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

• Author: Yonah Biers-Ariel
Title:  Complete Words and Superpatterns
Abstract:  This thesis surveys the most important results regarding complete words, by which we mean words containing as subsequences every permutation of some set, and then considers two variants of the problem of finding the shortest complete words. The first of these asks for the shortest superpattern, which is a complete word with some additional requirements, and the second asks for the expected number of timesteps before a particular random process generates a complete word.

• Author:  Sophie De Arment
Title:  2-Point Centroidal Voronoi Tessellations (available upon request)
Abstract:  A centroidal Voronoi tessellation (CVT) is a special kind of Voronoi tessellation such that the generating points of the tessellation are also the mass centroids of the corresponding regions. Due to their innate optimization properties, CVTs have applications in diverse fields; however, the theoretical nature of these tessellations is far from well understood. We approach some open questions about CVTs by looking in particular at 2-point tessellations of regular polygons with constant density. We show that for any CVT of an even-sided polygon, the Voronoi boundary or the centroids must lie on lines of symmetry, and anticipate that the same is true for odd-sided polygons. We predict which of these configurations are stable under Lloyd’s algorithm for computing CVTs. Finally, we illustrate basins of attraction for the stable CVTs under Lloyd’s algorithm.

• Author:  Sam Fischer
Title:  Integer Sided Triangles with Trisectable Angles:  Their Perimeters and Residual
Abstract:  Every triangle with integer sides and trisectible angles can be characterized by it’s perimeter and a square free number called a residual.  This paper describes the creation of a comprehensive list of triangles with trisectible angles and integer sides up to perimeter 10,000 and discusses how the perimeters of those triangles are related to their residuals.

• Author:  Nathan Fisher
Title:  Decision Trees
Abstract:  (none)

• Author:  Max Lloyd
Title:  Matrices Lie: An introduction to matrix Lie groups and matrix Lie algebras
Abstract:  This paper is an introduction to Lie theory and matrix Lie groups. In working with familiar transformations on real, complex and quaternion vector spaces this paper will define many well studied matrix Lie groups and their associated Lie algebras. In doing so it will introduce the types of vectors being transformed, types of transformations, what groups of these transformations look like, tangent spaces of specific groups and the structure of their Lie algebras.

• Author:  Halley McCormick

Title:  Construction of Centroidal Voronoi Tessellations Using Genetic Algorithms
Abstract:  Centroidal Voronoi tessellations (CVTs) are a way of partitioning sets, and genetic algorithms are a way of optimizing functions. In this paper, we discuss how to apply genetic algorithms to the problem of generating CVTs by minimizing a function associated with such partitions. We outline the ways of relating components of genetic algorithms to CVTs, we test implementations of a genetic algorithm that yields CVTs, and we compare the performance of the genetic algorithm to other methods of approximating CVTs.

• Author:  Alexander M. Porter
Title:  The Tutte Polynomial and Applications
Abstract:  This paper gives a detailed overview of the Tutte polynomial and proves results that are simply stated in most of the current literature. It covers basic graph theory and matroid theory, basic properties of the Tutte polynomial, the recipe theorem and two applications of the recipe theorem: one to the Potts model and Ising model of statistical physics and one to the HOMFLY polynomial in knot theory.

• Author:  Brett Porter
Title:  Cyclotomic Polynomials
Abstract:  If $n$ is a positive integer, then the $n$th cyclotomic polynomial is defined as the unique monic polynomial having exactly the primitive $n$th roots of unity as its zeros. In this paper we start off by examining some of the properties of cyclotomic polynomials; specifically focusing on their irreducibility and how they relate to primes. After that we explore some applications of these polynomials, including proofs of Wedderburn’s Theorem, and when a regular $n$-gon is constructible with a straightedge and compass.

• Author:  Aanand Sharma
Title:  Cancellation Properties of Summands of Direct Products of Groups
Abstract:  This paper investigates the cancellation properties'' of a group $C$ where, for some groups $A$ and $B$, $A\oplus C \cong B \oplus C$ implies that $A\oplus B$. Guided by the results of the Krull-Schmidt Theorem and Walker's Cancellation Theorem, this paper claims and proves by induction on the order of group $C$ that $A \cong B$ if $C$ is of finite order. Then, this paper introduces a discussion regarding the class of other possible cancellable structures.

• Author:  Jessica Shatkin
Title:  Exploring Fibonacci Numbers
Abstract:  This paper will illustrate a multitude of properties involving the Fibonacci and Lucas numbers. In an attempt to cover an array of different properties, this paper will include concepts from Calculus, Linear Algebra, and Number Theory. It will also include three distinct derivations of the Binet form for the $n$th Fibonacci number. To begin, a brief discussion of historical background is offered.

• Author:  Austin Sloane
Title:  Sudoku Puzzles and Mathematical Expressions
Abstract:  (none)
Abstract:  In literature, the Ornstein-Uhlenbeck process, a CAR(1) process, has been used extensively for data molding. We expand the classical OU process to be driven by a general Brownian motion. When such a process is observed at discrete times $0, h, 2h, \ldots, [N/h]h$, the sampled process $Y^{(h)}_n$, $n = 1, 2,\ldots, N$ and the approximation for the unobserved driving process (noise) are used to estimate the unknown CAR(1) parameters. These estimators are tested through simulations.