My path to math research began during an Honors Ordinary Differential Equations class taught by Dr. Thomas Hales. During office hours, I sought advice from Dr. Hales on how to get exposed to math research. Serendipitously, he was working on the Reinhardt conjecture, reformulated as an optimal control problem heavily reliant on ODEs. He shared with me the book in preparation with his doctoral student Koundinya Vajjha. Despite not fully understanding them at first, I found the equations good-looking. I spent a spring reading up on the topic and then a summer doing research as a Painter Research Fellow supported by the math department. This experience made me appreciate the effort behind solving those seemingly simple math problems, such as Dr. Hales’s proof of the 400-year-old Kepler conjecture. Initially, I thought the solution to packing spheres was straightforward, something even a greengrocer would know (how to stack oranges). But as I delved deeper into research, meeting weekly with Dr. Hales, presenting at Mathematical Association of America’s MathFest and the Joint Mathematics Meetings, learning about other mathematicians’ work, I realized that the simplest questions often demand the most profound thinking. In fact, common sense like I had only suggests that if we empirically know something to be true, seeking a formal proof appears redundant or unnecessary. This echoes what Alfred North Whitehead said, ‘in creative thought common sense is a bad master. Its sole criterion for judgement is that the new ideas shall look like the old ones, in other words it can only act by suppressing originality.’

My passion for math research grew from this reshaping of my thoughts. Despite only switching my major to mathematics in the spring of 2023, coming from previous interests in philosophy and economics, I barely had any mathematical background. Dr. Hales’s patience and never giving up on me, and the supportive Pittsburgh Mathematics Department emboldened me. My research, particularly on the Truncated Octahedral Conjecture for my undergraduate thesis, deepened my understanding of discrete geometry. Studying the Kelvin problem, the proof of the Kepler conjecture, the Reinhardt conjecture, as well as the Ulam conjecture has offered me a clearer view of both the highest peaks and achievable goals within discrete geometry. This exploration has crafted a roadmap for my journey in the field and sharpened my professional goals, making discrete geometry a solid starting point and initial professional direction for my career.

My main advice to undergraduate peers interested in math research is that regardless of your mathematical background, the transition from undergraduate coursework to real math research can be overwhelming, but do not hastily conclude from this that math research does not welcome you. Whatever you do, as long as it involves knowledge you have never encountered and a learning process that makes you uncomfortable, you might feel the same unwelcomeness, but the presence of this bitter taste signifies that you are continuously progressing and becoming a better being. Of course, you would be very fortunate if in this process you can have a mentor like Dr. Hales, who is patient enough and capable of making high level math concepts cool and accessible.

I am not sure what kind of research opportunities are offered by your department or program, but if you are interested in undergraduate research in mathematics, consider reaching out to Dr. DeBlois, the undergraduate director as of 2024. You will learn the most current information about what research is out there, and what kind of funding and presentation opportunities the math department offers. Whether you are within the math department or not, I also encourage you to read and contribute to our newly created university journal, the Pittsburgh Interdisciplinary Mathematics Review (hosted at pimr.pitt.edu), where you can find feature articles by faculty, exposition of math-related topics by graduate students, the mathematical history of Pittsburgh, competition math puzzles, career advice, as well as research articles authored by your undergraduate peers as the outcome of their research experiences. Best of luck in your research explorations and have fun along the way!