Physical Sciences and Mathematics Commons^™

A Cohomological Perspective To Nonlocal Operators, Nicholas White

Honors Theses

Nonlocal models have experienced a large period of growth in recent years. In particular, nonlocal models centered around a finite horizon have been the subject of many novel results. In this work we consider three nonlocal operators defined via a finite horizon: a weighted averaging operator in one dimension, an averaging differential operator, and the truncated Riesz fractional gradient. We primarily explore the kernel of each of these operators when we restrict to open sets. We discuss how the topological structure of the domain can give insight into the behavior of these operators, and more specifically the structure of their …

Echolocation On Manifolds, Kerong Wang

Honors Theses

We consider the question asked by Wyman and Xi [WX23]: ``Can you hear your location on a manifold?” In other words, can you locate a unique point x on a manifold, up to symmetry, if you know the Laplacian eigenvalues and eigenfunctions of the manifold? In [WX23], Wyman and Xi showed that echolocation holds on one- and two-dimensional rectangles with Dirichlet boundary conditions using the pointwise Weyl counting function. They also showed echolocation holds on ellipsoids using Gaussian curvature.

In this thesis, we provide full details for Wyman and Xi's proof for one- and two-dimensional rectangles and we show that …

A Fractal Geometry For Hydrodynamics, Jonah Mears

Honors Theses

Experiments have shown that objects with uneven surfaces, such as golf balls, can have less drag than those with smooth surfaces. Since fractal surfaces appear naturally in other areas, it must be asked if they can produce less drag than a traditional surface and save energy. Little or no research has been conducted so far on this question. The purpose of this project is to see if fractal geometry can improve boat hull design by producing a hull with low friction.

Exploration Of Piccirillo's Trick On Low Crossing Number Knots, Gabriel Adams

Honors Theses

Piccirillo recently discovered a process that can be applied to an unknotting number one knot to convert it into a different knot called a Piccirillo dual. Piccirillo duals have been shown to have the same n-trace and the same sliceness. However, exploration and knowledge of this process is limited. We were able to generate the Piccirillo duals for several low-crossing number knots. We offer the foundation for and explain how to follow the Piccirillo process and generate Piccirillo duals. This talk assumes little knowledge of knot theory and concisely gives newcomers a clear introduction to get started working with Piccirillo …

Decomposing Manifolds In Low-Dimensions: From Heegaard Splittings To Trisections, Suixin "Cindy" Zhang

Honors Theses

The decomposition of a topological space into smaller and simpler pieces is useful for understanding the space. In 1898, Poul Heegaard introduced the concept of a Heegaard splitting, which is a bisection of a 3-manifold. Heegaard diagrams, which describe Heegaard splittings combinatorially, have been recognized as a powerful tool for classifying 3-manifolds and producing important invariants of 3-manifolds. Handle decomposition, invented by Stephen Smale in 1962, describes how an n-manifold can be constructed by successively adding handles. In 2012, Gay and Kirby introduced trisections of 4-manifold, which are a four-dimensional analogues of Heegaard splittings in dimension three. Trisection diagrams give …

Classification Of Torsion Subgroups For Mordell Curves, Zachary Porat

Honors Theses

Elliptic curves are an interesting area of study in mathematics, laying at the intersection of algebra, geometry, and number theory. They are a powerful tool, having applications in everything from Andrew Wiles’ proof of Fermat’s Last Theorem to cybersecurity. In this paper, we first provide an introduction to elliptic curves by discussing their geometry and associated group structure. We then narrow our focus, further investigating the torsion subgroups of elliptic curves. In particular, we will examine two methods used to classify these subgroups. We finish by employing these methods to categorize the torsion subgroups for a specific family of elliptic …

Category Theory And Universal Property, Niuniu Zhang

Honors Theses

Category theory unifies and formalizes the mathematical structure and concepts in a way that various areas of interest can be connected. For example, many have learned about the sets and its functions, the vector spaces and its linear transformation, and the group theories and its group homomorphism. Not to mention the similarity of structure in topological spaces, as the continuous function is its mapping. In sum, category theory represents the abstractions of other mathematical concepts. Hence, one could use category theory as a new language to define and simplify the existing mathematical concepts as the universal properties. The goal of …

Introduction To Computational Topology Using Simplicial Persistent Homology, Jason Turner, Brenda Johnson, Ellen Gasparovic

Honors Theses

The human mind has a natural talent for finding patterns and shapes in nature where there are none, such as constellations among the stars. Persistent homology serves as a mathematical tool for accomplishing the same task in a more formal setting, taking in a cloud of individual points and assembling them into a coherent continuous image. We present an introduction to computational topology as well as persistent homology, and use them to analyze configurations of BuckyBalls®, small magnetic balls commonly used as desk toys.

