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Proving Patterns in Primes: The History of the Quadratic Reciprocity Law 

This presentation outlines the historical developments in the conception and solidification of the Quadratic Reciprocity Law. Starting with Fermat in the 1500s, the idea that prime numbers had patterns continued to be pondered for over 300 years later, inspiring great mathematicians such as Euler, Lagrange, Legendre, Gauss, and Eisenstein. An analysis on the techniques used by each of these to prove that certain primes always exist in certain forms naturally leads to a discussion on the Quadratic Reciprocity Law and its usefulness both at the time and in the development of modern number theory.

The Evolution of Curvature

The intuitive notion of "curvature" is a very old one, but the effort to turn this intuition into a formal mathematical concept required the work of many individuals over several centuries. In this talk, we will meet five of the major players in this story: Huygens, Newton, Euler, Gauss, and Riemann. We will see how each mathematician's theoretical contributions to curvature were intricately connected to major applied problems of their time. Finally, we will briefly journey from the work of Riemann to the modern day, exploring how these developments have shaped the current field of mathematics.

From Discrete to Continuous

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Zahra Cheesman

Computational Complexity of the Wild Number of Edge-Labelled Graphs

This paper introduces the wild colouring of a graph, a novel concept in graph theory which was developed out of work on graphical algebraic splines. We discuss the notion of a graphical algebraic spline and revisit the seminal paper of Anders, et al, that determines when an edge-labeled graph admits only constant splines. We define the notion of the wild colouring of an edge-labelled graph and establish the computational complexity of determining whether a given edge-labelled graph with L colours has a wild edge set of size at most k. This work will help to establish a new branch of research in graph theory, while contributing to the list of known NP-hard problems and computational complexity. We further highlight the collaborative and inclusive nature of mathematical discovery, particularly the contributions of women mathematicians in this field and the value of mathematical research as a powerful avenue for social change.

Mustafa Sameen

FORMAL:  Democratizing Lean 4 Formalization Through Retrieval-Augmented Thinking and Agentic Feedback Loops

This thesis presents FORMAL (Feedback-Oriented Retrieval Method for Automated Lean translation), a novel system that automates the translation of natural language mathematical statements into formal Lean 4 code. By combining Retrieval-Augmented Generation (RAG) with an agentic feedback loop, FORMAL introduces Retrieval-Augmented Thinking (RAT), where formalized mathematics and Lean 4 proving tactics in a vector database (stored using ChromaDB) guides the generation process.
Unlike approaches that require large models or extensive cloud computing resources, FORMAL deploys relatively small, local language models (Llama 3.2 and DeepSeek-R1-8B), using Ollama, that iteratively refine outputs based on feedback from the Lean 4 theorem prover. The system incorporates tactic awareness to produce structured proofs using Lean's tactics.
The system on being evaluated on benchmarks including the FormL4 dataset, achieves 92% syntactic correctness and 86% semantic accuracy on the basic test set, compared to only 58% and 42% respectively for a pure LLM approach. The performance remains robust even on out-of-domain problems, where the system achieves 78% syntactic and 70% semantic accuracy. The agentic feedback loop is effective with most successfully formalized problems solved within just 2-3 iterations.
This work democratizes access to formal verification tools and techniques by demonstrating how contextual retrieval, tactics awareness, and compiler feedback can be used to significantly improve autoformalization without requiring a large amount of computational power. 

Zach Zerbe

Modeling Opinion Dynamics in Social Networks

In an age where we are constantly exposed to information from a variety of sources, what steps can we take to better understand the effect these sources have on the beliefs we and the peers in our social networks have? In this thesis, we explore the practicality, insights, and limitations of using mathematical models to describe belief dynamics in social networks. In particular, we use an agent-based modeling approach to the interplay between the effects of peer influence versus the influence of generally accessible information sources on various simulated network structures. We run several experiments with the model’s belief updating algorithm, and compare the results against an approximation of the system using an eigenvector based projection of system convergence. 

Tanner Flagg

Social Network Classification

Social networks have unique underlying structures that characterize the relationships they encode. Social network classification is a task that is focused on using our understanding of different networks to determine whether a network belongs to a larger one. This study discusses the structures that characterize a network and uses them to accurately classify social networks. 

Connor Wellman

Generalized Rosette Harmonic Mappings

A harmonic mapping f is a complex valued univalent harmonic function defined on a region in the complex plane. Rosette harmonic mappings are generalizations of the polynomial harmonic mappings through modifying the canonical decomposition with hypergeometric 2 F1 factors. We expand upon the Rosette Harmonic Mappings, which have analytic and coanalytic parts with exterior angles of 𝜋/2 at the nodes. By introducing a parameter q, we define Generalized Rosettes, which have analytic and coanalytic parts with exterior angles of q𝜋. For appropriate parameters, these Generalized Rosettes ‘lift’ through Weirestrauss Enneper equations to the Generalized Rosette Minimal Surfaces. At small values of q, these surfaces approximate the classical minimal surface known as Enneper’s Surface, creating a link between this classical surface and the triply periodic Rosette Minimal Surface. As q increases, the angles at the nodes of the surfaces become increasingly sharp, and the surfaces eventually become unbounded. Moreover, their projections are no longer univalent harmonic functions. By introducing a rotation angle 𝛽, with 0 ≤ 𝛽 ≤ 2𝜋, we find that seemingly unrelated Generalized Rosette Surfaces are in fact part of an associated family, suggesting they are conjugate surfaces.

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Nathan Curl, Alana Ermeus, Arez Khidr, Karla Prado, Dylan Shryer

Scholar 2 Mentor

Scholar to Mentor is designed to streamline the mentor-matching process for Greenhouse Scholars, which is currently done entirely manually and requires significant time. Our program automates this task while maintaining the accuracy of human-made matches. By analyzing scholar and mentor applications, it identifies key similarities and returns each scholar's ideal mentor matches. This not only accelerates the process but also frees up valuable time for Greenhouse Scholars to focus on their broader mission of supporting student success in college. 

