鈥淚 never expected it to change the way I think about music education,鈥� said Briggs, who teaches at Dedham School.

The courses asked Briggs to consider what happens when children do more than learn music selected for them. Even the youngest students, she found, could help decide what the music would become.

鈥淥rff opened my eyes to how powerful it is when even our youngest musicians have ownership of their music and are trusted to make musical decisions,鈥� she said.

This summer, Briggs and four other teachers became the first group to complete a Level 3 course offered in Maine. They took all three levels together at 91福利, beginning in 2024. 

The American Orff Schulwerk Association sponsors the three-level training, which introduces educators to a child-centered approach built around play, movement and active music-making.

For Kalyn Simpson, who teaches at Maranacook Community Middle School, giving students ownership can mean handing them buckets and drumsticks. Her students first learn techniques and experiment with sounds. Then they compose drumming cadences in groups.

One group built a cadence around the sounds made by scraping sticks across different parts of their buckets at different speeds. Simpson said the training gave her the skills to prepare students for that freedom and the confidence to give it to them.

鈥淚 don’t think pre-Orff training me would have been comfortable giving that much freedom or being able to provide the foundational skills necessary for them to reach that point,鈥� she said.

Simpson has brought the approach into band rehearsals, too. Students learn melodies and basslines by sound, then discuss how they want to interpret the music. She emphasizes the months spent listening and making decisions together, rather than the single performance at the end.

Dustin Levesque of the South Portland School Department also gives students more say in his lessons. The courses gave him room to experiment and make mistakes while developing as both a teacher and a musician.

鈥淢y lessons are now more centered around student choice, and I am now more of a facilitator in their growth vs. directing every decision they make,鈥� Levesque said.

The five teachers completed Level 1 in 2024, Level 2 in 2025 and Level 3 this summer. Along with Briggs, Simpson and Levesque, the group includes Amanda Marquis of the Old Town School Department and Carlena Trombley of Falmouth Elementary School. All but Simpson are 91福利 alumni.

Returning together each summer also gave the teachers colleagues to call on after the courses ended. Trombley said she gained both ways to foster creativity in her students and a network of educators to exchange ideas with.

鈥淚 now have a network of people who I can share ideas with and continue learning from,鈥� Trombley said.

Marquis described the training as a valuable framework for educators at any stage of their careers.

Laura Artesani, course director and an associate professor in 91福利鈥檚 School of Performing Arts Division of Music, completed all three levels out of state from 2005 to 2007. She said the experience transformed how she teaches general music methods to 91福利鈥檚 music education majors. 91福利 began offering the courses in 2024 after music educators across Maine requested the training.

Each level meets for two weeks in the summer and includes pedagogy, recorder and movement. Instructors have traveled from North Carolina, Nebraska, Massachusetts and Kentucky to teach at 91福利.

This summer鈥檚 Level 3 course was the first offered in Maine. Artesani said she was told it was also the first offered in New England in at least 15 years. Thirty-seven educators from Maine and beyond have participated in the courses at 91福利 since 2024. A new cohort of 16 began Level 1 this summer.

Simpson鈥檚 students had the skills to compose a cadence. What they made from the sound of sticks scraping buckets was their own idea.

鈥淭he Schulwerk trains the teacher to give up control and let the music happen organically,鈥� Simpson said.

Every two years, the Probability and Statistics section of the Annales de l鈥橧nstitut Henri Poincar茅 (AIHP) awards two prizes, of $1,500 each, for the best papers published in the journal during those two years. The selection of the best articles is a collective decision made by all the board members and the different papers which have been awarded over the last years reflect the variety of topics in probability and in statistics which are published every year in the journal.

The Qnity group was honored for their breakthrough UV 26GNF photoresist, related to semiconductor manufacturing. From the ACS website:

“Semiconductor manufacturing is the foundation of modern electronic devices and technologies such as computers, smartphones, solar cells, and automotive systems. Much of this manufacturing has relied on per-and polyfluoroalkyl substances (PFAS) for their heat-resistant and chemical-stable properties. Recognizing the environmental challenges and impacts of PFAS and regulatory concerns, the industry has been working to move to materials other than PFAS.

“UV 26GNF photoresist is a breakthrough in nonfluorine photoresists for semiconductor manufacturing. Qnity has managed to demonstrate the necessary acidity of photoacid generators, which are used to achieve high photosensitivity, without PFAS鈥攁dvancing sustainability and meeting performance needs.”

Read more .

Each student worked on a project with a faculty advisor, with a weekly research group lunch that created a cohort and allowed everyone involved to discuss the work being done. Read on to hear from the students themselves about the work they did and the experiences they had.

Thishanka Alahakoon – Sri Lanka

鈥淚 am currently a second-year master鈥檚 student in the Mathematics program at the University of Maine. I am currently working with Professor Gilbert Moss on a project to construct formal degrees for representations of GL(2,F) over rings in which the residue characteristic is invertible, where F is a non-Archimedean local field. Over the summer, I focused on understanding the representations of GL(2,F) and the classical definition of formal degree over the complex numbers. My summer research was supported by Professor Moss鈥檚 NSF grant.

