Adam Bendetson 

Evaluating the Impact of Salary Constraints on Competitive Balance in Major League Baseball

Abstract:

This project investigates the mathematical relationship between team payroll and on-field success in Major League Baseball (MLB) from 2005 to 2025. MLB currently lacks a ā€œhardā€ salary cap; instead it has used various forms of a tiered luxury tax system for over 20 years. The pending expiration of the Collective Bargaining Agreement at the end of 2026 presents a unique opportunity to model alternative economic structures. Using linear and linear-log regression analysis, this study models the marginal cost for each additional win.

These findings form the basis for a Monte Carlo simulation, developed to project league parity under three distinct competitive balance structures: a hard salary cap/floor, a tiered luxury tax system, and full revenue sharing. Competitive balance is evaluated using the Noll-Scully ratio and the Gini coefficient of team win totals. Initial findings suggest a diminishing return at higher payroll levels, and the simulation results form the basis for two proposals to improve competitive balance while still allowing player salaries to grow.

Rori Coomey

The Spread of Misinformation in Social Networks 

Abstract:

This project studies how a committed minority spreads an opinion through a social network using the Binary Agreement Model (BAM). The central question is: how does the tipping threshold pc, the critical fraction of committed agents required to dominate a population, change when we modify either how interactions are selected (algorithmic bias) or where committed agents are placed in the network (structural hierarchy)? Using stochastic simulations on Erdos-Renyi random graphs, I compare the baseline BAM with two modifications: (i) weighted interaction rules that create asymmetry in neighbor selection and (ii) structural placement of committed agents at high degree nodes. Results show that both algorithmic weighting and structural hierarchy significantly reduce the tipping threshold. These findings demonstrate that the critical minority size required for consensus is not fixed but depends sensitively on network structure and transition probabilities.

Bradan Craig

Large Language Models

Abstract:

Before ChatGPT, Gemini, or any other modern large language model, natural language processing was restricted to much more primitive models. Models like ELIZA and SHRDLU were some of the first instances of using machine learning to predict the next word in a sentence where as the Georgetown experiment used its own model to translate 60 Russian sentences into English in the 1950s. However, all of these models took large amounts of compute power and were very error prone. On June 12, 2017, the paper ā€Attention Is All You Needā€ was published and introduced a new architecture to natural language processing that would address all of these concerns.

In the paper, they described a new mechanism called ā€™Attentionā€™ that could be used in order to get a very context-rich understanding of a query in order to return a comprehensive and thoughtful response. This technique uses d dimensional matrices to represent each word in the query as well as three separate matrices containing the words meaning, the information it gives, and what it is looking for respectively. The other key aspect of this new architecture is the positional encoding matrix. This is a matrix that is concatenated with the words vector representation in order to give further context of where each word lives in the sentence. Within this paper, we will be specifically studying the effectiveness of this positional encoding matrix and using it to make inferences on how robust the architecture is at receiving grammatically incorrect statements and if the model can still properly comprehend the query.

Emma Dionne

The meta-analysis of the relationship between mathematical academic achievement and metacognition

Abstract:

Throughout recent human history, it has become widely agreed upon that some people, especially children, are particularly skilled with processing and understanding unfamiliar mathematical concepts, while others, though provided with the same resources, struggle. The fastest rate at which a student can achieve mastery of a school subject is often naively attributed to factors that are ā€œout of our controlā€; a popular belief is that a personā€™s nature is responsible for how their mind works and its potential, but without understanding why that is, thereā€™s little we can consciously do to change that. However, in recent decades, the theory that the capacity for learning is quantifiable, even improvable, has grown large in popularity through the idea of metacognition. In this paper, we will begin by discussing the standards used to assess metacognition and the meaning and purpose of a meta-analysis. We will then discuss the methods and results of a meta-analysis of the relationship between mathematical academic achievement and metacognition, as well as the characteristics of the studies involved. Additionally, we will be looking at two recent multi-year studies in an effort to look closer at the correlation between the development of these skills over time. Finally, we will consider the significance of these findings and determine what adjustments to our education system would be beneficial for improving the overall comprehensibility of various mathematical topics.

Killian Gribben

Maximizing the Success of Penalty Shootouts in Sports

Abstract:

Penalty shootouts are one of the most stressful, intense, and dramatic moments in all ofĀ  sports. It is used as a method to determine the winner of tied matches in the knockout stages of the FIFA World Cup. Despite their frequent characterization as a game of fortune, statisticalĀ  analysis suggests that a variety of factors may influence the probability of success in these high-pressure situations. By examining historical data from previous World Cup penalty shootouts, we will be looking at how teams can maximize their chances of success. Using statistical analysis and probability models, the analysis examines patterns in conversion rates and investigates what factors have a direct correlation to an advantage in these shootouts. Things that will be examined are Shooting Order, First Mover, Age of Penalty Takers, and Placement of Shots. There has been ongoing speculation that the team that shoots first has a higher chance of winning shootouts than those shooting second. Despite this, the First Mover misconception has been disproven using data collected from previous shootouts, and insufficient evidence to claim that this is true. Instead, we will look at how the Age of the penalty takers has the biggest impact on winning penalty shootouts compared to the other factors that have been examined in this analysis.

