During the start of my Spring 2025 semester at UT, I applied for the opportunity to work on a mathematical project of my choice with a graduate student mentor, Aaron Benda. I chose to research dynamical systems, and the textbook we studied was Randomness and Recurrence in Dynamical Systems by Rodney Nillsen.
The textbook proved to be much denser* than I'd expected, and I was exposed to many graduate-level+ mathematical concepts. This included:
- Irrational number properties & applications to circle rotations
- Probability, in the measure theory/real analysis sense
- Randomness and average recurrence times (think infinite monkeys typing Shakespeare)
- Outer measure, σ-algebra (sigma-algebra) sets
- Birkhoff's Ergodic Theorem
and of course, Benford's Law.
I met with my mentor once a week, and we'd discuss the topics in the textbook. Usually I came in with notes I had taken on my iPad throughout the week; I've attached them at the bottom of this page.
This was my first ever experience at math research, so I wasn't sure what to expect coming in. One perspective shift was that I have a lot more respect for grad students now; the work they do is definitely not for the faint-hearted. At one point my mentor showed me his PhD candidacy presentation on ergodic theory, and I thought my textbook topics were hard to digest enough already.
Overall, this program was an interesting and intellectually enriching experience, and I got some valuable research skills out of it, especially from having to prepare a technical talk using LaTeX. I do hope to continue doing research in the future, in some form or fashion. Future areas I'd like to research include computer science, statistics, finance, or even economic theory.
*Real analysis joke. (ha)