George Beck, Henry Howe, Abe Lipson, Stuart Sessions, Conor Wellman

Smart Bettor

SmartBettor.ai is a website started by a CC Alum which aims to provide users with statistics on favorable sports betting opportunities. In this project, we built out a new trends page on their website which finds trending bets and displays relevant statistics to users. This project involved refactoring old code, including the completion of a new Identity Access Management system.

Yael Homa, Evelyn Needham

SpyderBrats

Pikes Peak Performance Project: Optimization of the Spyderbat Analytics Engine

The Spyderbrat’s project aimed to enhance the performance of the Spyderbat analytics engine by leveraging the lower-level programming language Rust. Spyderbat, a cybersecurity company based in Austin, Texas, automates threat hunting by identifying the root cause of malicious events. Spyderbat’s Analytics Engine, which is written in Python, is responsible for parsing a machine’s records and mapping relevant data. When handed over to the team, it faced performance bottlenecks that limited its ability to deliver real-time threat analysis effectively. After reviewing the engine’s codebase, the Spyderbrats determined that the existing implementation was already highly optimized within the constraints of Python and Spyderbat’s design decisions. To enhance performance, they profiled the engine and rewrote critical components in Rust within the constraints of maintaining readability and the design choices of the company.

Kalie Chang, Judy Gonzalez, Grace Mun, Tori Zhu

Language Exchange

Language Exchange is a platform designed to help Colorado College students, faculty, and staff develop verbal language skills through interaction with the campus community. Due to the Block Plan at Colorado College, students often have limited time to learn and practice a language. Outside of formal language courses, there are few opportunities to maintain and improve language proficiency. Language Exchange aims to fill this gap by providing a resource for students to enhance their speaking skills while connecting with others interested in developing language skills. By fostering these connections, the platform supports students in building confidence and proficiency in their target languages.

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George Beck

Statistics of Professional Cycling

Professional Cycling is a unique and exciting sport that combines strength, endurance, teamwork, tactics, and skills. In 2020, Zwift, a mixed-reality cycling platform, hosted a Virtual Tour de France over six weekends in the summer. There were six stages of racing each with a different course that resulted in about an hour
of virtual racing. As the professional riders raced virtually from all over the world, the outputs from their power meters were broadcast for the world to see.

This paper aims to answer two questions: What power output metrics are the most significant in describing a rider’s finish time? And, can we predict a rider’s finish time using their power metrics? The data from the Virtual Tour de France is modeled with a linear regression and multilevel models. Through our analysis, we found the peak 20-minute relative average power and the peak 15-second relative average power to be the most significant metrics. This informs us that both a rider’s ability to sustain an effort and their sprinting ability are important in mixed-reality competitions. The mean squared error and root mean squared error through 10-fold cross validation are computed to assess the predictive performance of the models. We show that multilevel models have better predictive performance than linear regression.

Kenna Grenier

Understanding the Kakeya Set

The Kakeya needle problem asks a simple question: what is the smallest possible area in which a unit-length needle can be continuously rotated through a full $360^\circ$? While simple shapes, like a circle or a deltoid triangle, seem like natural answers, more surprising constructions allow the needle to rotate in arbitrarily small regions.

We explore a construction called a Perron tree, a fragmented shape built by iteratively shifting and overlapping triangles. This peculiar structure allows a needle to turn while occupying an area that can shrink as much as we like, leading to the striking conclusion that Kakeya sets can have practically zero measure.

This raises deeper questions: If a Kakeya set can be arbitrarily small in area, what is its true dimensionality? Do their infinitely fine details give them full dimension, or are they ‘thin,’ existing in a lower-dimensional space despite their complexity? 

Hunter Markowich

Kaktovik Numerals

I will explore some mathematics from indigenous cultures, featuring counting, operations on numbers, and various applications in calendars, mapping, kinship, and other everyday concepts. I will mainly focus on the Inupiaq people of Northern Alaska, and their recently-invented number system that began as a middle school class project.
My paper will begin with a survey of the varying number systems and approaches to mathematics from different indigenous communities, and reflect on how different systems have arisen based on the applications and needs in these communities. I will also examine a specific indigenous number system: the Kaktovik numerals. Then, I will discuss what the system is used for, and how it differs from the widely-used base-10 system.
I will conclude with a closer look at how the spoken Inupiaq language links to the Kaktovik numeral symbols, in place-value, addition, subtraction, multiplication, and division.

Brendan McCune

Explorations in Category Theory

ℤThe concept of completion plays a fundamental role in mathematical analysis and algebra. The classical real numbers arise as the completion of the rational numbers with respect to the usual absolute value, but an alternative class of completions, the $p$-adic numbers, emerges when one instead considers the $p$-adic norm. This paper introduces the construction of $p$-adic integers and $p$-adic numbers as respective completions of ℤ and ℚ, with respect to the 

Porter Barnes

Topological Data Analysis of Weather Patterns

The El Niño-Southern Oscillation (ENSO) is one of the climate phenomena with the greatest impacts on weather across the globe. Though it is formally defined by average sea surface temperature (SST) increase or decreases across a large span of the Pacific, this paper investigates whether or not the shape of the sea surface temperature is also correlated to the ENSO phase. To that end, this paper introduces the fundamental concepts of Persistent Homology in order to use Topological Data Analysis to answer that question. Using Monthly mean SST data from a grid of points in the Pacific, we compute persistence diagrams for each month and ENSO phase, and then use bottleneck distances to compare those persistence diagrams. The result is clear evidence that the shape of SST data, beyond the anomaly in SST's total mean, is indicative of the ENSO Phase.

