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

• TITLE: THE ORDER IN CHAOS

AUTHOR: Jeremy Batterson

ABSTRACT: A fractal is a mathematical structure with extremely complex geometry that is created from just a few simple recursive maps. They appear everywhere in the real world, from ferns and clouds to coastlines and computer games. This paper presents the foundations for studying fractals through iterated function systems and chaotic dynamical systems. There are a number of algorithms presented for the purpose of making fractals on a computer. All of this leads to the main focus of the paper which is a proof of the correctness of the random iteration algorithm for producing fractals. We also provide references for real world applications for fractals including computer graphics and image compression.

• TITLE: MOST OF GÖDELS INCOMPLETENESS: A SUMMARY OF THE $\omega$-CONSISTENCY THEOREM

AUTHOR: William Crawford

ABSTRACT: In this paper, we closely examine the fundamental assumptions that we make in mathematics: axioms. Any set of axioms we use to prove propositions is an axiomatic system. Ideally, we seek an axiomatic system which is not only internally coherent, but also capable of proving theorem relevant to those axioms. We call the former of these two ideal criteria consistency and the latter completeness. Gödel's Incompleteness Theorems illustrate the relationship between these two criteria, and show that finding an axiomatic system that satisfies both criteria requires the system to be unusual. Our final goal is to prove the $\omega$ Consistency Theorem, a powerful result regarding incompleteness. On the way, we will show that Peano Arithmetic, a common axiomatic system for basic Arithmetic, is incomplete.

• AUTHOR: Whitney Griggs

ABSTRACT: Regression analysis is a branch of statistics that examines and describes the relationship between different variables of a dataset. In this paper, we investigate penalized spline fits, a nonparametric method of regression modeling, and compare it to the commonly used parametric method of ordinary least-squares (OLS). Using data from our neuroscience research, we demonstrate several different applications of these penalized splines as compared to linear regression. We conclude the paper with some exploration of statistical inference using bootstrapping and randomization of our spline models.

• AUTHOR: Mzuri Handlin

ABSTRACT: We explore how the simple concept of a conic section can be generalized to higher dimensions.  Two definitions of the conic sections, algebraic and geometric, are given, allowing the development of two distinct generalizations of conic sections from three dimensions to four.  These two generalizations are the quadric surfaces and conic surfaces. After finding that all conic surfaces are quadric surfaces, we use tools from differential geometry to study quadric surfaces.  The most significant result that we develop is a theorem placing important constraints on the curvature of quadric surfaces.  The full theorem is developed using a tool called the Gauss curvature.  One implication of this theorem is the following: if one point on a quadric surface is a saddle point, then every point on that surface is a saddle point.

• TITLE: PROOFS YOU CAN COUNT ON

AUTHOR: Helen Jenne

ABSTRACT: Benjamin and Quinn's (2003) proofs by direct counting reduce the proof of a mathematical result to a counting problem. In comparison to other proof techniques such as proof by induction, proofs by direct counting are concrete, satisfying, and accessible to an audience with a variety of mathematical backgrounds. This paper presents proofs by direct counting of identities involving the Fibonacci numbers, the Lucas numbers, continued fractions, and harmonic numbers. We use the Fibonacci numbers and Lucas numbers primarily to introduce proofs by direct counting. We then present Benjamin and Quinn's combinatorial interpretation of continued fractions, which allows us to reduce identities involving continued fractions to counting problems. We apply the combinatorial interpretation to infinite continued fractions, and ultimately present a combinatorial interpretation of the continued fraction expansion of $e$. We conclude this paper by discussing Benjamin and Quinn's combinatorial interpretations of harmonic numbers and a generalization of the harmonic numbers called the hyperharmonic numbers.

• TITLE: SPECTRA OF SIMPLE GRAPHS

AUTHOR: Owen Jones

ABSTRACT: Spectral graph theory concerns the connection and interplay between the subjects of graph theory and linear algebra. We assume that the reader is familiar with ideas from linear algebra and assume limited knowledge in graph theory. In this paper we begin by introducing basic graph theory terminology. Then we introduce the adjacency and Laplacian matrices and explore the spectra of some basic types of graphs. Next, we look at the relationship between spectra, cliques and colorings of graphs. The paper concludes with a discussion on regular graphs and algebraic connectivity.

• AUTHOR: Steven Klutho

ABSTRACT: Decision problems are an inescapable part of everyday life, and it can be useful to have mathematical tools to deal with them. This paper gives an overview of one such tool, the Analytic Hierarchy Process. The paper gives a background of the process at work and the theory behind it, and then goes to discuss how we know the results we obtain are valid. Using this groundwork, we examine possible scenarios where the process can run into trouble, and different ways these paradoxes can be explained. Finally, we present a real world example of a decision problem that was worked through using the Analytic Hierarchy Process.

• AUTHOR: Devin Kuh

ABSTRACT: This paper will discuss the constructability of regular $n$-gons. The constructions will follow the rules of Euclidean Constructions. This question of which regular $n$-gons are constructible stems from the same era of Ancient Greek questions like doubling the cube and squaring the circle. The paper will examine both the Abstract Algebra theory and the physical constructions. The theory will center on Gauss' theorem of constructible regular $n$-gons and a larger result of which Gauss' theorem is a specific case of (although proven independently). Our physical constructions will look at the regular pentagon, 17-gon, 15-gon and 51-gon as specific examples to illuminate these possibilities.

• TITLE: PURPOSES OF PROOFS WITHOUT WORDS: AN EVALUATION OF VISUAL PROOFS USING THE PHILOSOPHIES OF FREGE AND LAKATOS (selected outstanding senior project, available upon request to Albert Schueller)

AUTHOR: Lauren Kutler

ABSTRACT: Since 1975 the Mathematics Magazine has published strictly visual proofs under the heading "Proofs Without Words" (PWWs). PWWs claim to demonstrate mathematical results without employing the typical logical structure of proof, a difference which raises deep philosophical questions. In addition to surveying 79 PWWs, this paper uses the philosophies of Gottlob Frege and Imre Lakatos in order to examine the purpose of proof. Although I argue that PWWs fail to fully accomplish the aims of proof according to either Frege or Lakatos, I show that they are nonetheless pedagogically useful devices for understanding.

• TITLE: AN INVESTIGATION OF INTEGER TRIANGLES WITH TRISECTIBLE ANGLES(copies of this paper are available upon request to Russ Gordon)

AUTHOR: Jaclyn Rudd

ABSTRACT: This paper explores the properties of trisectible angles, first drawing upon a result from the 19th century of an ancient geometric problem: that it is impossible to trisect an arbitrary angle to absolute accuracy using only a compass and straightedge. After giving some background on Euclidean constructions, we investigate properties of trisectible angles of integer value. Then, for the majority of this paper, we study angles with rational trigonometric values and investigate conditions such that different types of triangles have both integer side lengths and trisectible angles.