Eva Junge

A proof of Burnsideā€™s Theorem using Representation Theory

Abstract:

This capstone provides an introduction to the representation theory of finite groups. Representation theory extends group theory to linear algebra by connecting group homomorphisms with vector spaces, providing powerful tools for proving results from group theory. After covering preliminary results, a proof of Burnsideā€™s “pq” Theorem is stated. This theorem establishes that every group of order pįµƒqįµ‡, for primes p and q, is solvable; a group G is solvable if it possesses a series 1 = Hā‚€ āŠ“ Hā‚ āŠ“ ā€¦ āŠ“ Hā‚› = G such that each factor Hįµ¢ā‚Šā‚ / Hįµ¢ is abelian. This is a fundamental result for the classification of finite simple groups.

Menelik Mekonen

Reachability of Nonsmooth DAEs

Abstract:

Reachability analysis asks: given a set of initial states, what other states can a dynamical system reach over time? This question is central for assessing safety of engineered systems such as power systems and chemical process systems. Many of these models are differential-algebraic equations (DAEs), which couple differential dynamics with algebraic constraints and often include nonsmooth features such as min-max functions that correspond to the system switching between modes under changing operating conditions. These nonsmooth DAEs fall outside the scope of most existing reachability results. In this project we extend recent reachability theory for nonsmooth ordinary differential equations to nonsmooth, semi-explicit DAEs by using generalized derivatives to analyze the underlying dynamics and obtain trajectoryā€‘separation bounds for both the differential and algebraic variables. These theoretical bounds lay the groundwork for numerical algorithms that overā€‘approximate reachable sets for safety assessment of nonsmooth power system models.

Andrii Obertas 

Control of Chaos in Discrete Dynamical Systems using the OGY Method

Abstract: 

Chaotic dynamical systems are famously difficult to analyse due to limited predictability and high sensitivity to initial conditions. This motivates the development of methods for controlling such systems. This project investigates applications of the Ottā€“Grebogiā€“Yorke (OGY) method, which allows for stabilization of unstable periodic orbits in discrete dynamical systems. Chaotic dynamical systems are not random but rather possess an inherent structure of embedded unstable periodic orbits that influence the local dynamics. The OGY method exploits this structure by stabilizing selected orbits through small perturbations of system parameters. This project focuses on the idea that complex chaotic systems can be guided towards the neighborhood of a desired state with minimal external influence without eliminating chaotic behavior when the system trajectory is sufficiently close to that state. The method relies on the local linearization of the dynamics and selective adjustment of parameters to achieve a desired behavior of the system. The ideas are demonstrated using simple discrete-time models that retain features of nonlinear chaotic dynamics when subjected to control. Analysis shows that minimal changes can result in large changes in complex systems. These ideas are relevant for potential real-life applications where stability must be maintained in inherently unstable environments.

Alex Sharon

Classifying Satellite Imagery using Topological Data Analysis

Abstract:

Classifying structural patterns in satellite imagery is a problem of significant importance in climate and atmospheric science, yet traditional machine learning approaches often sacrifice interpretability for accuracy. This paper applies persistent homology, a tool from Topological Data Analysis that tracks how connected structures and loops form across an image, to classify the Sugar, Fish, Flowers, and Gravel (SFFG) shallow cloud dataset. Pairwise support vector machine classifiers are trained across all six class combinations, yielding a systematic map of which cloud texture pairs are separable by topological features. An analysis of the classifier decision boundaries reveals that separability is governed by which topological dimension drives classification: pairs where connectivity structure dominates are well-separated, while those where loop structure dominates are far harder. This decomposition provides a transparent, interpretable account of where and why topological methods succeed and fail in cloud texture classification.

Wayne Shaw

Exploring Some Diatonic Scale Group Actions

Abstract: 

This paper explores the group actions of the diatonic twelve tone western musical scale. We can lay out all twelve musical tones in what is called the Circle of Fifths. This common musical object is used to highlight the common tones with each key center. If we rearrange the Circle of Fifths and describe it as it is, a 12-gon, we can get some intuition about how this object should be described with group actions. In fact, here we sort out a set of group operations that when acted upon an element can generate a group that is isomorphic to the dihedral group of the 12-gon, D24. This can be done by looking at diatonic triads as a subset of the collection of all possible 3-point combinations of the 12-gon. This is visually a set of inscribed triangles on the 12-gon. The idea can then be furthered by looking at 4 note collections called Seventh Chords. This explore the objects that these actions generate.

Eugene Smalley

Effects of Home-Field-Advantage on Win Probability and Penalties in the National Football League

Abstract:

With the addition of a 17th game, as well as having several division winners with a sub-.500 record, home-field-advantage has been a hotly debated topic in the NFL in recent years. While the AFC and NFC used to have 8 home games each, an additional inter-conference game each year has caused one conference to host 9 games, while the other hosts only 8. And with the possibility of a 4th seed with a lower win rate hosting a higher-rated 5th seed in the postseason, some teams have called for rule changes to seed the playoffs based on record alone, with division winners no longer getting automatic home-field advantage. This study uses statistical testing in R to see whether home-field-advantage has a significant effect on win rate, penalties, and penalty yardage, based on play-by-play data from the 2025 NFL season. The findings suggest that the number of defensive penalties, defensive penalty yardage, and points scored are affected by home-field advantage. Those results could influence the competition committee’s decisions about playoff seeding going forwards, as well as the league and NFLPA’s negotiations towards adding an 18th game.

