Tim Leafblad did a Research Experience for Undergraduates at Kansas State University during the summer of 2018. He studied invariants of welded knots.
How did you find out about your research experience, and what was the process for applying?
I talked to professors and was pointed to the NSF site, where a list of mathematics REUs with information and links can be found (see https://www.nsf.gov/crssprgm/reu/list_result.jsp?unitid=5044). REUs also exist for a number of other fields, including chemistry, biology, engineering, physics, and others (see https://www.nsf.gov/crssprgm/reu/reu_search.jsp).
For the K-State REU in particular, an application consisted of emailing five items: an unofficial transcript, a brief survey of personal information (name, address, phone number, etc.), a personal statement (up to 2 pages), two letters of recommendation (to be emailed directly to K-State by the letter writers), and a letter (no more than one page) describing educational and professional goals as well as reasons for applying. Application processes do vary among math REUs to some degree; but from what I’ve seen, this is a good representative of what to expect.
Can you summarize your project and what your role was?
Our project was in knot theory. It started with a very broad goal, and no particular path in mind to get there: find an invariant (hopefully an interesting one) of extended welded knots, which are a new, fascinating, and somewhat mysterious variety of knots existing in the literature for less than two years now. As a layman’s description of our project, we found some properties that help us tell whether two of these strange knots are distinct.
You might think that with four people on a project, they’d tend to gravitate toward certain roles. We had a different group dynamic, though, and everyone ended up doing a good share of everything. Toward the beginning, there was a lot of reading about general knot theory and techniques for constructing invariants. None of us knew a whole lot about the field coming in, so it was a bit strange being thrust into doing research straight off the bat. Fortunately, we came across an approach that was used to find an invariant of classical knots that looked like we might be able to adapt to our needs. After this, there was a lot of algebra and verifying (and re-verifying, and re-re-verifying) certain properties. There were several hurdles along the way, where our more linear path of computational work was halted by the need for some sort of theoretical insight or trick; but with the help of our mentor, we managed to accomplish our original goal.
I should also mention that we gave (casual) progress presentations as groups to the other math REU participants each Friday, as well as full 20-minute presentations over our entire projects at the end of the REU. These final presentations were given at an organized event that was open to the public. I had the interesting experience of being the moderator for this. Additionally, we were to write a paper in the style appropriate for submission to an academic journal.
How did you benefit from this experience?
There are so many ways in which I benefited from the REU!
I learned how to work on a difficult research-level math problem with a team under an adviser. I learned a significant amount about knot theory (consequently algebra and topology), as well as the topics researched by the other groups (total variation flow and modelling the effects of nanoparticles on cell behavior). Our group was fortunate enough to obtain publication-worthy results and we’re in the final stages of preparing our paper for submission to the Journal of Knot Theory and its Ramifications. This is in itself exciting, but it will also look good on a resume and increase my appeal to graduate schools. Another benefit is that we managed to leave a good enough impression on our adviser, a well published mathematician, that he’s willing to write us letters of recommendation in the future. The project also strengthened my motivation to continue to pursue mathematics, and in particular it confirmed my belief that math research is what I want to do.
Aside from some of these more “practical” benefits, I had the time of my life at K-State. It was an incredible environment full of incredible people. Of the eight other participants, I expected to become close to maybe two or three of them; but it worked out that we all got along incredibly well, and I now call each one of them a friend. So although my primary interest going into the REU was the academic and professional benefit, I think what I value most about it now is the relationships I built.