\overline{0 \frac13}, \overline{0 \frac23})\) and describe the corresponding countable family of possible critical portraits \((\overline{c}', \overline{d}')\) and associated with them distinct laminations. These results can be useful for the construction of a model of the cubic connectedness locus.

Michaela Carter (Lander University)

Multiplying Rabbits

This presentation will cover the main components of Julia sets and their laminations. Laminations are a topological way of modeling the dynamics of Julia sets. It begins with defining fundamental components of laminations. Next, images are provided in reference to these definitions, as well as some guidance on how to read them. Following this is a deeper explanation of how to identify the correct Julia set when given a lamination, focusing primarily on the Douady Rabbit Julia set and building off of it. More diagrams have been created to assist with visualization and stress the significance of making the correct identification. This joint work has been done with Brittany Duncan and Chase Worley.

Luke Cooper (Nipissing University)

An introduction to Thompson's group \(F\)

Thompson's group \(F\), introduced by Richard Thompson in unpublished notes from the mid-1960s, has become one of the central examples in geometric group theory. It is particularly striking because it is both easy to define and unexpectedly complex, while admitting multiple useful models that help to illuminate its behaviour. Thompson identified \(F\) early as a possible counterexample to the von Neumann conjecture, since it contains no nonabelian free subgroup of rank 2. Although the conjecture was later disproved by other counterexamples, the amenability of \(F\) is still an open question.

We will survey the standard definitions of \(F\), including its finite presentations and its characterizations in terms of piecewise-linear homeomorphisms of the unit interval and tree-pair diagrams. We will also discuss the subgroup structures that make \(F\) such a rich example, and address some of its finiteness properties; for instance, \(F\) contains arbitrarily large free abelian subgroups, which prevents it from having a finite-dimensional classifying space.

While the amenability question remains unresolved, it has motivated the study of many of the properties we will discuss and remains a central reason for the enduring interest in Thompson's group \(F\).

Brittany Duncan (University of North Georgia)

Hyperbolic Laminations To Their Julia Sets

This talk will discuss previous methods of finding Julia sets from hyperbolic laminations with the intention of setting the groundwork for talks by Michaela Carter, Kent McCathern, and Robert Risdon. We use Mathematica and Matlab to model and solve a system of equations that represent the lamination in order to find the unique corresponding Julia set. Issues surrounding this method will also be discussed.

Patrick Hartley (University of Alabama, Birmingham)

Connecting a simplicial structure of rotational sets to criticality

Common objects of study in complex dynamics are rotational sets. In this talk we review how Julia sets and rotational sets are represented in laminations; what it means for a rotational set to be unicritical or multicritical, both globally and locally; what it means for rotational sets to “go together”; and a known simplicial structure behind which sets “go together”.

All polynomial maps of degree \(d\) with a connected Julia set are known to be conjugate to a power map, \(z \mapsto z^d\) on the outside of the Julia set. We represent this as a \(d\)-fold map, \(\sigma_d\), from the circle to itself. An arc, or union of arcs, of the circle that maps one-to-one onto the circle is known as a critical sector. We represent rotational sets as finite polygons of chords of the circle whose vertices map to each other while preserving circular order. Rotational sets where all criticality sits on one side of the polygon are known as unicritical, and rotational sets that have criticality on more than one side are known as multicritical. It is possible for two rotational sets to go together in such a way that they can be treated as one larger rotational set.

Goldberg described a characterization for all rotational sets for polynomials of any given degree \(d\). Using this characterization, it has been shown by McMullen that there is a simplicial structure behind which rotational sets can go together, in the above sense. We discuss connecting this simplicial structure to known results regarding unicritical rotational sets, as well as extending them to include multicritical rotational sets.

Forrest Hilton (University of Alabama, Birmingham)

Computing Arbitrarily Infinite Invariant Gaps

Every lamination contains an invariant gap. These gaps are important to the boundary of the Principle Hyperbolic Domain and by renormalization, important to the boundary of every hyperbolic domain. In this talk we describe surjections onto sets of invariant gaps. We show how to concretely compute these in terms of \(d\)-nary digits.

This largely solves the question of correctly parametrizing invariant gaps. For a fixed degree, the global topology of invariant gaps is unclear; however the methods described in this talk allow us to homeomorphically parameterize any set of invariant gaps that are monotonically semi-conjugate to each other.

