Digital Commons Network^™

Local Connectedness Of Bowditch Boundary Of Relatively Hyperbolic Groups, Ashani Dasgupta

Theses and Dissertations

If the Bowditch boundary of a finitely generated relatively hyperbolic group is connected, then, we show that it is locally connected. Bowditch showed that this is true provided the peripheral subgroups obey certain tameness condition. In this paper, we show that these tameness conditions are not necessary.

An Analysis And Comparison Of Knot Polynomials, Hannah Steinhauer

Senior Honors Projects, 2020-current

Knot polynomials are polynomial equations that are assigned to knot projections based on the mathematical properties of the knots. They are also invariants, or properties of knots that do not change under ambient isotopy. In other words, given an invariant α for a knot K, α is the same for any projection of K. We will define these knot polynomials and explain the processes by which one finds them for a given knot projection. We will also compare the relative usefulness of these polynomials.

A Structure Theorem For Bad 3-Orbifolds, Rachel Julie Lehman

Graduate Theses and Dissertations

We explicitly construct 10 families of bad 3-orbifolds, X , having the following property: given any bad 3-orbifold, O, it admits an embedded suborbifold X ∈ X such that after removing this member from O, and capping the resulting boundary, and then iterating this process finitely many times, you obtain a good 3-orbifold. Reversing this process gives us a procedure to obtain any possible bad 3-orbifold starting with a good 3-orbifold. Each member of X has 1 or 2 spherical boundary components and has underlying topological space S2 × I or (S2 × S1)\B3.

Clustering Methods For Gene Expression Data Of Oxytricha Trifallax, Kyle Houfek

USF Tampa Graduate Theses and Dissertations

Clustering is a data analysis method which is used in a large variety of research fields. Many different algorithms exist for clustering, and none of them can be considered universally better than the others. Different methods of clustering are expounded upon, including hierarchical clustering and k-means clustering. Topological data analysis is also described, showing how topology can be used to infer structural information about the data set. We discuss how one finds the validity of clusters, as well as an optimal clustering method, and conclude with how we used various clustering methods to analyze transcriptome data from the ciliate Oxytricha …

A Spider's Web Of Doughnuts, Daniel Stoertz

Graduate Research Theses & Dissertations

This dissertation studies an interplay between the dynamics of iterated quasiregular map-

pings and certain topological structures. In particular, the relationship between the Julia set

of a uniformly quasiregular mapping f : R 3 → R 3 and the fast escaping set of its associated

Poincaré linearizer is explored. It is shown that, if the former is a Cantor set, then the latter

is a spider’s web. A new class of uniformly quasiregular maps is constructed to which this

result applies. Toward this, a geometrically self-similar Cantor set of genus 2 is constructed.

It is also shown that for any …

Complete Bipartite Graph Embeddings On Orientable Surfaces Using Cayley Maps, Madeline Spies

Honors Program Theses

We explored how effective Cayley Maps are at embedding complete bipartite graphs onto orientable surfaces, such as spheres and tori. We embedded the graphs onto surfaces using Cayley Maps with the intent of finding rotations that result in the graphs’ optimal genera. Because there are only three groups that can be used to describe Kp,p, where p is prime, we chose to focus on this specific graph type. This paper explains how we determined the genera when using a Cayley Map, provides general theorems for surface face sizes and the Dihedral Group, and discusses our results for Kp,p up to …

Periodic Points On Tori: Vanishing And Realizability, Shane Clark

Theses and Dissertations--Mathematics

Let $X$ be a finite simplicial complex and $f\colon X \to X$ be a continuous map. A point $x\in X$ is a fixed point if $f(x)=x$. Classically fixed point theory develops invariants and obstructions to the removal of fixed points through continuous deformation. The Lefschetz Fixed number is an algebraic invariant that obstructs the removal of fixed points through continuous deformation. \[L(f)=\sum_{i=0}^\infty (-1)^i \tr\left(f_i:H_i(X;\bQ)\to H_i(X;\bQ)\right). \] The Lefschetz Fixed Point theorem states if $L(f)\neq 0$, then $f$ (and therefore all $g\simeq f$) has a fixed point. In general, the converse is not true. However, Lefschetz Number is a complete invariant …