STEP Project: Undergraduate Research

Name: James Enouen
Type of Project: Undergraduate Research

My STEP project was doing undergraduate research in mathematics with a group of peers over the summer of 2018. This included enriching my personal knowledge by attending weekly talks and seminars about different topics in mathematics. The majority of my summer was spent in a small group with three other peers, researching a mathematical object called Stanley’s symmetric chromatic polynomial.

This research experience gave me a deeper look into how math research works which I wouldn’t say is the most obvious. At one point in human time we didn’t know about the quadratic equation or Euler’s identity but ultimately some mathematician discovered these facts. This process is still ongoing today but obviously people are reaching deeper and deeper into their respective field. As my life goes on, I am starting to realize more and more that I would love to be on that list. I would love to make a meaningful discovery that gets used by humanity decade after decade. Math is one of the basic sciences which fuels the fire of the whole array of sciences which is why I believe it is so fundamental and so important. This research group helped me also the value of math for its own sake and strengthen my belief in research in general.

I am currently a computer science and engineering major as well as a mathematics major. This experience reinforced the idea that my computer science knowledge would help me study mathematics. I was able to code software which would generate the symmetric chromatic polynomial for trees with up to 10 vertices which would be a ridiculously tedious calculation on hundreds of trees. This gave us a lot more data to work with when analyzing the open problem called the tree conjecture on Richard Stanley’s symmetric chromatic polynomial. Although the problem remains unsolved, we feel we made steps in the right direction towards understanding how Stanley’s polynomial is able to distinguish trees. I am very pleased that I was able to put both of my majors together into one project and make so much progress.

Over the summer, we learned about knot theory and developing knot invariants; this is a strategy of how to research knots in mathematics. Then we took a look at graph theory and some theorems about them and how this would influence our study of Stanley’s polynomial which is a polynomial defined on a graph. We then studied a lot of the interplay between these two topics and some other related results. In addition to these seminars which occurred multiple times every week, we also had weekly meetings with our adviser. Here we would have a personal conversation about what we had studied on the polynomial over the past week. We were ultimately able to accomplish our adviser’s goal of developing a B_n symmetric type version of the A_n symmetric chromatic polynomial. In a certain sense we changed from only using positive colors for the chromaticity of our polynomial to using both positive and negative colors on our polynomial. After we accomplished this goal, we further generalized slightly in the same direction by putting an arbitrary group on the edges of a finite graph in what is called a voltage graph or a gain graph. We did not push this idea too far because it didn’t seem particularly fruitful.

After this, we studied some of the deeper results which had been discovered on Richard Stanley’s symmetric chromatic polynomial and after we understood these results, we were able to come up with the parallel results for our signed polynomial we had constructed. This was notably prolific because the more we understood both Stanley and our polynomial, the easier it was for us to understand a result about the chromatic polynomial, and the easier it was for us to make the generalization into the signed polynomial. Once we felt we had covered a sufficient array of results about our newly created signed chromatic polynomial, we moved on to some other material surrounding Stanley’s symmetric chromatic polynomial. Most notably, we studied the conjecture that given a two different trees which are not the same, their symmetric chromatic polynomial will also not be the same. At the end of the summer, we applied for and attended a youth conference at the Ohio State University where we presented our findings to a group of peers.

It felt very rewarding to discover something entirely new, even if it really wasn’t of the greatest importance to the majority of math. Just going through the process of discovery was interesting enough on its own. This process helped me further realize that what I enjoy even more than designing a product for a particular person is discovering a result for the entirety of humanity. This has inspired me to focus even more on academia because I think that even outside of just math I would like to study for a large proportion of my life. This experience makes me feel the desire to do research in computer science as well. I want to study the machine learning and computer vision on the forefront of the wave of technology for the future. I want to be right in the thick of developing new techniques and new algorithms to let artificial intelligence pave its way through our society. Ultimately, I see so much value in learning new things that humanity is yet to discover and I am itching with excitement as I move through my undergraduate career to find the opportunities to study in this way. Inside of class, outside of class, inside of industry, outside of industry, there are so many ways to participate in this kind of research. I hope that as I go into my junior and senior years of my undergraduate degree at The Ohio State University I am able to find these chances in both computer science and math to further my education and to develop fields which are yet to be fully fleshed out. My future plans have been altered to align with this aspiration to have a meaningful contribution to the world. I strongly feel I can best help the planet through research in both mathematics and computer science, so this is not only what I yearn to do, but what I need to do.