## 2020 Mathematics Capstone Presentations

Talks will take place via remote videoconference. Talks are 20 minutes long, followed by 5 minutes for questions.

 DATE NAME/TIME TOPIC ABSTRACT Mon, 4/13 10:00am Spencer Ward Simplifying Nonstandard Analysis with Alpha-Theory Nonstandard analysis is an attempt to simplify analysis by formalizing infinity and infinitesimals as fully fledged numbers. For the most part, it has succeeded in doing so, but only with a prohibitive level of complexity that has given many mathematicians reason to stick with more traditional analysis. This talk will introduce Alpha-Theory, a relatively new approach to nonstandard analysis that may be simple enough to compete with tradition. Wed, 4/15 10:00am Matthew Donovan Representations of Reals as Continued Fractions The continued fraction is a useful tool in various ways for approximating and representing real numbers. Algorithms to find a number’s continued fraction will be explained, as well as an intuition to prove that infinite continued fractions converge to real values. 10:25am Jens Hansen Bounding Prime Gaps using Sieve Theory Prime gaps have long been an interest of study for mathematicians. We have long known that there are infinitely many primes; but what can be said about gaps between primes as they get very large? A long-standing conjecture is that for every even positive integer k, there are infinitely many pairs of adjacent primes satisfying $p_{i+1}-p_i = k$, with k=2 being the twin primes conjecture. In modern mathematics, it is known that there exists some $k \le 245$ for which this statement is true. In this paper, we seek to answer a related but weaker result about the gaps between primes; that is that $\displaystyle \liminf\limits_{n\rightarrow\infty} \frac{(p_{n+1}-p_{n})}{\log{p_n}} \le 1.$ This result, first proved by Erdös in the mid 20th century, is proven in the same thematic way as the more powerful results of modern mathematics. As such, it provides much insight into the mechanisms of modern sieve theory. Fri, 4/17 10:00am Noah Car The Discrete Fourier Transform and applications to Gaussian Blurring of Images The aim of this talk is to introduce the audience to the concept of discrete-position signals, within the context of image analysis. We will then explore the foundations of Gaussian blurring, including image convolution and the discrete Fourier transform. These mathematical concepts will then be applied to the Gaussian blurring operation, and we will explore how it removes random noise from images. 10:25am James Burkhart Taking the Pythagorean Theorem into Hyperbolic Space This talk will present the Euclidean Pythagorean theorem and its proof. It will then address what sorts of considerations need to be made in the Poincaré disk to accommodate the theorem. Then a proof of the hyperbolic Pythagorean theorem will conclude the talk. Mon, 4/20 10:00am Betsy Spear Sports Ranking: The Colley Matrix Method This talk will introduce the Colley Matrix Method, a method of ranking in sports. We will discuss its uses, its strengths and weaknesses, and the motivation behind the equation. 10:25am Aidan Regan Simulating a Football Game Using Markov Processes and Other Methods In this project, we will be looking at how to simulate a game of American football and determine the winner using a few different methods. Firstly, we will factor in the strength of each team’s run offense, pass offense, run defense and pass defense and use Markov chains to find the probability of each team reaching the end zone. Then we will develop a simple model to determine the winning probability of the home team given its winning percentage and the away team’s winning percentage. Wed, 4/22 10:00am Rene Yost Hypergeometric Series with Applications My talk will introduce hypergeometric series, and then demonstrate examples of applications in differential equations and combinatorics. These examples will be comprehensively explained, so that they will be easy to follow for someone seeing hypergeometric equations for the first time. 10:25am Jack Schnurr The Wallis product Fri, 4/24 10:00am Taye Brown Generalizations of Fermat’s Last Theorem 10:25am Abigail Hayward The Exploration of a Peg Puzzle Say you have a linear row of red pegs and blue pegs that are separated by a single space in the middle. You are allowed to jump over one peg into an empty spot and slide a peg into an empty spot. How do you move the red pegs to the blue side and the blue pegs to the red side? Techniques from discrete mathematics can help answer questions like how to get there in the least number of moves, show that there is a pattern in the movements, and other related questions. Mon, 4/27 