John C. Mayer (University of Alabama, Birmingham)

Combinatorial Structure of the Connectedness Locus for Cubic Polynomials

Understanding the dynamics of complex polynomials involves two approaches: understanding the Julia set for a given polynomial and understanding the parameter spaces of families of polynomials. In this talk we review some of what is known about the structure of the hyperbolic part of the parameter space of quadratic and cubic polynomials and reflect on the combinatorial connectedness of the “first layer” of hyperbolic parameter domains just adjacent to (and meeting) the principal hyperbolic domain of quadratic polynomials, unicritical cubic polynomials, and cubic polynomials in general.

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, Douady/Hubbard, and others since the 70's. 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 unicritical cubic polynomials of the form \(z\mapsto z^3+c\) is \(\mathbb C\) where \(c\) is the critical value of the polynomial. Schleicher studied these unicritical polynomials of degree \(d>2\) calling the subset of the parameter space which represented \(c\) values with connected Julia set, the \(d\)-multibrot set. In a thorough analogy with the Mandelbrot set, the combinatorial structure of each multibrot set is also a tree, the dual graph of a minor lamination more thoroughly described by Bhatacharya and co-authors. However, the combinatorial connectedness of the cubic connectedness locus for cubic polynomials, in general, is more complicated.

The corresponding parameter space of cubic polynomials of the form \(z\mapsto z^3-3bz^2+c\), representing up to conformal conjugacy all cubic polynomials, 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. 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. We show that there is more than one combinatorial bifurcation path to the same hyperbolic cubic lamination in the first layer outside the principal hyperbolic domain, and consequently, the combinatorial connectedness of it is a graph which is not a tree.

Kent McCathern & Robert Risdon (Lander University)

Investigation of Julia Sets

This presentation shows the process of finding a corresponding Julia Set from a particular lamination, as well as a look into previous research conducted and where we have developed further. The presentation delves deeper into the understanding of a specific Julia Set, the “Sea Turtle,” as well as into the periodic nature of Julia Sets. We will cover the manipulation of the Sea Turtle, aka the “\(d\)-Armed Turtle,” as well as a showcase of period points, pre-periodic and periodic orbits, as well as the Mathematica code used to dynamically generate the Julia Sets.

Desiree Paczay (Nipissing University)

An Introduction to Persistent Homology with Applications to Hyperspectral Imaging

Topological Data Analysis (TDA) has emerged as a powerful framework for extracting meaningful structure from complex, high-dimensional data. In particular, persistent homology is widely used for its ability to quantify multiscale topological features while exhibiting robustness to noise.

In this talk, I will introduce some of the mathematical background underlying TDA, with a particular focus on persistent homology. Topics will include simplicial complexes, filtrations, and persistence diagrams. The goal is to provide an accessible introduction to how topological information can be extracted from data and interpreted in practice.

Additionally, I will discuss some preliminary research which involves applying persistent homology to hyperspectral images of retinal tissue. Due to the high dimensionality of hyperspectral image data, TDA provides a promising framework for analysis and interpretation. We use persistence-based summaries in dimension 0 to study structural differences between albino and pigmented tissue, with the goal of informing more rigorous imaging procedures and evaluating whether these techniques can detect differences in pigmentation. Preliminary results suggest that these summaries may provide a useful approach for analyzing this type of data.

Chase Worley (Lander University)

Some Algebraic Conditions on Complex Polynomials and Their Associated Julia Sets

In this presentation, we will begin by discussing unicritical polynomials of the form \(p_c(z)=z^d+c\) where \(c\in \mathbb{C}\). We (re)derive the condition on \(c\) such that for each \(n\in\mathbb{N}\), \(p_c^{\circ n}(0)= \overbrace{(p_c\circ p_c\circ\cdots\circ p_c)}^{n\text{ times}}(0) = 0\). We are also able to give the exact number of solutions to this equation. We then provide examples of Julia sets realized by these complex polynomials for low degree, \(d\), and period, \(n\). If time provides, we will then extend from unicritical polynomials to symmetric cubic polynomials of the form \(p(z)=z^3+3cz\) along with some examples of Julia Sets realized by these polynomials of small periods.