Fellows Scholarship - 2020 Davidson Fellows

Biography

My name is Jason Liu, and I recently graduated from the Davidson Academy in Reno, Nevada. I first discovered my love of mathematics through math competitions, and since then, I’ve learned all sorts of random math and beautiful theorems. While many tend to think of math as a robotic field of calculations and tedium, it's actually a very beautiful field that starts with basic axioms and finds an endless variety of results through simple logical proofs. I am incredibly honored to be named a Davidson Fellow and to join the ranks of many amazing Fellows before me.

My project was assigned to me at the 2019 Research Science Institute. It is (as far as I know) only the second paper on Quantum Integer-Valued Polynomials, following one in 2016 by Nate Harman and Sam Hopkins. As such, the field of study is very new, and my project will be very useful for future research in the area as it provides a characterization of certain types of Quantum Integer-Valued Polynomials, giving mathematicians a starting point for further research. My project works on q-deformations, which study various properties of the q-integers: a generalization of regular integers that turn them into polynomials in an auxiliary variable q. These q-deformations have relations to quantum mechanics, and oftentimes there are natural generalizations, or q-deformations, of results for regular integers that will hold for q-integers. What I’ve done is essentially to find characterization of polynomials that send positive q-integer inputs (corresponding in some sense to positive integers) to outputs that are polynomials in q, and negative q-integer inputs to outputs that are polynomials in q’s inverse.

Although mathematical research doesn’t require any lab or field work, it’s by no means a walk in the park. Even before I started researching in earnest, I encountered my first difficulty: actually understanding the previous paper by Harman and Hopkins. As such, I would like to thank my mentor, Robert Burklund, a graduate student at MIT Mathematics, for helping me through the paper and letting me bounce ideas off of him throughout my research process. He gave me all sorts of ideas and motivation that eventually led to my final results, and he helped to edit my paper and clear up the language and notations I used. Without his help, I wouldn’t have been able to get anywhere near as far as I have on this project. I would also like to thank the Center for Excellence in Education for giving me the opportunity to research at RSI 2019 and to work with Robert. Another difficulty I encountered was simply in trying to figure out how to represent my characterization of Quantum Integer Valued Polynomials in a reasonably nice format.

As a math project, the main quality of life improvement that my research will bring in the short term is simply making things easier for other researchers. In the long term, an understanding of these polynomials could provide a simpler understanding of quantum groups (the set of all quantum integer-valued polynomials is closely related to quantum groups, and it seems likely that the more specific types are related in some way to some other quantum groups), which are in turn fundamental in quantum mechanics. While I don’t know exactly what benefits the public would get from science’s increased understanding of quantum mechanics, a better understanding of fundamental physics would almost certainly allow for improved technologies and a better standard of life in one way or another.

I attended the Davidson Academy, which allows for both an accelerated high school curriculum and dual enrollment at the local university, University of Nevada, Reno. As a result, I took courses like Calculus and Physics significantly earlier than most high schoolers and took numerous college courses for dual enrollment, including Groups, Rings, and Fields (which relates a lot to my project, although I didn’t take it until after I mostly finished my project…), Complex Analysis, Introductory Topology, Quantum Mechanics, and Computer Vision. In the fall, I will be attending the Massachusetts Institute of Technology, where I plan on majoring in some combination of mathematics, computer science, and physics.

I participate a lot in math and physics competitions, leading to my acceptance into the national math and physics camps: MOP and US Physics Team. I also had a lot of fun in my school’s Science Bowl team, and we actually managed to get top 16 at the Nationals competition this year. This project has also given me the honor of being a Regeneron Science Talent Search top 40 finalist, where I met some of the most amazing teenagers in the world. I’ve also played the piano for about the last 14 years, and with my teacher’s help, I’ve learned to play both classical Western pieces and more traditional Chinese pieces (usually adapted from other instruments). Before COVID-19 started, I would also go swimming (at the behest of my mom) for about 2000 yards three times a week.