Sam Johnson

Explorations in Political Geometry, Mathematical Redistricting

Redistricting is critical in the democratic process, determining political representation by defining electoral district boundaries. However, the practice is often subject to manipulation through gerrymandering, where district lines are drawn to favor specific political parties or groups. This thesis explores computational methods for evaluating redistricting plans for partisan bias using ensemble analysis and Markov Chain Monte Carlo (MCMC) techniques.

By leveraging GerryChain, a computational tool designed for redistricting analysis, this study generates large
ensembles of districting plans to assess whether a given redistricting configuration is an outlier. A key focus is
the Recombination (ReCom) algorithm, which modifies district boundaries while maintaining contiguity and equal population constraints.

A case study of El Paso County, Colorado, is presented to demonstrate these methodologies. The analysis
compares the current districting plan against a large ensemble of possible configurations, revealing whether the
existing map exhibits signs of partisan bias. 

Zoe Harrington

Hyperbolic Tilings on Triply Periodic Minimal Surfaces

Hyperbolic tilings and triply periodic minimal surfaces (TPMS) seem to have little in common at first glance, however hyperbolic tilings can be mapped onto TPMS similarly to regular tilings on Euclidean surfaces. This project explores these connections by explaining important features of hyperbolic tilings and TPMS, delving into the specifics of orbifold symmetry notation and the tools from topology that map the tilings onto TPMS. Finally, we explore examples of hyperbolic tilings mapped onto TPMS, particularly the P-surface. Applications of TPMS range from materials science and engineering to biomineralization due to their unique structure. Many of these natural occurrences are not yet well understood by scientists, hence better understanding minimal surfaces and connections to hyperbolic geometry furthers understanding of mysterious natural and physical structures.

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Yunus Bolat, Zahra Cheeseman, Tanner Flagg, Nick Thomas, Zach Zerbe

Caravan

Caravan is an iOS app intended to help plan and enhance the group road trip experience through its integration of real-time navigation and group coordination. Unlike existing tools that commonly isolate navigation from communication, Caravan allows users to track friends' live locations, collaboratively add and manage stops along a shared route, and plan customized trips in a centralized interface. Caravan makes road trips more organized, social, and memorable.

Luke Anderson, Jimmy Andrews, Primera Hour, Shamdeed Kabir, Mustafa Sameen

AFK

Working with the Southern Collegiate Athletic Conference (SCAC)—Colorado College’s Division III athletic conference—our team AFK (an acronym which gamers know as "Away From Keyboard") built a platform to transform Esports tracking across member schools. By combining a centralized website, database, and Optical Character Recognition (OCR), we replaced manual scorekeeping for games like Apex Legends and Valorant with a rapid screenshot-to-database pipeline. Beyond improved competitive transparency and streamlining the scoring process, our solution gives SCAC staff and players exactly what our name promises: more time "away from keyboard"—freeing them to focus on strategy, growth, and their shared passion of collegiate Esports.

Talon Carballeira, Isaac Greenwald, Sawyer King, George Sowles

Tiger One

Tiger One is a product designed to give teachers and school administrators the ability to access data on how large language models are being used in the classroom. The product consists of a chat page, which allows the student to interact with a large language model. The question and response of the student is logged in a database and displayed on the Admin page accessible by teachers. The Admin page also displays usage statistics. These statistics include usage over time, sentiment analysis, and common themes.

Kiernan Nesslar, Kazu Shimotake, Isabelle Wagenvoord

Autograder

Grading for CC Computer Science classes can be tiring, time-consuming, and just a bit obnoxious with the Canvas interface, even for very trivial assignments. This takes the time of busy professors and results in students having to wait long periods for their grades, which itself leads to piling backlogs of incomplete and incorrect assignments. CATS is a configurable command-line auto-grading program that can build custom testing pipelines for each Canvas assignment. Each time an assignment is submitted, CATS automatically downloads, performs custom tests, grades, and posts detailed feedback on student assignments to Canvas. For more subjective criteria, CATS provides output to streamline grading. CATS simplifies the grading process for all involved, saving valuable Block Plan time, and rescuing students from Week 4 headaches.

Willa Polman, Kaylie Stuteville, Ronan Takizawa, Anna Vu

AssessAI

AssessAI is a desktop software tool designed to help users evaluate and test the performance of large language models (LLMs) specifically in summarizing custom datasets. The tool allows clients to upload their datasets and assess how well various LLMs summarize their content. With this, users can determine how effectively these models could potentially serve their own projects.

AssessAI supports various popular LLM summarizers and provides evaluation metrics using BLEU, METEOR, ROUGE-Score, BERTScore, and G-EVAL. This allows for a detailed view of how each model performs, allowing the user to see which model best fits their needs.

Mira Giles-Pufahi, Oliver Kendall, Oliver Ramirez, Leigh Rose Walden, Dan Schmidt

Big Nasty

Understanding how votes add up matters. It matters for big races, but it matters for smaller, down-ballot races too. The Colorado down-ballot vote visualizer is a tool for understanding down-ballot voting trends in Colorado races. The visualization tools coloradovotevisuals.com offers are meant to lend a hand to journalists covering these races and help voters seeking more information about how support for state-level issues has changed over time. The site quickly generates three different types of visuals for any down-ballot race in Colorado that can be exported in their entirety and used in local election reporting. All the data for these visualizations comes from the Colorado Secretary of State’s elections data.