10:00am Abram Karam Second-order generalized derivatives of nonsmooth functions Derivative information is a fundamental tool used in classical theory and methods for equation solving, differential equations, and optimization. However, many real-world problems, such as those found in process systems engineering and civil engineering, exhibit nonsmoothness in the form of discrete events, which often means that derivative information is unavailable or inaccurate. Motivated by applications from engineering, the aim of this project is to extend the theory of generalized derivatives and classical differentiation to evaluate second-order generalized derivatives in an accurate and automatable way, for the purpose of improving accuracy and convergence of nonsmooth methods. 10:25am Crockett Lalor The p-adic Absolute Value and Ostrowski’s Theorem An absolute value on a field induces a metric space topology. Up to equivalence of the induced topologies, we may wonder how many absolute values can be defined on a given field. In this talk we will discuss Ostrowski’s theorem, which fully classifies the absolute values on Q, and take note of some interesting topological properties of those absolute values. Wed, 4/29 10:00am Zoey Zuo A statistical analysis of the effect of various hotel amenities on the hotel’s user ratings. My goal is to analyze data sets through a combination of statistical theory and programming. We analyze the effect of various hotel amenities on the cumulative review score and room rate for a double room. Considering cumulative review score and room rate for a double room as a linear model, we are going to use the hypothesis test to infer the effect of hotel amenities on the model. 10:25am Matthew Ryckman The Fractal Properties of Pascal’s Triangle Modulo 2 In this short presentation, we will illuminate the fractal properties of Pascal’s Triangle modulo 2. To begin, the “self-similar” properties of Pascal’s Triangle will be highlighted; this is achieved by applying an interesting theorem from number theory, attributed to Edouard Lucas. Finally, we will map the entries of Pascal’s Triangle to the unit square, so that we may take its box-counting dimension (allowing us to further investigate its fractal properties), introducing some required definitions and theorems along the way. 11:00am Tavo Harrsen An overview of Combinatorial Game Theory This presentation will cover how to determine who wins games of perfect information and no chance and by what margin, as well as how games can be added together and compared via relation (equal to, greater than, fuzzy to). We will discuss what it means for a game to be a number, a nimber, up, down, or hot, and how adding games like these together affects the game’s outcome. Fri, 5/1 10:00am Davis MacDonald 10:25am Ian West An In-Depth Look at a Proof of Stirling’s Formula In April 2014, Thorsten Neuschel published an article in The American Mathematical Monthly entitled “A New Proof of Stirling’s Formula”. He takes the typical proof of Stirling’s Formula and replaces a critical step with a partial fraction expansion of a tangent function. I will be taking an in-depth look at Neuschel’s proof, and presenting an overview, highlighting the important aspects of the proof, as well as providing some examples of the applications of Stirling’s Formula. Mon, 5/4 1:30pm Abdourazack Hassan Analysis of the basic reproduction number of an epidemic model. In this presentation we will investigate the basic reproduction number R0 which in epidemiology represents the expected number of secondary cases that an infected individual can produce in a susceptible population. We will illustrate that with an epidemic (Cholera) that occurred in South Africa (between 2000 and 2001). 1:55pm Rudy Chase The Braess Paradox The Braess Paradox is about how adding roads to a network does not necessarily make the network more efficient. In fact, when drivers try to decrease their own travel time, the addition of new routes may actually increase travel time for the entire network of drivers. In this presentation, I aim to give the audience an understanding of what the Braess Paradox is, the theories behind it, and how these theories are what makes planning a road network difficult. 2:30pm Liam Griffin Normality and Infinite Randomness This presentation will investigate the properties of numbers normal in a base b and visualize those properties for different constants. Knowing what makes a number normal, we will prove that almost all real numbers are normal. Finally we will discuss the peculiarities of the non-constructive nature of this proof and its implications toward computing. 2:55pm Henry Geoffrion Packing tetrahedra