2025 Mathematics Capstone Symposium

Friday, May 2nd
Join the Mathematics Department in Morken 214 and Morken 216 to hear the senior capstone presentations. If you’re unable to join us in-person, we welcome you to attend via Zoom. Links will be shared on May 2nd.

Morken 214

12:30-12:50pm – Pythagorean Triples
Stacie Spahr
We show how Pythagorean triples arise naturally from Gaussian integers as well as rational points on the unit circle, establishing a complete characterization of both primitive and nonprimitive triples.

1:00-1:20pm – Cardano’s Formula
Andrew Sandoval
This presentation explores Cardano’s Formula, also known as the cubic formula, focusing on deriving the formula and interpreting discriminant cases. Through worked examples, we will demonstrate how certain solutions require techniques from complex analysis to obtain all roots of the cubic equation.

1:30-1:50pm – Fractals
Emmanuel Obikwelu
The Mandelbrot and Julia sets are famous fractals that reveal complex patterns through simple mathematical rules. We will explore what they are, how they are related, and how they can be visualized using complex numbers.

2:00-2:20pm – Fourier Series
Luke Sunderman
We use audio signals as an application to explore the Fourier Transform and develop an intuition for how and why the Fourier Transform works.

2:30-2:50pm – Break

3:00-3:20pm – Fourier Series
Brandon Monson
In this presentation we examine components of the Fourier Transform to gain insight into its rationale while drawing connections to signal processing. Optimizations and developments of this process will be also be discussed.

3:30-3:50pm – Circuit Analysis
Tyler Stratton
In this presentation, we introduce the topic of Circuit Analysis, discuss its relation to complex analysis involving Euler’s formula, and prove Ohm’s Law.

Morken 216

12:30-12:50pm – Möbius Transformations
Samuel Shigematsu
We will discuss Mobius Transformations, including what they are, some important proofs regarding their properties, and some applications.

1:00-1:20pm – Complex Limits
Robert Marti
We explore the concept of limits in the complex plane. We start with a review of real limits to establish a foundation. We then introduce the formal definition of a complex limit and compare it to its real counterpart. Through visualizations and examples, we illustrate how complex limits must approach the same value along all possible paths to a point for the limit to exist.

1:30-1:50pm – Cauchy Integral Formula
Thalisa Saldivar
The Cauchy Integral Formula (CIF) is foundational to complex analysis, showing that a function’s values inside a region are fully determined by its behavior along the boundary. This presentation introduces the core ideas of analytic functions, contour integration, and singularities before proving CIF and exploring its applications. Through examples, we’ll see how CIF simplifies integration and leads to one of math’s most famous results: the Fundamental Theorem of Algebra.

2:00-2:20pm – Taylor Series
Wajid Ramzan
We explore the Taylor’s series (TS) of a complex function, first reviewing the TS of a real function, then moving on to the complex case, including a few examples showcasing how to find them.

2:30-2:50pm Break

3:00-3:20pm – Laurent Series
Drew Kraft
We introduce and break down Laurent series and their role in classifying isolated singularities while presenting a proof of Picard’s Great Theorem.

3:30-3:50pm – Residues & Poles
Chloe Cadelina
Poles and residues are two fundamental concepts in the study of complex analysis. We explore their definitions, properties, and mathematical significance, and illustrate these ideas through a selection of examples. Central to this discussion is Cauchy’s Residue Theorem–a powerful tool that uses the residues of a function at its isolated singularities to compute line integrals over closed contours. We present a proof of the theorem and demonstrate how it enables the evaluation of integrals for functions that are analytic on and within a contour, except at those singular points.