Kyle Moriarty, Oliver Moscow, Caleb Piemann

Front Range Lights

With increasing urban development and expansion previously undeveloped areas, the cultural and scientific impacts of light pollution have been studied extensively. It is true that more urban development leads to more light pollution, but quantitatively determining the amount of light pollution for any area depends on a number of factors, from the types of infrastructure present, implementation of light pollution policies, and density of light sources. Working with State of the Rockies, we believed it would be important to focus on developing a tool that would help visualize and minimize the impacts of light pollution locally. This project aims to use VIIRS data and machine learning algorithms to predict the changes in nighttime sky brightness based on the businesses and zoning data in Colorado Springs. We collected data on businesses from the ArcGIS business suite and zoning data from SpringsView. This data was used to train the model on the amount of light emitted per business so that city planners can use our tool to calculate the expected increase in light pollution when adding businesses to an area.

David Wine

An Analysis of Racial Disparity in Police Stops in Illinois

In this paper, I will perform analysis of a large data set containing information on police stops in the state of Illinois in an attempt to find potential racial biases. This is done by comparing the rate at which races are stopped to the rate at which those stops are ‘successful’, or which of those stops turned up contraband or led to further investigation or arrest. I found that Black or Hispanic drivers are stopped at a higher rate while also having a lower rate of success, indicating bias. I also completed the ‘Veil of Darkness’ test, comparing stop rates at a time of day when it is light outside during some part of the year and dark outside during another.

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Yousheng Tang

Harmonic Mappings and the Hexasquare

This study explains the foundational principles required for the creation of the "Hexasquare Minimal Surface". This study covers the main theorems, proofs, and examples related to analytic functions, conformal mappings, and Riemann Mapping Theorem. Building upon the fundamental theories, this study also includes a systematic exposition of the Schwarz-Christoffel Mapping, Poisson Integral Formula, Sheil-Small Theory, and their relation to the Hexasquare. Using harmonic functions defined on the unit disk, this study explains the mapping to hexagons that extends the properties of the harmonic mapping connected with the classical Scherk minimal surface. The new Hexasquare surface changes its boundary heights more frequently, and expands our understanding of Jenkins-Serrin Surfaces and their geometric properties.

Trey Crawford

Beyond Buzzer Beaters

This thesis provides an analysis of National Collegiate Athletic Association (NCAA) basketball stats and their relationship with winning percentage as determined by the outcome of individual games. It strives to identify the "Four Factors" that contribute the most to the outcomes of Division 1 (D1) College Basketball games during the 2016 season. Using a mathematical technique called \textbf{Logistic Regression}, the method can solve for weights or coefficients that test how important each stat relates to wins and losses. The findings of this study will further our understanding of the complex and dynamic nature of college basketball as the game keeps evolving and provide insights into the crucial statistics that influence game outcomes.

Silas Blanchard and Quinn Sebso

Esports Computer Availability

Our program aims to address a fundamental problem with the Colorado College esports lab; there is no way to know how busy the lab is without being there. For many students, particularly in cold months, the walk to the lab could be a huge wasted effort if they find it full. Our project is a website, built using node js, that displays which computers are in use and which are free. Additionally, it displays the calendar, something only available through Discord currently, and supports customization by gaming lab staff. Available to anyone on the Colorado College campus (and using a Colorado College network), this project is currently deployed and visitable at http://esportscomm.coloradocollege.edu.

Eric Uerling, Ethan Fuentes

Boys and Girls Country App

This project is to make an IOS app for Boys and Girls Country of Houston (BGC), a nonprofit children’s home. The app contains content about donating, placing a child, with the goal of creating an emotional connection and encouraging users to get involved with BGC. Another feature was the Clothing Closet, which allows BGC to upload their stock of donated clothes so that children and staff can easily view and order clothes.

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Gwen Hardwick

Supersingular Isogeny Diffie-Hellman Cryptography

In today's digital era, the need for secure communication is prominent. While traditional systems like RSA and ECDH fend off classical threats, quantum computers pose a new challenge. The Supersingular Isogeny Diffie-Hellman (SIDH) protocol emerged as a promising candidate, being one of three finalists in the NIST Post-Quantum Computing replacement campaign, prior to the groundbreaking Castryck-Decru Attack being published in July 2022. Its foundation lies in traversal over an isogeny graph made of supersingular elliptic curves.

Esa Chen, Junhao Qu

MonsterVault: Budgeting App

In an era where effective financial management is paramount yet challenging for many, MonsterVault, an iOS app, revolutionizes financial management by incorporating a virtual pet care system, encouraging expenses and income tracking with features like receipt scanning, object detection, and manual entry. For each recorded tracking, users accumulate points that can be exchanged for pet food, connecting users’ budgeting actions with the growth of pets. This gamification aspect makes financial management a more engaging and rewarding process, aiming to cultivate lasting financial discipline among users. MonsterVault enhances financial literacy through insights into spending patterns by offering a visual representation of spending habits and budget status. MonsterVault turns the routine task of budget management into an enjoyable activity, fostering financial discipline and literacy uniquely and interactively.

Fremont Fosberg

Async Co.

Async Co. is a software application designed for a theoretical Venture Capital company, inspired by a past work experience. Aysnc Co. is meant to be a centralized workplace for multiple workflows. The application allows the user to see data stored in a database about Investments, Investors, the relationship between Investors and Investments, as well as KPI data that would theoretically be acquired each quarter from the Investments to track growth.

Louisa Penrice, Blanche Stora

RoCCy Rides 

As many of us know, the closest major airport to Colorado College is in Denver, an hour and fifteen minute drive away. Ride share applications, shuttles, and long term parking are expensive, and those with their own cars are forced to waste gas (and money) on a ride for themselves. Every school break, student group chats are flooded with inquiries hoping to connect with others flying in or out of the airport at similar times. RoCCy Rides is an iOS application built in Swift that streamlines this process by allowing students to post, chat, and connect with one another regarding airport transportation based on flight times. Thanks to this app, students will have one less thing to worry about when traveling.

