Senior Capstones

Mandy Buell
When Earthquakes Hit, Buildings React: Modeling Building Response to Earthquakes Using Differential Equations and Eigenvalues
When an earthquake hits, buildings often have some kind of displacement. In some instances the displacement may cause the building to fall down, while, in other cases, the building may be unharmed. The movement of a building as a response to an earthquake can be modeled using differential equations and analyzed using eigenvalues. In this talk, we will go through an example that allows us to understand the relationship between the differential equations, eigenvalues, and movement of a building. Additionally, we will see why even a small earthquake force causes a building to take an excursion.

Carolina Concannon
Markov Chains and Their Criteria For Regularity
A Markov chain is a system of states with rules and probabilities for moving between any two states. These Markov chains can be represented by a matrix, or in a more visual way in the form of a graph. We call certain Markov chains regular if a power of their matrix makes all entries positive. Although this is a good way to determine regularity, there are ways to determine this property more visually through looking at a chain's graph. In this presentation, we will look at possible ways a chain can be regular and some criteria for regularity by looking at the graphs of Markov chains.

Meghan Gerber
Hat Games
There are several recreational mathematics problems with very useful applications hidden within. One such problem is known as a hat game. In it, players are given one of two hat colors, and must work together to guess their own hat color correctly. For this talk, I will explain the connection between the hat game and covering codes, an important topic in coding theory. I will also prove the maximum win percentages for games played with fewer than 9 people.

Ryan Matzke
Fixed and Furious: Fixed Points in Mathematics
The study of Dynamical Systems involves looking into how a system changes over time. When determining the behavior of a dynamical system, one might wonder if it has any fixed points. There are many different kinds of dynamical systems, for example: complete metric spaces with contractions, compact spaces with continuous maps, partially ordered sets with order-preserving functions, and finite sets with permutations. For different kinds of dynamical systems, we need different ways of determining if they have fixed points.

Alice Mitnick
For When Life Isn't Fair: Methods in Fair Division
Fair division has many applications in the real world: from divorces and inheritance to dividing up a cake or a stash of Halloween candy. This topic has inspired many methods dating back to the 1940s. In this talk we explore these methods, specifically looking at divisions of goods that are finite, continuous, and divisible such as division of a cake. We define a method as fair if it creates partitions that are Proportional, Envy Free, and Pareto-Optimal. Using this definition, we aim to determine what method is the best depending on the number of participants.

Christina Moore
The 2008 Financial Crisis & The Gaussian Copula: What Went Wrong?
The financial crisis of 2007-2008 is considered to be the worst financial meltdown since the Great Depression. Behind the story of housing bubbles, mortgage loans, and complex securities is a probability formula called the Gaussian copula. We will explore why, mathematically, the choice to implement this particular model was a poor one. How did Wall Street believe it could really reduce correlation of hundreds of assets to just one number?

Tyler Reckner
p-adic Analysis and the Maximum Modulus Principle
First introduced by Kurt Hensel in 1897, the p-adic number system was originally formalized as a means of extending power series into the field of number theory. Nowadays, however, the p-adics have far-reaching applications throughout mathematics, including in algebra, analysis, and classical number theory. This paper  introduces the p-adic number system, explores the field of p-adic analysis, and compares p-adic analysis to classical analysis by offering a p-adic result analogous to the Maximum Modulus Principle - a fundamental result in classical complex analysis.

Sirius Xu
Nash Equilibria in Non-Cooperative Games
If you have watched the movie A Beautiful Mind, you should be familiar with the name John Nash. Nash Equilibrium, named after John Nash, is a fundamental concept in game theory, a field that studies strategic interactions between rational players in games using mathematical models. In this talk, I will introduce the concept of Nash Equilibrium in the context of a type of game called non-cooperative game. I will also revisit Nash's proof of the existence of Nash Equilibria in non-cooperative games assuming Brouwer's fixed point theorem.