I have always been drawn to computers and games. Some of the fondest memories of my youth were receiving Pong (to play on a black and white TV), Mattel’s handheld Football II, the Intellivision gaming system, the 2k of RAM Timex Sinclair 100 (I saved files to a cassette player), and later the Commodore 64 (with a big old floppy drive). There were not many opportunities to use computers in the schools. In eighth grade we had a project in which we wrote a computer program using punch cards and in ninth grade I took the one computer course that was available.
After high school I attended St. Olaf College, which has a tradition of graduating robust classes of mathematics majors but did not have a computer science major at the time. I was thinking of mathematics and English as majors but eventually decided on just mathematics.
I enjoyed proving things, as is the brunt of the upper-level mathematics course work. In particular, there was something very satisfying in being able to show that something was absolutely true. My Ph.D. adviser, Jon Simon of the University of Iowa, once said that it’s the desire to take some small bit of the universe and make it perfect that drives the mathematician. Others postulate that coffee and cookies are the main fuel.
At the end of my undergraduate time, I had to figure out what to do with my life. I had toyed with the idea of high school teaching and enjoyed a teaching practicum I had taken at St. Olaf, but it did not feel quite right. In another course, I had done a short research project on sound compression and I enjoyed the freedom of working on something that no one ever had thought about. I imagined that graduate school might be interesting in this same way, so I headed to the University of Iowa. After passing through the qualifying exams, I decided that I wanted to work in topology (which is like geometry for hippies). I met with the different topology professors and eventually teamed up with Jon Simon. Jon had done some work on mathematical problems in chemistry, had a network of colleagues in different disciplines and was not afraid to use a computer in his research. In particular, Jon is a knot theorist (like knots in your shoes) and had been working on studying what knots look like when they are pulled tight. His research sounded like fun.
For my dissertation, I wanted to use a computer to simulate the tightening of knots. But we could not simulate real ropes on a computer since real ropes have, in some sense, an infinite amount of information and computers are finite. However, computers do like polygons, so my first task was to translate this tightening problem into something we could analyze with polygons and then show that what we get from computer simulations would correlate with the real ropes problem. I got to work on the mathematics, took a couple of programming courses, and later started to play around with writing code to simulate knot tightening. I really enjoyed combining computations with mathematics. I went to a couple of interdisciplinary conferences during these years and found the talks by researchers from other disciplines intriguing (even when I did not understand all of the details). I thought it would be enjoyable to do work that might have applications.
When I started my first professor job I did not have plans for a big research career. I thought that I would publish a paper related to my dissertation and maybe one or two follow-up papers. That would give me enough for tenure and then I would be in the clear. Research is a lot of work, with lots of ups and downs and long periods in which you feel like you are putting in a lot of effort and getting nothing in return. Balancing research with a new job, teaching new courses, adjusting to a new city, etc., seemed overwhelming, so the thought of spending a bunch of time on research was not appealing.
At the same time, I wanted to at least try to do research beyond my dissertation. I was afraid that if I did not try it right away then I probably never would. I did not know where to start, so I asked Jon what I should do to get more connected to the research community. He suggested emailing people that I had met at conferences to express interest in staying in touch. I did that.
I received several emails back but one in particular stood out. It came from Ken Millett from the University of California, Santa Barbara. He said he was interested in my work and suggested that maybe we could collaborate on a project. Ken is very well known in the mathematics community, so I was both thrilled and scared. After communicating by email for several months, Ken invited me to visit Santa Barbara over my spring break. I remember getting to the airport and thinking “I do not even know this guy and now I am going to spend a week with him at his house talking math.” But I went and I loved it. In fact, Ken and I are still collaborators to this day.
A couple of years later, Ken and I decided to organize a special session at a math conference. There were upcoming meetings in Hoboken, N.J., and Las Vegas. Ken suggested Hoboken but I thought that Las Vegas would be a better draw for our foreign invitees. We invited the top people in our field from all over the world hoping we could get at least a couple of international participants. To my surprise, nearly everyone said yes. This was one of the best things I ever did. It really connected me to the research community. It has been 12 years since I organized that first meeting and this November will mark the 15th meeting I have organized.