Clay Arnold, Omar Castro-Frederick, Tom Heffernan, Jackson Kaib

A Machine Learning Approach to Detecting Credit Card Fraud

Credit card fraud poses a challenge for financial institutions, leading to losses and erosion of customer trust. We have developed Fraud Finders, a machine learning-based system designed to detect fraudulent credit card transactions. With models including Random Forest and XGBoost, our system looks at fraud data to identify patterns of fraud. We use Machine Learning Models trained on historical data, and a dashboard to visualize the results. We aim to make a solution that is scalable and secure. Our system hopes to mitigate risk and protect consumers.

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Jack Dresser, Lucy Flanagan, Jay Moran

TigerList

TigerList is a website for Colorado College students to buy, sell, or exchange goods and services. Mimicking an online marketplace, TigerList allows only Colorado College students to use the platform. Unlike other online marketplaces, TigerList guarantees products are within walking distance, are in safe locations and are only amongst other students. This improves safety and equity, while also encouraging good sustainability practices. Currently, Colorado College students have no single interface to message each other about buying and selling goods. Messages end up in class GroupMes, on Instagram stories, or by word of mouth, where everything typically gets drowned out, and also creates an equity concern of who gets what items. TigerList is a confluence for students to buy and sell goods without going to extreme measures.

Mai Nguyen, Khawla Douah, Calvin Than, Dylan Chapell

CoralLabeler: A better way to label large datasets

We built the CoralLabeler application to assist our clients, Dr. Cory Scott from Colorado College and Dr. Amber Stubler at Occidental College, in labeling a large dataset of underwater corals. We built tools into this application to manually label parts of the image and developed an API to allow a configurable machine-learning model to predict image labels. Once the labels have been produced, they will be used to train more effective machine-learning models for coral classification and generate statistics relevant to coral research. 

Haoru Yang

The Winning Strategies of RISK: the Game Dynamic of Rolling More Dice with s Sides

The board game RISK has many interesting but complicated probabilistic problems. The classic version of RISK involves 3 attacking and 2 defending dice with 6 sides. This work investigated the modified versions of RISK with t × m battles, which involves t attacking dice and m defending dice with s sides. The paper presents the generic formulas for all possible outcomes of a single attack in 3 × 2 and 3 × 3 battles, and the simulated results of a single attack in other t×m battles. These results for each attack allow the estimation of actual conquer probabilities by normal approximation, and the estimation helps find the winning strategies and discuss game dynamics in modified games. The winning strategies show the number of attacking armies needed to conquer an enemy territory with approximately 50% and 80% chance of successful conquer, for example, the attacker needs 1.71 times of the defending armies to have over 50% chance of success in 3 × 3 battles. The paper also introduces a concept called ”balance of power” that measures the equivalence between the attacker and defender, for example, 3 × 2 battles with 6-sided dice favor the attacker and 3 × 3 battles with 6-sided dice favor the defender. The balance of power helps players understand the game dynamics and make reasonable decisions.

Obie Kahn

Chaotic Orbits and Random Fibonacci Sequences

From sequences (of numbers, of events, of movements) there can emerge a great deal of information. This thesis explores two types of sequences: (1) sequences, or "orbits," generated by repeatedly applying the same function to an initial value, and (2) variations of the famous Fibonacci Sequence, which we define recursively. By studying "period three" orbits in particular, we demonstrate how mathematical "chaos" arises in deterministic systems that have seemingly random behavior. We then harness randomness to construct variations of the Fibonacci Sequence with one random process and explore their growth rates, using these rates to offer a guess for the growth rate of a Fibonacci Sequence generated by two random processes.

Jingyi Liu

Comparative Analysis of Frequentist and Bayesian Variable Selection Methods on Mental Health Datasets

Variable selection remains an important problem researchers encounter when building a statistical model. While frequentist methods have been widely used in many fields, the Bayesian variable selection received less attention due to its complexity. In this paper, we review variable selection methods from both frequentist and Bayesian perspectives. We evaluate the performance of such algorithms via simulation studies using cross-validation. We argue that Bayesian variable selection is advantageous as it shows superior performance in identifying the true underlying process. The predictive performance of frequentist and Bayesian variable selection was assessed on international students' mental health during the pandemic as an application in psychology.

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John Lê

Bayesian Optimization: Theory and Applications

Optimization is a crucial process in mathematics, especially used in real-world scenarios such as economics, engineering, and machine learning. This is because optimization can give us information about an ideal solution in a scenario, underlying phenomenon given specific parameters, and etc. However, when dealing with black-box functions where we do not know anything about their usages and characteristics, Bayesian Optimization is a powerful tool that can be used to find global optimums in unfamiliar environments. This thesis provides a theoretical framework to Bayesian Optimization by understanding the foundations of Bayesian Inference.  Alongside the theory, this thesis explores the implementation of Python to employ Bayesian Optimization in various computational problems. Specifically, we will look at data in optimizing parameters for ferroelectric materials. 

Zhiqi Yao

Comparison of the Predictive Ability of Different Financial Models for Stock Prices

This study compares the predictive abilities of two time series analysis models, the Geometric Brownian Motion (GBM) model and the Autoregressive Integrated Moving Average (ARIMA) model, in forecasting the stock prices of the S&P 500 index over both long-term and short-term periods. The performance of each model is evaluated using two error measures: the Root Mean Square Error (RMSE) and the Mean Absolute Percentage Error (MAPE). The results show that the ARIMA model outperforms the GBM model in predicting stock prices for both long-term and short-term periods.

