Emma Richey - The Journey of Mathematics Research and Liminality

This past semester, I attended the Math Department’s Colloquium: Solving the wrong problem on purpose: subproblems, setbacks, and strategy in mathematical research. Greg Robson—a graduate student from the University of Primorska—delved into the graph isomorphism problem, demonstrating his research process and how he was able to break down the problem into smaller, more attainable pieces. To start, the graph isomorphism problem asks whether two graphs, which may look different, are structurally identical, essentially prompting the question, “When do I know if I have two of the same graphs?” In the field of mathematics, this problem is extremely difficult to deal with, not having a set answer. With this, Robson, instead, looked at the Cayley digraph to use that as an easier question to answer (however, this became too easy and simple of a problem). Next, he looked at the Bicayley Isomorphism (BCI), then reduced the BCI to the CI Problem, finally ending at the (A)BCI Problem. For this abelian problem, he was able to come up with a partial solution (a corollary from his research), but this does not reduce ABCI to CI problem. Therefore, this research does not allow for Robson to make a “big theorem” for nonabelian groups.

Although all of Robson’s work was well beyond my capabilities as an undergraduate math student (especially one who is only a sophomore), the journey of his research and how he continuously changed problems in order to find questions to answer was very intriguing to me. As I want to be a secondary math teacher some day, this helped me understand how I can best increase my students’ growth and learning. For example, if a student is struggling with factoring, one way to break down the problem is to ask about multiplication facts; if I am able to find the line where the student understands the content as well as where they are getting stuck with multiplying polynomials, then I can understand what extra help and assistance they truly need.

This method of research has also strengthened my understanding of liminality in the field of mathematics. There are certain moments where discovery and understanding is at a roadblock—especially in situations like Robson’s where certain problems are too simple and others are too complex to solve. With research, transforming the "in-between" stages of investigation (i.e. ambiguity and confusion) into productive, creative, and critical analysis effectively utilizes liminality in a way that is not overwhelming and defeating.

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