Around the same time as the first meeting, Ken encouraged me to apply for a National Science Foundation grant. I did not think I had a chance but figured that I would get some good feedback and that maybe I would have a better chance in the future. I applied to a program focusing on funding faculty at undergraduate institutions, so in my first proposal I included money to pay undergraduates to work with me. Much to my surprise, the project was funded.
Right now, I am amidst my fourth NSF grant.
This funding has helped to launch my research career. In addition to providing computing equipment and opportunities to travel to conferences to see old colleagues and meet new ones, it has been instrumental in including undergraduates in my research. Through the years, 31 undergraduates have been supported by these grants (many of them for two or more years), including 12 since I arrived at UST in 2006. Over a third of these students have gone on to graduate school.
While it is fun to have the students go on to graduate school, a research experience is valuable for any student. As such, I have worked with students with varying interests, majors, and abilities. It is always satisfying to see these students develop as they tackle difficult problems, and I always learn more from the students than they learn from me. In fact, seven of my publications have undergraduate coauthors.
This summer I had a special experience. I took six students to Dennison University in Granville, Ohio, for the UnKnot 2012 Conference. One of my former students, Tom Wears of Longwood University, also participated in the conference. I gave a one-hour invited lecture and three of my students gave contributed talks. It was particularly satisfying to see my current students interact with Tom, possibly peering into their futures.
The focus of my research always has been on studying knotting in objects imbued with some physical properties, beginning with the tightening problem from my dissertation. The tightening work continues today although the focus has changed a little. About 10 years ago, I began a collaboration with Jason Cantarella from the University of Georgia to improve the software to tighten knots, resulting in the program RidgeRunner. Since then, we have written three papers with one more submitted. The most recent work is a collaboration with Jason and two physicists Tom Kephart (Vanderbilt University) and Roman Buniy (Chapman University). Kephart and Buniy have proposed that subatomic particles called glueballs are tightly knotted and linked, tube-like objects. In creating a catalog of tight knots and links, we are (in theory) creating a catalog of these subatomic glueballs.
In the early 2000s, I also shifted my main focus from understanding the structure of tight knots to understanding the structure of polymers. Polymers (like rubber, styrofoam, proteins, and DNA) are long chemical chains and can
be modeled by polygons with some physical properties. We wanted to study what the polymers looked like when they were moving about at random, focusing on knotted polymers. For example, we wanted to know whether the polymers could most easily be placed inside the shape of a pencil, rugby ball, a basketball, M&M candy, etc. This work has been in collab- oration mainly with Ken and Andrzej Stasiak (a molecular biologist from the University of Lausanne, Switzerland who I first met as a graduate student).
A few years ago, I saw a very interesting talk by Joanna Sulkowska (then a physics post-doc at the University of California San Diego and currently of the University of Warsaw) about proteins. Proteins are chains of amino acids that perform many functions within cells. They are created as linear chains and then fold into their functional form (called the native state). These proteins need to fold reprocibly to the same form (some diseases, for example, come from misfolded proteins), and preferably quickly. Some proteins have knotting in their native states. Since knotting should complicate the folding process and make it slower, why would proteins form knots? In particular, is this an accident of nature or does the knotting serve some purpose? The answers to these questions are unknown, so Ken, Andrzej, Joanna, Jose Onuchic (Joanna’s post-doc adviser at UC San Diego, now at Rice University) and I decided to search for answers. Our resulting paper appeared in the Proceedings of the National Academy of Sciences, a high impact journal spanning all of the sciences. Our paper made a bit of a buzz on the Internet too. After the release of a press statement, the story was covered by Wired.com.
These types of cross-disciplinary collaborations now drive my research and continued learning. My collaborators and I combine our skills to tackle problems that none of us could solve on our own. Indeed, the whole is greater than the sum of the parts (which is a tough proposition for a mathematician). These many years later, I am still drawn to computers and games. However, instead of playing Pong on a black and white TV, I now play with supercomputers.
From Exemplars, a publication of the Grants and Research Office.