Titles are hyperlinked to pdf copies of the final project write-up. Course coordinator: Barry Balof

• Author:  Lane Barton

Title:  Ramsey Theory

Abstract:  Ramsey theory is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a specific size. This paper will explore some basic definitions of and history behind Ramsey theory, but will focus on a subsection of Ramsey theory known as Ramsey numbers. A discussion of what Ramsey numbers are, some examples of their relevance in real-life scenarios, and a computational method for determining Ramsey numbers will be provided in an attempt to create an accessible, easy to understand look at an interesting topic.

• Author:  Marissa Childs

Abstract:  This thesis considers the open problem in topological graph theory: What is the largest connected graph of minimum degree 3 which has everywhere positive combinatorial curvature but is not in one of the infinite families of graphs of positive combinatorial curvature? This problem involves the generalization of the notion of curvature (a geometric concept) to graphs (a combinatorial structure). The thesis presents past progress on the upper and lower bounds for this problem as well as graph operations that may be used in the future for improving the lower bound.

• Author:  Deborah C. DeHovitz

Title:  The Platonic Solids

Abstract:  The five Platonic solids (regular polyhedra) are the tetrahedron, cube, octahedron, icosahedron, and dodecahedron. The regular polyhedra are three dimensional shapes that maintain a certain level of equality; that is, congruent faces, equal length edges, and equal measure angles. In this paper we discuss some key ideas surrounding these shapes. We establish a historical context for the Platonic solids, show various properties of their features, and prove why there can be no more than five in total. We will also discuss the finite groups of symmetries on a line, in a plane, and in three dimensional space. Furthermore, we show how the Platonic solids can be used to visualize symmetries in $\mathbb{R}^3$.

Title:  Data Mining

Abstract:  Data mining is the art and science of discovering new, useful and fruitful relationships in data. This paper investigates unsupervised learning models of data mining. It sheds the light on various techniques used to find meaningful patterns in data, and analyze each technique to highlight its advantages and limitations. Furthermore, the paper gives detailed overview and analysis for the algorithms used in unsupervised learning models.

• Author:  Alec Foote

Title:  The Boids are Flying!

Abstract:  Differential equations can model the competition between species fairly well. But what if we modeled populations with moving shapes instead? Join me as I take you on a journey into the world of boids (that is, digital triangle birds), as they eat food, reproduce, and might even generate an effective simulation of population interference!

• Author:  Kathrin Liane Gillespie

Title:  2-Point Centroidal Voronoi Tessellations (available upon request)

Abstract:  In this paper, we discuss important properties of the Centroidal Voronoi Tessellation, and the geometry of CVTs on ellipses. We discuss some definitions, algorithms, and applications for CVTs. We then proceed to prove the main result of this paper: that the only CVTs of ellipses with two generators are those where the boundary between the two Voronoi regions is a line of symmetry of the ellipse. We also generalize this result to a similar class of shapes-particularly, convex shapes with rotational symmetry of order 2 whose boundaries mirrored across any line through their point of rotation intersect the original boundary in exactly four locations. To achieve both of these proofs, we first prove an important theorem which states that the Voronoi boundary of CVTs with two generators on convex shapes with order 2 rotational symmetry must intersect the origin.

• Author:  Teylor Greff

Abstract:  Do we actually need calculus to solve maximum/minimum problems? Optimization problems are explored and solved using the AM/GM inequality and Cauchy Schwarz inequality, while simultaneously finding trends and evolutions in these optimization problems as we look at a textbooks ranging from 1902 - 2015.

• Author:  Nina Henelsmith

Abstract:  If we want to throw a projectile as far as possible, at what angle should it be launched? This paper focuses on how the answer to this question changes depending on the situation. We look at launching projectiles onto differently shaped hills, as well as how varying initial velocity and height affect the launch angle. Finally, we add air resistance to the projectile problem and compare two different models: air resistance proportional to the projectile's velocity and air resistance proportional to velocity squared.

• Author:  Gregory Reid Holdman

Abstract:  The theory of error-correcting codes has historically been most useful in the context of linear codes. Such codes may be viewed as vector spaces over Galois fields carrying with them many familiar and well-studied properties. A generalization of Galois fields is the concept of Galois rings. It is therefore natural to consider codes over Galois rings to study which properties such codes maintain in the move to a more general setting. This thesis will present two separate expositions on coding theory and Galois rings. After this, the intersection of these topics will be considered: codes over Galois rings.

• Author:  Hannah Horner

Abstract:  A Graeco-Latin Square of order $n$ is an $n \times n$ array of unique ordered pairs whose entries are numbers between $1$ and $n$ such that every number appears in each row and column once for each coordinate of the ordered pair . In 1782, Euler predicts that Graeco-Latin Squares of order $n$ exist if and only if $n \geq 3$ and $n \neq 2 (\mod 4)$. This prediction, known as Euler's conjecture, is eventually proved almost entirely wrong in the following 180 years. In this paper we will explore the mathematics and history behind the conjecture.

• Author:  Tate Jacobson

Abstract:  In this paper we will characterize the differences between non-chaotic and chaotic dynamics. The first section will introduce dynamical systems in general, providing vocabulary and highlighting why our naive example is so predictable. The second section will then provide a general definition of chaos and give an example of a chaotic dynamical system. The third and fourth sections will then
explore a powerful method for proving that dynamical systems are chaotic: topological conjugacy.

• Author:  Casey Schafer

Abstract:  A neural network is a powerful mathematical model combining linear algebra, biology and statistics to solve a problem in a unique way. The network takes a given amount of inputs and then calculates a specified number of outputs aimed at targeting the actual result. Problems such as pattern recognition, linear classification, data fitting and more can all be conquered with a neural network. This paper will aim to answer these questions: what is a neural network, what is its mathematical background, and how can we use a neural network in application?

• Author:  Karen Vezie

Abstract:  This paper provides an overview of the Mercator map projection. We examine how to map spherical and ellipsoidal Earth onto 2-dimensional space, and compare two paths one can take between two points on the earth: the great circle path and the rhumb line path. In looking at these two paths, we will investigate how latitude and longitude play an important role in the proportional difference between the two paths.

• Author:  Ryan Wallis

Title:  Drawing Rectangles in Rectangles: A Euclidean Construction Problem (available upon request)

Abstract:  (none)