Dan Conlin, Alisha Bloom, Elliot Triplett

Black and Pink Resource Portal and Data Visualization Page

The Southern Colorado chapter of Black and Pink is a prison abolition group that focuses on providing support for members of the community who have recently been unincarecerated. This project adds two pages to their website: one for folks to find reentry resources, and one aimed at educating people about the importance of prison abolition. Some of the resources include housing equity, LGBTQ+ centers, centers for people transitioning out of prison, food resources, and employment opportunities. The portal is easy to use, and easy for Black and Pink volunteers to update. The data visualization page provides clear visual portrayals of racial and other inequalities in the prison industrial complex within Colorado and the greater United States.

Dan Phuong, Teva Tannenbaum, Gwen Hardwick

College Marketplace App

To address the issue of college students’ wasteful behavior and transportation challenges, we developed a marketplace iOS app tailored for student needs. The app, designed with sustainability in mind, offers a platform for students to buy and sell items conveniently. With a user-friendly interface and a swiping mechanic reminiscent of popular social media apps, it encourages engagement and addresses safety concerns by facilitating transactions within the student community. A variety of other features are also included to contribute to the user experience.

Hayley Heineken

Pattern Formation in Arid Grasslands

Arid grasslands cover an estimated 35% of the earth’s surface and exist in nearly every continent (Archibold 2012). Low rainfall amounts and limited growth characterize these landscapes. As the changing climate threatens a continued decrease in rainfall, ecologists and applied mathematicians seek to quantify the point of ecosystem collapse and explore the effect of different conditions on observed patterns. We offer a model consisting of two partial differential equations that interact in a Turing manner to instigate pattern formation. Ultimately, we were able to quantify the point of ecosystem collapse under different parameter conditions, figures that will contribute to future conservation efforts. Additionally, we offer an in-depth analysis of the effect of different rates of water advection and diffusion on the types of patterns produced. This paper seeks to add to the existing ecological and mathematical knowledge on this topic through the introduction of a simple differential model that captures the most important aspects of more complex models on this topic.

Daniela Santillan

Pattern Finding: A Curriculum for the Stroud Scholars Program

This project consists of building a 3-week mathematics curriculum on pattern-finding for the quantitative reasoning course on the third and final summer of the Stroud program, which is a program that provides under-resourced high school students the experience of academic preparation and the opportunity to earn admission to Colorado College. The objective of the curriculum is for students to identify patterns, describe them mathematically, and demonstrate how pure math can be applied in the real world. Patterns we explore include number sequences, symmetry of figures, and fractals.

Tiia Shea

Integrating more real-data into Calculus I courses: An exploration and adaptation of Maximum Likelihood Estimation for math curriculum

How can students be motivated to invest in their own learning? This paper addresses the lack of real-world data in pure math curricula by adapting the maximum likelihood estimation method for usage in a Calculus I class. This is achieved through the creation of a small MLE-based calculus module.

Maximum likelihood estimation (MLE) is a statistical procedure used to estimate the parameter(s) of a probability distribution that best describes a data set. By using optimization of single-variable equations, MLE can be used to bring more real data into math classrooms. 

Results of module implementation in Blocks 3 & 4 primarily show that students' perceptions about math applicability (where and how math can be useful in the real-world) and self-efficacy improved. Work in mathematics education is crucial to improving math curricula and making mathematics more applicable and accessible to all students.

Leo Fries

Positive Steady-State Varieties of 2-Reaction Chemical Reaction Networks

Chemical reaction network theory is a field of applied mathematics concerned with modeling chemical systems. It is used in many areas of science, for example in systems biology to understand cellular signaling networks. This research seeks to understand networks’ biologically relevant equilibrium points through the lens of algebraic geometry, by translating the networks into systems of polynomial equations and computing the positive portion of their solution sets. Currently, there is no algorithm to calculate this positive steady-state variety in general, or to determine if the variety is nonempty. This thesis provides the conditions for 2-reaction networks to produce a nonempty positive steady-state variety and a more in-depth classification of the varieties produced by 2- and 3-species, 2-reaction networks, grounded in combinatorial and algebraic properties.

Walt Jones and James Settles

Heat Sheet Cheat Sheet

The Accurate Heat Sheets project will allow a NCAA college track coaches to see competitor information for any race in one button click. The coach will upload a heat sheet to accurate-heat-sheet.com and will receive every runner’s personal best time, season-best time, and other relevant information about the runners so that the coach may make accurate race strategy decisions for their athletes. Initially, this will be deployed as a web application, and the database will support the 800, 1500, 3,000 steeplechase, 5,000 and 10,000 meter distances. 

Kaija van Zante, Kathleen Shea, Lizzie Blaschke, and August Knox

Cutler Publications App

Get access to all of Culter’s publications on-the-go! Currently, Cutler sponsors five different publications, all of which have different release schedules and limited physical copies, making it difficult to keep up with the variety of writing and art that students are producing. Our iOS application allows users to view content from The Catalyst, Cipher, Leviathan, Anamnesis, and The Disincentive. Exciting features include favoriting articles for easy access later, the ability to search by keywords and article content across all publications, and receiving notifications from authors/publications users subscribe to. By bringing all of Cutler’s writers and artists together under one mobile platform, we hope to boost readership and retention for Colorado College’s numerous publications. 

Tyler Chang and Simay Cural

Preserve: Reduce Waste at Home

The FDA indicates that the typical household wastes around 32% of purchased foods. Preserve, an iOS mobile application, serves as a solution by keeping track of ingredients within refrigerators, pantries, and freezers accompanied by reminder notifications of when an item is about to go bad. Users can import food from the physical world to the digital world through barcode scanning, receipt scanning, or manual entry. Each imported item is assigned an estimated shelf life based on a dataset from the USDA. Users are then reminded of products that are about to or have expired. Additionally, Preserve recommends recipes that use produce that are soon to go bad in the inventory. With these tools, Preserve provides intuitive ways to prevent food waste.

