2026 Mathematics Capstone Symposium

Friday, May 1st
Join the Mathematics Department in Morken 214 to hear the senior capstone presentations. If you’re unable to join us in-person, we welcome you to attend via Google Meet. To view presentations, use this link.

Morken 214

1:30-1:50pm – Pythagorean Triples: A Complex Analytic Perspective
Sydney Reisner
Pythagorean triples are integer solutions to the equation a2 + b2 = c2. In this presentation, we explore how these triples can be understood and generated using complex numbers, specifically Gaussian integers. We show how squaring a Gaussian integer leads to a formula that produces Pythagorean triples, and we prove that this formula always satisfies the Pythagorean equation. We also define primitive and nonprimitive triples and discuss how all Pythagorean triples can be generated from this construction. Finally, we connect these ideas to geometry by showing how Pythagorean triples correspond to points on the unit circle.

2:00-2:20pm – Möbius Transformations
Anna Chaffee
Möbius transformations, also known as linear fractional transformations, play a central role in complex analysis. Möbius transformations are important in complex analysis because they describe many ways in which we can transform the complex plane. In this paper, we introduce Möbius Transformations and demonstrate their three basic transformations. These transformations include translation, dilation, and inversion. We prove that any Möbius transformation can be expressed as a composition of these three basic transformations. We also prove through these transformations that lines and circles in the complex pla

2:30-2:50pm – Fractals and Complex Iteration: How Simple Rules Create Infinite Complexity
Biniam Hailu
We introduce fractals as geometric objects that exhibit self-similar complexity at every scale. We then develop the notion of Hausdorff dimension as a rigorous way to assign non-integer dimensions to fractal objects, and we compute the Hausdorff dimensions of some examples. We conclude with applications of fractal geometry to nature, medicine, and financial modeling.

3:00-3:20pm – Rigged by Design: Modeling Electoral Districts with MCMC Methods
Meghan Buchanan
This project explores the use of the Markov Chain Monte Carlo (MCMC) method to generate ensembles of valid electoral district maps. By comparing these generated maps to existing plans, this project investigates how district boundaries can influence election outcomes and provides a computational perspective on gerrymandering.

3:30-4:00pm – Break

4:30-4:50pm – Isomorphic to C
Junior Quenga
We have all wondered about the set of complex numbers ℂ, in particular the imaginary unit i as we are taught to be defined as the square root of minus one, i.e. i = √−1. In addition, Algebraists like Galois have long searched for answers regarding the structure of the real numbers ℝ and complex numbers ℂ. One particular question that arises is: How do these structures relate to ℂ? Not just the real numbers, but groups, rings, and possibly fields. In this paper, we will explore what structures are isomorphic to ℂ. The purpose of this paper is to provide an overview of some basic facts about Isomorphic structures in Group Theory and Ring Theory, specifically the inner-workings between certain algebraic structures and the complex numbers.

5:00-5:20pm – How to Steal an Election
David Luna Güitron
In the United States some representation is more equal than others. Gerrymandering refers to the manipulation of electoral district boundaries in order for political parties to maximize their political advantage in the House of Representatives. We explore a simplified version of electoral maps via a grid-based puzzle commonly known as Gerrymandering Puzzles where cells represent voters’ political affiliation. The objective is to partition the grid into contiguous districts that maximize wins for a chosen party. We analyze how strategies analogous to real-world gerrymandering practices emerge in these puzzles, characterize valid puzzle configurations, and investigate combinatorial methods for constructing optimal districts. We also examine the feasibility of enumerating all valid region partitions to identify maps that maximize political advantage. These models provide a foundation for understanding the political strategies underlying gerrymandering while also serving as an accessible educational tool.

5:30-5:50pm – From Waves to Frequencies: Understanding Fourier Analysis in Signal Processing
Naufal Alavi
We introduce the core ideas of Fourier analysis by first examining how periodic functions can be expressed as sums of sinusoidal components through both the Fourier series and its complex form, building an intuitive understanding of frequency decomposition. This idea is then extended to non-periodic functions using the Fourier transform, which allows general signals to be represented in terms of their constituent frequencies. We then consider discrete data by introducing the discrete Fourier transform (DFT) and addressing its computational limitations through the fast Fourier transform (FFT), which allows efficient large-scale analysis.Finally, we briefly look at how Fourier methods are used in areas like physics, data science and medicine to better understand real-world signals.

6:00-6:20pm – Frequency Response of Finite Discrete Filters
Andrew Lee
This project studies how a simple finite filter changes a discrete signal, and why the best way to understand that change is through the filter’s frequency response. The main idea is that filtering is done by convolution in the time domain, but in the frequency domain that same operation becomes multiplication. That makes it much easier to see which parts of a signal are weakened, preserved, or emphasized. The project also math 499b: abstracts looks at what happens when the same filter is applied repeatedly, and explains how the frequency response can be viewed using a complex-variable function H(z) evaluated on the unit circle. Overall, the goal is to give a clear mathematical explanation of how finite discrete filters work.