Sarah Turmel

Optimization Methods in Deep Learning 

Abstract

Optimization lies at the heart of deep learning, governing how neural networks learn from dataĀ  and directly influencing convergence speed, stability, and generalization performance. ThisĀ  research investigates the standard optimization methods used in training neural networks,Ā  primarily stochastic gradient descent (SGD) and backpropagation, examining why they haveĀ  become foundational techniques and how their limitations have motivated the development ofĀ  modified and experimental approaches. Stochastic gradient descent addresses the computational expense of full-batch gradient evaluation by approximating the gradient using individual or small subsets of training samples, significantly reducing per-iteration cost while enabling scalability to large datasets. Backpropagation efficiently computes the partial derivatives of the cost function with respect to network weights and biases by exploiting the layered structure of neural networks, making gradient-based optimization tractable even in high-dimensional parameter spaces. Building on this foundation, this capstone project explores a range of adaptive optimization methods, including Adam, Momentum, Mini-Batch SGD, Shampoo, Lion, Prodigy, and Sample Gradient Descent. These methods aim to improve convergence speed, numerical stability, curvature adaptation, and robustness to hyperparameter selection. The research analyzes how these algorithms modify gradient updates, the theoretical motivations behind their design, and empirical reasons for their adoption or rejection within the machine learning community. Through comparative evaluation of standard and modified techniques, this project seeks to identify structural trade-offs, such as computational overhead versus convergence efficiency, and assess whether emerging methods offer principled improvements over traditional approaches. Ultimately, this investigation aims to clarify the criteria that define a ā€œstandardā€ optimization method in deep learning and to highlight promising directions for future methodological refinement.

Haleigh Young

Nash Equilibrium in Rock Paper Scissors and the Prisoner’s Dilemma

Abstract: 

In the 1940s, John von Neumann and Oskar Morgenstern developed a new branch of mathematics that used games to model economic behavior. This branch of mathematics and economics is now known as game theory. Game theory is the study of strategic decision-making among rational agents, where each participantā€™s payoff is determined by the actions of others. After a brief history of the development of game theory, this project will examine the mathematics behind game theory, primarily analyzing John Nashā€™s contribution, the Nash equilibrium. The Nash equilibrium describes the phenomenon that occurs when participants reach a state in which no player can increase their position in the game by unilaterally changing strategies. Focusing on two-player games, I will use the prisoner’s dilemma and rock-paper-scissors to describe pure and mixed equilibria.

MS’s – Logan Harkins, Kyle Pease

BS’s – Madison Oliver, Riley Day, Lee Schmidt, Kim Spears

The Chain Reaction Award:  Sarah Lindahl

The Catalyst for Clarity Award:  Natalie Machamer

Below: William Gramlich

Mathematical

MAT 329 – Problems Seminar II, 1 credit, Wednesdays 1:00-1:50pm, Bradley

MAT 362 ā€“ Linear Algebra II, MWF 2:00-2:50pm, Pitt

MAT 425 ā€“ Introduction to Real Analysis I, MWF 1:00 ā€“ 1:50pm, Crisp

MAT 453 ā€“ Partial Differential Equations I, MWF 10:00 ā€“ 10:50am, Patel

MAT 463 ā€“ Introduction to Abstract Algebra I, MWF 12:00 ā€“ 12:50pm, Moss

MAT 471 – Differential Geometry, MWF 9:00-9:00am, Atzema

MAT 487 – Numerical Analysis, MWF 11:00-11:50am, Stechlinski

MAT 500 – Topics in Graduate Mathematics: Number Theory, MWF 9:00-9:50am, Buttcane

MAT 523 – Real Analysis I, MWF 10:00-10:50am, Miller

Statistical:

STS 434 ā€“ Probability Theory, MWF 2:00 ā€“ 2:50 pm, Wang

STS 437 ā€“ Statistical Methods in Research, MWF 10:00-10:50am, Hiebeler

STS 533 – Stochastic Systems, MWF 11:00-11:50am, Parekh

Fall 2026 Enrollments Dates

Graduate Students & Seniors ā€“ Week of March 30 at 7:00am

Student Athletes & Juniors ā€“ Week of April 6 at 7:00am

Sophomores ā€“ Week of April 13 at 7:00am

First Years ā€“ Week of April 20 at 7:00am

Non- Degree Students ā€“ Begins May 4

Sabrine Alcock 
Puck Allen
Katherine Baker
Lily Berard
Riley Day
Connor Dever
Brody Edwards 

Anne Heflin
Katie Horton
Lee Schmidt
Lilia Seekamp
Allison Sweetser
Clara Thompson