The summer before my senior year at Cal Poly, I did undergraduate research in graph theory + dynamic network models. It turned into a peer-reviewed paper on how you can represent certain rational Pick functions using graph-based models (via Cauchy transforms and tools like Schur complements). I remember thinking how strange this type of math was: it looks abstract at first, but it's extremely structural underneath. Once you build the model, you can literally see how changing the graph changes the function's behavior.

The published paper is here: https://www.sciencedirect.com/science/article/pii/S0024379525004525?via%3Dihub

the problem (in plain English)

We wanted a clean, concrete way to understand and represent a specific class of functions (Pick functions) using objects you can reason about visually and structurally (graphs). Instead of treating the function like a mysterious black box, the goal was to give it a graph "skeleton" that makes its behavior more interpretable and easier to analyze.

what i worked on

• Researched the underlying theory around graph-based models and how they connect to rational Pick functions
• Helped build and validate the framework connecting graphs ↔ function representations
• Worked through the math tools that make the connection precise (including Cauchy transforms and Schur complements)
• Contributed to writing and polishing the final paper, with an emphasis on clarity and structure