Cadi Howell 鈥� Maine

鈥淚 graduated from the University of Maine with a BS in Mathematics in 2024 and have just graduated with my MA in Mathematics this summer (2026). For my MA thesis, I worked with Professor Peter Stechlinski to perform a sensitivity analysis of a nonsmooth Wilson-Cowan neural mass network model as well as develop a nonsmooth singular value sensitivity theory that I then used to determine the parametric influence to the singular values of the weight matrix of the nonsmooth Wilson-Cowan model. This summer, the Mathematics Department provided me with the opportunity to finish writing and defend my thesis which allowed me to graduate with my MA degree. Additionally, their funding support allowed me to revise and resubmit a journal article, based on the work I completed in my Honors Thesis, for . I am grateful to have been able to complete these accomplishments, and my experiences at 91福利 have also earned me a spot at the University of North Carolina in Chapel Hill where I am currently pursuing my PhD in Bioinformatics and Computational Biology.鈥�

Henry Johnson 鈥� Maine

鈥�In 2025 I graduated with my BA in math from 91福利 to continue in the graduate program the next fall as a 4+1 student. After I finish my MA, I plan to move to work in industry, hopefully in a field that lets me incorporate some of the applied learning from my thesis. This summer I worked under Dr. Peter Stechlinski studying non-smooth bifurcation theory as part of my thesis work. The math department’s funding allowed me to spend several weeks focused on this work after the end of the spring semester and gain considerable progress on finishing my thesis. The aim of this work was to develop some theoretical test based on non-smooth sensitivity theory that could detect the presence of bifurcations. An important component of the research was for the theoretical tests to be reproduced in more complicated real-world models in order to apply them to the GR4J, a non-smooth discrete time rain runoff model commonly used in hydrology.”

Eva Junge 鈥� Rhode Island & Maine

鈥淚 just completed a double degree from 91福利 in English and Math, and now I am joining the math MA program as a 4+ student. This summer I worked with Professor Andrew Knightly on representation theory research, focusing on unitary discrete series representations of GL(2,R). We defined a compactly supported function to take the place of the non-integrable matrix coefficient for weight 2 discrete series representations, which will have an impact on modular forms research in number theory. This work will provide the basis for my MA thesis and give an important foundation for continued research in my academic career.  My work this summer was supported both by the department and Professor Knightly鈥檚 AMS/Simons research grant.鈥�

Nhi Luong – Vietnam

鈥淚鈥檓 entering my second year of the MA in Math program at 91福利. Before this, I was a math major at Carleton College in Minnesota! This past summer, I worked with Professor Jane Wang on the topic of cutting sequences on translation surfaces, with financial support from Professor Wang鈥檚 NSF grant. The experience has given me a different mode of engaging with math and made the prospect of doing a thesis more manageable.鈥�

Menelik Mekonen – Maryland

鈥淚 just graduated from 91福利 with a degree in mathematics and am entering the math MA program this fall. This summer, with financial support from the Math Department, I worked with Professor Peter Stechlinski on extending reachability results for nonsmooth ordinary differential equations to nonsmooth parametric differential-algebraic equations, drawing on lexicographic derivative theory. Reachability theory is concerned with estimating where a system can end up after some amount of time, but current methods exist for nonsmooth ODEs and for smooth DAEs separately, not for systems that are both. This matters because many real-world systems, such as power systems and those used in process engineering, are naturally described by differential-algebraic equations and are often nonsmooth. The project grew directly out of my capstone research, supported by a CLAS fellowship last year, and has since shaped the direction of my MA thesis.”

Laura Thornburg – Illinois

鈥淚 am starting my second year of the master鈥檚 program here at University of Maine. I completed my undergraduate degree at Mount Holyoke College in Massachusetts. This summer I worked with Professor Tyrone Crisp getting a head start on my thesis project. We developed a new construction of Hopf algebras of representations of combinatorial automorphism groups, and as we continue into the school year, we will study the antipode function on these Hopf algebras.  This work was supported by Professor Crisp鈥檚 AMS/Simons research grant.  I am motivated to complete a thesis because I am preparing to apply to doctoral programs to join academia as a professor.鈥�

(Note: In addition to the students featured above, Alejandro Jimenez also worked on a project with Professor Jane Wang.)

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 鈥渉ard鈥� 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 鈥滱ttention Is All You Need鈥� 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 鈥橝ttention鈥� 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 鈥渙ut of our control鈥�; a popular belief is that a person鈥檚 nature is responsible for how their mind works and its potential, but without understanding why that is, there鈥檚 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 of聽 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, statistical聽 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 Burnside鈥檚 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 Burnside鈥檚 “pq” Theorem is stated. This theorem establishes that every group of order p岬僸岬�, for primes p and q, is solvable; a group G is solvable if it possesses a series 1 = H鈧� 鈯� H鈧� 鈯� 鈥� 鈯� H鈧� = G such that each factor H岬⑩倞鈧� / H岬� 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 trajectory鈥憇eparation bounds for both the differential and algebraic variables. These theoretical bounds lay the groundwork for numerical algorithms that over鈥慳pproximate 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 Ott鈥揋rebogi鈥揧orke (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 data聽 and directly influencing convergence speed, stability, and generalization performance. This聽 research investigates the standard optimization methods used in training neural networks,聽 primarily stochastic gradient descent (SGD) and backpropagation, examining why they have聽 become foundational techniques and how their limitations have motivated the development of聽 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 鈥渟tandard鈥� 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 participant鈥檚 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 Nash鈥檚 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