Olivia Bouthot

An Exploration of Symmetry Groups 

Humans are drawn to patterns, whether that be in the reflective symmetry of butterflies or repetition of musical beats. In exploring pattern classification through geometric means, it becomes clear how math always informs the structure of our world, including beauty. This thesis explores the classification of frieze and wallpaper groups. We follow the methodology of William Barker and Roger Howe in their book Continuous symmetry: From Euclid to Klein. We categorize 7 isomorphism classes for frieze groups, and 17 for wallpaper groups. The method is similar for frieze and wallpaper groups, illuminating the possibility of higher dimensional pattern classification.

Tim Somerset

Prior sensitivity analysis of Bayesian hidden Markov models for hospital infection data

Analyzing hospital infection data presents a number of difficulties from the structure of the data – sparse, low counts, and auto correlated – to the nature of the data generation process – transmission through largely unobserved infections. A natural solution is to use a hidden Markov model. This talk focuses on a Bayesian methodology to this model, a key step of which is to define our prior belief of key transmission parameter values. Due to the size of the dataset, our prior belief has a large impact on the output of the analysis. What is this impact? And is there a predictable pattern we can identify?

Topics covered in the talk include: Bayesian/frequentist statistics, Monte-Carlo Markov chain sampling, and hidden Markov models.

Edie Brazil

Mathematical Model of Stochastic Differential Equations of Population Recovery Dynamics with a Non-Constant Carrying Capacity

Stochastic models integrate randomness into the model which allows us to analyze systems and processes and their dynamics in the presence of noise. We use the methods outlined by Gillespie in his paper, “The Chemical Langevin Equation” to develop a mathematical model of stochastic differential equations modeling population dynamics with a nonconstant carrying capacity in the wake of an assumed population devastating event. We are interested in how the aggregation of noise contributed by the individuals of a population affects population recovery dynamics. This presentation will include an overview of our model and its derivation and underlying assumptions and an analysis of our simulation results.

Jessica Hannebert, Moises Padilla, Giang Pham, Pralad Mishra

The Quantitative Reasoning Center Scheduling System (QSS) 

The QSS is an online application that will be used by the directors of the Quantitative Reasoning Center (QRC) for scheduling tutors’ walk-in shifts. The QSS will utilize a variety of tutor inputs to determine the schedule by allocating the best set of shifts to each tutor. The website will allow the heads of the QRC to manage all aspects of the scheduling while giving the tutors the ability to select which shifts they would like to sign up for. With this application, blockly meetings between QRC tutors and administrators will become much more efficient and organized.

Will Barber, Tucker Hale, Bryan Moreno, Ronak Patel

Find Your Fun: Building Your Calendar With A Little Help from AI

Search engines today have made it extremely easy to find anything and everything, too easy in fact! When it comes to adding to your personal schedule, too many options on what to do can make adding events and planning unnecessarily cumbersome. Our web application seeks to make the process of finding things to do in your local area easy and convenient. Our web app offers a concise selection of events its users can chose from. We use Natural Language Processing (Python spaCy) to look through the descriptions of events hosted on our website to assign it the most relevant tags to further tailor user experience.

Marcus Behenna, Quattro Musser, Moses Solomon

Visualizing Sodomitical Paris

Accounts of homosexual practices in 18th century Europe are few and far between. The scarcity of these accounts often limits historians to analysis of individuals, obscuring larger scale community dynamics. 18th century records of the Parisian Police give historians a rare opportunity to study homosexual practices on the level of communities, through a wealth of data on masculine homosexual practices, and the policing thereof. The project “Policing Male Homosexuality in 18th Century Paris,” (PHS) founded in 2016, is working to translate these documents into English, publish them online, and supply the tools necessary for community scale analysis of these records. For our thesis project we developed two such tools: a web map and a data dashboard. The web map allows users to visualize, query, and filter all locational data the project has translated. The data dashboard allows users to explore non-locational data and create their own charts with custom filters.

William Holtz, Daniel Lewinsohn, Ben Modlin, Max Perozek

scSHARP: Python and R Packages for Robust Single-Cell RNA Sequencing Cell Type Annotation

Single-cell RNA sequencing (scRNA-seq) data, annotated by cell type, are useful in a variety of downstream biological applications, such as profiling gene expression at the single-cell level. However, manually assigning these annotations with known marker genes is both time-consuming and subjective. We present scSHARP, a combination of Python and R packages that is easily installable and usable for bioinformatics researchers. Our R package, R4scSHARP, implements five state-of-the-art cell type annotation tools which allow us to find cells with highly confident assignments through consensus. Our Python package, scSHARP, provides a semi-supervised Graph Convolutional Network (GCN) and other methods to spread these confident labels and interpret the results. As a result, scSHARP provides highly accurate and interpretable cell type annotations in an easy-to-use format.

Henry Jones

Explorations in Diffusion and the Mean First Passage Time

The subject of interest in a variety of applied and theoretical fields, the Mean First Passage Time (MFPT) is a solution to a particular Poisson equation deeply connected to the classical diffusion equation. Within the scope of this thesis, the MFPT is interpreted as the average time for a diffusing particle or random walker to arrive or 'react' at an absorbing boundary. This thesis aims to contextualize the study of diffusion and random walks and examine comparisons between the discrete and continuous. We begin with several foundational derivations before delving into some analytical and numerical results for a particularly interesting MFPT problem.

Na’ama Nevo

Error Correcting Codes

Error Correcting Codes are algorithms used to maximize accurate data transmission in networks between a sender and receiver. Data transmission is often hindered by random errors which can flip bits of a message to the wrong symbols. The goals of error correcting codes is both to be able to detect as many errors as possible and to be able correct as many errors as possible, while also maximizing the length of possible messages and minimizing the amount of storage space required. Hermitian-Lifted Codes were first described in a paper by Lopez, Malmskog, Matthews, Pinero-Gonzales, Wootters, and have many advantageous properties such as maximal length and good locality and availability. Additionally, the Hermitian-Lifted Code has a large dimension, which is an improvements from previous similar codes.