Extensions Of The Morse-Hedlund Theorem, Eben Blaisdell

Honors Theses

Bi-infinite words are sequences of characters that are infinite forwards and backwards; for example "...ababababab...". The Morse-Hedlund theorem says that a bi-infinite word f repeats itself, in at most n letters, if and only if the number of distinct subwords of length n is at most n. Using the example, "...ababababab...", there are 2 subwords of length 3, namely "aba" and "bab". Since 2 is less than 3, we must have that "...ababababab..." repeats itself after at most 3 letters. In fact it does repeat itself every two letters. …

The Axiom Of Choice In Topology, Ruoxuan Jia

Honors Theses

Cantor believed that properties holding for finite sets might also hold for infinite sets. One such property involves choices; the Axiom of Choice states that we can always form a set by choosing one element from each set in a collection of pairwise disjoint non-empty sets. Since its introduction in 1904, this seemingly simple statement has been somewhat controversial because it is magically powerful in mathematics in general and topology in particular. In this paper, we will discuss some essential concepts in topology such as compactness and continuity, how special topologies such as the product topology and compactification are defined, …

An Investigation Of The Four Vertex Theorem And Its Converse, Rebeka Kelmar

Honors Theses

In the study of curves there are many interesting theorems. One such theorem is the four vertex theorem and its converse. The four vertex theorem says that any simple closed curve, other than a circle, must have four vertices. This means that the curvature of the curve must have at least four local maxima/minima. In my project I explore different proofs of the four vertex theorem and its history. I also look at a modified converse of the four vertex theorem which says that any continuous real- valued function on the circle that has at least two local maxima and …

Tying The Knot: Applications Of Topology To Chemistry, Tarini S. Hardikar

Honors Theses

Chirality (or handedness) is the property that a structure is “different” from its mirror image. Topology can be used to provide a rigorous framework for the notion of chirality. This project examines various types of chirality and discusses tools to detect chirality in graphs and knots. Notable theorems that are discussed in this work include ones that identify chirality using properties of link polynomials (HOMFLY polynomials), rigid vertex graphs, and knot linking numbers. Various other issues of chirality are explored, and some specially unique structures are discussed. This paper is borne out of reading Dr. Erica Flapan’s book, When Topology …

Some Examples Of The Interplay Between Algebra And Topology, Joseph D. Malionek

Honors Theses

This thesis presents several undergraduate and graduate level concepts in the fields of algebraic topology and topological group theory in a manner which requires very little mathematical background of the reader. It uses non-rigorous interpretations of concepts while introducing the reader to the rigorous ideas with which they are associated. In order to give the reader an idea of how the fields of algebra and topology are closely affiliated, the paper goes over five main concepts, the fundamental group, homology, cohomology, Eilenberg-Maclane spaces, and group dimension.

Normal Surfaces And 3-Manifold Algorithms, Josh D. Hews

Honors Theses

This survey will develop the theory of normal surfaces as they apply to the S3 recognition algorithm. Sections 2 and 3 provide necessary background on manifold theory. Section 4 presents the theory of normal surfaces in triangulations of 3-manifolds. Section 6 discusses issues related to implementing algorithms based on normal surfaces, as well as an overview of the Regina, a program that implements many 3-manifold algorithms. Finally section 7 presents the proof of the 3-sphere recognition algorithm and discusses how Regina implements the algorithm.

General Relativity And Differential Geometry, Harry Hausner

Honors Theses

N/A

Domain Representability And Topological Completeness, Matthew D. Devilbiss

Honors Theses

Topological completeness properties seek to generalize the definition of complete metric space to the context of topologies. Chapter 1 gives an overview of some of these properties. Chapter 2 introduces domain theory, a field originally intended for use in theoretical computer science. Finally, Chapter 3 examines how this computer-scientific notion can be employed in the study of topological completeness in the form of domain representability. The connections between domain representability and other topological completeness properties are subsequently examined.

Odd Or Even: Uncovering Parity Of Rank In A Family Of Rational Elliptic Curves, Anika Lindemann

Honors Theses

Puzzled by equations in multiple variables for centuries, mathematicians have made relatively few strides in solving these seemingly friendly, but unruly beasts. Currently, there is no systematic method for finding all rational values, that satisfy any equation with degree higher than a quadratic. This is bizarre. Solving these has preoccupied great minds since before the formal notion of an equation existed. Before any sort of mathematical formality, these questions were nested in plucky riddles and folded into folk tales. Because they are so simple to state, these equations are accessible to a very general audience. Yet an astounding amount of …

Two Views Of The Projective Plane, Rebecca J. Thomas

Honors Theses

The projective plane is a mathematical object which can be defined in two ways. In the following paper, I will explain the two definitions and show how they are equivalent by establishing a homeomorphism between the two objects.

Modern Art Through Geometric Eyes, Janice M. West

Honors Theses

When tourists--even homefolks--go through a modern art museum, many opinions are accumulated. Some people may have chills when they see a certain painting, while others get a sick feeling of dizziness when they see the same one. In fact, if there were an opinion box at the exit of an art show, I imagine you could almost accurately count the different opinions by counting the total number of people who viewed the show. Yet, there is one opinion that most 'ole foggies' (and I use the term loosely) would agree upon, and that is this: "Why that's nothing but a …

The Regular Polyhedra: A Study In Visual Aids For Teaching Geometry, Sammye Halbert

Honors Theses

Traditionally, mathematics, past simple addition, subtraction, multiplication, and division, has been taught of as being so boring, irrelevant, and in short, one of the unavoidable evils of school. An advertisement in The Mathematics Teacher expressed the general attitude of many students when it said, "mathematics was invented by an old magician in the desert who, with the help of his talking monkey, bakes equations and cupcakes in the hot sun." It seems that many students think mathematics is just one problem after another that has some mystical answer floating around in the air somewhere. The object is to get that …