Research mathematics can seem inaccessible to undergraduates, mathematicians prefer to state results in their most general setting, so the theory and terminology can be very dense. I learned through my experience that it is unrealistic to try to understand every mathematical concept related to your research, at least at the undergraduate level. There is an incredible amount of mathematics, one can spend a lifetime simply learning it. The best research mathematicians don’t necessarily know the most mathematics, a surprising reality that speaks volumes. To really benefit from undergraduate research in math it is necessary to jump in and explore the problem and the objects involved somewhat naively. A strong theoretical foundation helps of course, but it is not a necessity. Much of undergraduate math research involves programming, simply because the skills necessary to do analytical math research are advanced. If you have the opportunity to do research with more of an analytical flavor, take it! Programming, making figures and so on is of course an important piece of much of modern mathematical research, however the intuition and experience you gain through analytical research is priceless. I found that the most important resource I had was my mentor. In our meetings we would work through the problem together and I could observe how he approached a question. Know that math is a massive field and the more you dig into a specific research question the less information exists on the internet. Furthermore you will very quickly know more about the particular problem than mathematicians who are much more experienced. A math course is typically very general and covers many theories and examples, while math research is frequently focused on a very specific case. You will come to know that case thoroughly, and through understanding the special case you begin to understand the field comprehensively. Mathematical research has increased my ability to form connections between subjects, I constantly relate topics from my courses back to my research. It has improved my ability to succeed academically as well as in a research setting. I believe it is quite different from classroom math and an invaluable opportunity to sharpen your critical thinking.
While the CURF is over I would still like to explore dispersive quantization further. I am beginning to draw connections between abstract algebraic concepts and quantization, which excites me and leads me to believe there is a different way of viewing the behavior mathematically. This summer I will be taking part in the University of Pittsburgh TECBio REU in computational and systems biology. As a rising senior I am focused on the process of applying to graduate schools. I also plan to apply for the NSF GRFP and potentially other fellowships if time permits. The CURF, along with my other research experiences has allowed me to be all the more prepared for what lies ahead.
As I discussed above, a lot of undergraduate research in mathematics involves a significant programming component. Dr. Chen intentionally made sure that I was also involved in the analytical aspect of the research. That being said, numerical experiments can be a massive asset for verifying results as well as looking for potentially interesting areas to study further. I leveraged the linearity of the equations I am studying to create functions in Mathematica which I could use to plot numerous different initial conditions. This allowed for ease of experimentation. Included below is an animation illustrating the convergence to quantization, in the one dimensional case, as the time grows closer to a rational multiple of the period. The cover photo was obtaining using the linear combination method I described.