Emerson Worrell

An Exploration of Connect Sums of Knots Using the Trip Matrix

In the field of knot theory, we use knot invariants to determine if two knots are the same or distinct. The trip matrix provides a method of computing a knot invariant known as the Jones Polynomial that requires only linear algebra. The encapsulation of so much information in a matrix over Z_2 provides an interesting opportunity to see what other tasks the trip matrix can be used to perform. We utilize the trip matrix method to give an alternative proof that the Jones Polynomial is multiplicative under connect sums, and then use the structure of the trip matrix itself as a method to determine if a given knot diagram with minimal crossing representation is prime or composite. 

Davidson Cheng

Lattice-based Cryptography on Quantum Computer

Lattice-based crypto systems are currently the prime candidates for quantum-secure crypto systems. This talk will cover an introduction to hard lattice problems, which are at the core of lattice-based cryptography. I will also illustrate how a specific instance of a hard lattice problem might be solved efficiently on a general purpose quantum computer.

Mike Romer

Exploring the Rubik’s Cube Group

The Rubik’s Cube was invented in 1974 by architecture professor Ernő Rubik. Since then, it has become the most popular puzzle in history according to the national museum of play. Despite the cube’s popularity, its underlying algebraic structure is rarely discussed outside of college math departments. This is understandable as configurations of the Rubik’s Cube can be represented by a mathematical group with over forty-three quintillion elements. This project investigated the structure of this group and its subgroups and used them to create an original solution to the puzzle.

Casmali Lopez

Phylogenetic Networks: Combinatorics and Algebra

This presentation will demonstrate the applications of combinatorics and algebra to the field of phylogenetics. Phylogenetics is the study of the evolutionary relationships between organisms. The goal of phylogenetics is to use biological data from a collection of individuals or species to infer a tree or network that describes how they are related evolutionary. Phylogenetic networks, an expansion of phylogenetic trees, are more accurate in certain biological circumstances but provide increased complexity and therefore increased mathematical challenges. The first section of this presentation will work through counting results related to phylogenetic networks. The second section of this presentation focuses on the statistical algebra of phylogenetic networks and their use in inferring phylogenetic networks.

Cooper Doe

Effect of Randomness on Behavior of Gene Transcription in Certain Motifs 

Transcription of genes within organisms can be characterized as a Gene Regulatory Network of different smaller patterns, called motifs. The Feed Forward Loop (FFL), a three-gene motif, is characterized by a first gene X that has a direct effect on second gene Y, and a direct effect on final gene Z. Gene Y also has a direct effect on the final gene Z. FFLs are 'incoherent' if the indirect and direct pathways have an opposite effect (e.g. upregulation or downregulation). The first IFFL is the most biologically abundant incoherent loop. We find that certain deterministically predicted functions of the I1-FFL are not reproducible in our models with added stochasticity, and some functions are heavily dependent on intra-cell gene particle density. 

Ethan Lebowitz, Tony Mastromarino, Mac O’Brien

QuickCheck: Rapid Formative Assessment for K-12 Teachers

There's been a push in the education sphere towards formative assessment; as opposed to tests and quizzes, which happen at the end of a unit, formative assessment establishes an understanding of student knowledge at the time of teaching. This allows teachers to adjust their lesson plans and ensure that the entire class is on the same page. Like most things within the education, there's been little support with this; so, working with science teaching coach Monica Tino, we developed a web app that allows teachers to set up their classes and assessments on their computers and quickly perform the assessments while teaching on their phones, capturing data in a way that empowers teachers to do what they do best.

Davidson Cheng, Hset Hset Naing, Richard Wang

Mesh Feature Learning

It is difficult for computers to understand “shapes”: an apple and a plane both appear as a sequence of bits. But there is one specific
way to convey “shape” information to a computer: graphs! In our project, we developed a model that extracts structural information from graphs using adversarial machine learning, and we put it to the test by comparing our feature extraction method with a traditionally robust feature extraction method: heat kernel signature.

Lena Fleischer, Ellen Moore, Miranda Hunter

Sun Chase: A Machine Learning-Based iOS App Predicting the Beauty of Sunrises and Sunsets

Have you ever woken up early to watch the sunrise, only to be disappointed by gray skies? Our mission is to create an iOS application which predicts the aesthetic quality of sunrises and sunsets, encouraging users to get outside when skies are beautiful. The app takes in user location and gathers corresponding weather data which it passes into a machine learning algorithm. We have trained a neural network to recognize weather patterns that are highly correlated with beautiful skies, using geotagged image data (Flickr API) in tandem with historical weather data (Visual Crossing API). This model, trained in Python (PyTorch), is then packaged into an iOS application built using Swift. The final product, Sun Chase, is an intuitive application which displays sunrise and sunset predictions.

Liz Seero

Recovery Connection

According to the most recent SAMHSA National Survey on Drug Use and Health (NSDUH) detailing
Mental Illness and Substance Use Levels in 2021, 46.3 million people meet the DSM-5 criteria for having
a substance use disorder in the past year. The same study found that in that same year, 94% of people
aged 12 or older with a substance use disorder did not receive any treatment. The project aimed to
focus on recovery through the lens of Human Computer Interaction utilizing an Agile Design process.
Recovery Connection is a life tracking and reflection application that allows for users to record and
analyze information about relationships, activities, and substances. To test usability, I conducted two
user studies and implemented program changes inspired by the results. Available on both mobile and
desktop, Recovery Connection is a tool that was built to protect integrity, provide a non-judgmental
place to store and revisit private information, and above all, provide help to under-served communities.