Jesse Hamer entered the Mathematics Graduate Program at the University of Iowa believing he wanted to be a pure researcher.

Midway through his graduate student career, however, the native of Lee’s Summit, Mo., veered toward a teaching-oriented academic position. He completed the graduate certificate in college teaching and served as a teaching assistant for one of the UI’s graduate-level mathematics courses.

In addition, Hamer, a recipient of a Ballard and Seashore Dissertation Fellowship from the Graduate College, has become more interested in the applications of mathematics to the real world, which is a change from the pure-math focus he had entering graduate school.

Hamer’s research is motivated by artificial neural networks (ANN). These statistical models, whose structure is inspired by that of the human brain, are used on intuitive tasks like computer vision and natural language processing, such as Facebook’s face recognition and Siri’s voice recognition. He defended his dissertation on November 27, 2018.

**Q: Describe your research in non-expert language?**

**A:** Broadly speaking, I study knots. Knots can be thought of as idealized pieces of string in 3D-space, twisted and interwoven about themselves, with ends tied together to form a closed loop. Knots have applications outside of math, such as in biology (DNA), physics (string theory), and chemistry (some molecules have a knotted structure). In my thesis research, I study a certain subclass of knots defined by a certain kind of homogeneity in the types of crossings that they admit. A crossing is where one strand of the string passes over another strand; it is a constituent element of the “interweaving pattern” of the knot.

**Q: What impact has your work had on the field/world? What impact do you hope to have on your field/world?**

**A:** My advisor (Keiko Kawamuro) and I have been able to use our work to contribute significantly to the website KnotInfo, which provides a massive database of nearly 3,000 knots and related invariants—an auxiliary mathematical object like a number or polynomial used to distinguish two knots. So, if a mathematician needs a knot with certain invariants, they can consult KnotInfo, or conversely if a mathematician is dealing with an unknown knot for which they know several invariants, they can use KnotInfo to narrow their search and try to pinpoint exactly which knot they’re dealing with.

**Q: Why did you pursue graduate school/become a researcher?**

**A:** I decided to pursue a Ph.D. in pure math during the second semester of my college career—the only semester where I did not take a math course. I realized during that time how drawn I was to the orderly, clean universe (relatively speaking) in which pure math problems are carried out. Pure math also seemed to be the subject for which I had the most natural aptitude.

**Q: What was your experience like this summer while on the NSF Math Sciences Graduate Internship (MSGI) at Fermilab?**

**A:** My job at Fermilab, therefore, was to take a dataset of artificial neural networks, which were trained for the purpose of event location, and convert them into a meta-dataset of architectural information for each network, and then analyze this meta-dataset for any clustering patterns or correlations to test accuracy. From what my mentor and I could tell, few people, if any, had ever attempted such a meta-analysis. Thus this work, small in scope as it is, represents some sort of first step towards what we hope will be a better understanding of the relationship between neural network architecture and task performance. The NSF-MSGI Program was undoubtedly a big moment in my career development.

**Q: What programs or resources (on or off campus) have influenced or supported your academic goals?**

**A:** The UI’s Mathematics Department has been immensely helpful in my development as a graduate student. I’ve been financially supported every semester via rewarding TA experiences. I’ve had opportunities to deepen my understanding of a wide range of ideas via participation in various department seminars—some of which were official, and others of which were spontaneously created by small groups of enthusiastic professors and students.

**Q: Do you have role models, mentors, or inspirational people who have encouraged you to pursue your work?**

**A:** My advisor, Keiko Kawamuro, has been my greatest role model, mentor, and inspiration during my career here at Iowa. Her keen insight has helped to refine my ideas about mathematics and carry out my research. Her open heart and mind has listened to and accommodated the many shifts in my goals throughout my graduate career, without sacrificing expectations of my quality of work.

**Q: If you could go back to a time at the beginning of your graduate career, what advice would you give yourself?**

**A:** My path through the graduate program has by no means been straight or streamlined, but at the end of the day I view all of the twists and turns I’ve taken as essential in my development, and I am thankful for all of the opportunities I’ve had. That said, I think the best piece of advice I could give my past self is to wake up routinely at 6 a.m., exercise, and eat a large breakfast. I’ve only started doing this recently and it has done wonders for my health, mood regulation, and productivity.