### 2020 Mathematics Capstone Symposium

**May 1st**

Join the Mathematics Department to hear the senior capstone presentations.

#### Session 1

**1:00-1:25pm – Gödel’s First Incompleteness Theorem**

*Clara Elizabeth
*Gödel’s First Incompleteness Theorem is a result from logic that stunned the mathematical world. Gödel challenged the concept that the right givens (axioms) and the right reasoning could allow a mathematician to obtain all mathematical truths. Specifically, Gödel showed that any system capable of arithmetic that is consistent and has a decidable set of axioms must be incomplete — meaning it contains a statement that is unprovable. Gödel’s proof itself is as beautiful as the First Theorem: it involves a brilliant use of mapping and the creation of a formal self-referential statement. What can be learned from the proof’s methods can be just as informative as the First Incompleteness Theorem itself.

**1:30-1:55pm – Computer Science, Mathematics, and the Classroom: How Do They Benefit From One Another?**

*Coral Hunter
*In this presentation we investigate what mathematics has to offer computer science and how the two disciplines work together in a variety of ways. We begin by introducing the concept of correctness, introducing the Loop Invariant Theorem and using it to mathematically prove the correctness of the Euclidean Algorithm. We will then dive into specific examples of how computer science can be used in a secondary math classroom to enhance math curricula that relates to specific Common Core Standards.

**2:00-2:25pm – Polynomial Rings and Connections with Common Core Math Standards**

*Andrew Curran
*We begin by introducing the Common Core State Standards in Mathematics and their history, as well as bringing in some examples of high school math standards. We then introduce abstract algebra concepts, defining terms such as rings, fields, and polynomial rings, as well as showing how arithmetic works in polynomial rings. Next, we introduce and prove the Division Algorithm. We conclude the paper by discussing some final connections between the abstract algebra concepts provided and the Common Core Standards in the high school classroom.

**2:30-2:55pm – Group Actions: Burnside’s Counting Theorem**

*Corey Ng*

In this talk, we prove Burnside’s Counting Theorem and provide a geometric example of the use of the theorem.

**3:00 – 3:30pm Break**

**3:30-3:55pm – Cryptography and Proofs of Security**

*Nathan Hohnbaum
*Cryptography is an important branch of applied mathematics for computer security. Although there is a lot of beautiful mathematics behind systems such as AES and RSA, they are useless unless they actually provide security. Although the security of these systems is still an open question to researchers, there is a cipher called the one-time pad that is known to achieve perfect secrecy. But perfect secrecy comes at a cost, and this cost significantly limits the usefulness of the one-time pad.Cryptography is an important branch of applied mathematics for computer security. Although there is a lot of beautiful mathematics behind systems such as AES and RSA, they are useless unless they actually provide security. Although the security of these systems is still an open question to researchers, there is a cipher called the one-time pad that is known to achieve perfect secrecy. But perfect secrecy comes at a cost, and this cost significantly limits the usefulness of the one-time pad.

**4:00-4:25pm – Quaternions and What You Can Do With Four Dimensions**

*Quinn Kiel*

**4:30-4:55pm – A walkthrough of Tricolor Pyramid Puzzles**

*Aja Hauff
*In this presentation we examine Tricolor Pyramid puzzles. We talk about how to solve them, present patterns which are especially important for us to consider, and discuss how you may create your own puzzles. This talk provides a fun way to see linear algebra through a new lens.

#### Session 2

**1:00-1:25pm – Food Access in Washington State: Using Levene’s Test to Inform 3D Visualization**

*Chris Anderson
*Packages developed for the statistics programming language R are used to display data in three dimensions using LEGO bricks. Data is rounded to the nearest integer to accommodate the discrete height of the bricks, introducing some inaccuracy in the display. Levene’s test for equality of subgroup variances is applied to verify that this rounding error is small relative to other sources of variation in the data. Results of Levene’s test, together with visual inspection, provide evidence that the rounding error in the display is appropriately small.

**1:30-1:55pm – Regression Discontinuity Analysis and UnPLUgged**

*Daniel Hatchet
*In this talk, we discuss regression discontinuity analysis, and explain why it is a viable method for determining the effects of a program when performing a controlled randomized experiment is not possible. We apply this method to investigate electricity usage changes during UnPLUgged, an energy conservation campaign directed toward students in PLU residence halls. Next we discuss several models to account for occupancy and seasonal effects in residence hall electricity usage. These models will be both linear and sinusoidal in nature. After selecting the best models, we use regression discontinuity analysis with and without covariates to investigate whether UnPLUgged appears to have an effect on electricity usage.

**2:00-2:25pm – An Introduction to the Singular Value Decomposition (SVD)**

*Aidan Liddiard
*In this talk, we introduce the singular value decomposition (SVD), a type of decomposition generally not included in introductory linear algebra courses. The SVD allows you to decompose a rectangular matrix into component matrices, unlike spectral decomposition (or eigendecomposition) which can only be applied to square matrices. The component matrices provide interesting and important information about the original matrix, which is why the SVD has many applications, including its use in principal component analysis.

**2:30-2:55pm – Hedging: A Method of Financial Risk Management**

*Alex Ellis*

Investors spend money with the hopes of making a profit on their investments; however, the market sometimes behaves in ways that cause them to lose money. While the market can’t be forced by an individual to act one way or the other, economists have developed ways to help manage the risks taken on by investing and prevent losses. In this presentation we will explore one such method of managing financial risk, called hedging, in the context of bond investments.

**3:00 – 3:30pm Break**

**3:30-3:55pm – One-Dimensional Chaotic Dynamical Systems**

*Lance Coyer II
*The family of quadratic functions F

_{μ}(x) = μx(1 − x) is often cited as the classic example of how chaos occurs in dynamical systems. In this talk, we investigate the behavior of F

_{μ}under iteration. Restrictions of the parameter μ will be determined algebraically and visually using graphical analysis.

**4:00-4:25pm – Particle in a Hypercube: An Examination of the Schrödinger Equation
**

*TJ Mallos*

The Schrödinger equation is explored. We begin by reviewing the underlying math behind quantum mechanics, then familiarize ourselves with the principles of quantum mechanics. We then introduce the particle in a box problem and derive a one-dimensional solution. We conclude by examining higher-dimensional boxes and derive a solution to the three-dimensional box problem.

**4:30-4:55pm – Origamia**

*Steven Lopez
*There exists a vast and diverse plethora of Origami folding games that revolve around constructing animals and other cool shapes. The solutions to these games are comprised of the paper structures that result from the folds. The marketplace, however, is devoid of 2D folding games whose solutions are comprised of the points and lines that result from single folds. It turns out there is a lot that can be done with the results of single folds. These possibilities are described by the Axioms of Origami. The purpose of this capstone project is to derive the equations necessary to create an Origami folding game based on the first 6 Axioms of Origami.

**5:00-5:25pm – An Exploration of the Gaussian Integers**

*Tylor Rabina*

In this talk, we discuss the Gaussian integers, a subring of the ring of complex numbers. A Gaussian integer is a complex number of the form a+bi, where a and b are integers. We introduce Gaussian primes and compare them to the usual integer primes, which will lead to a discussion about which of the usual integer primes are prime in the ring of Gaussian integers. We conclude by using a KenKen Puzzle to discuss a concept of factorization within the ring of Gaussian integers.

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