Mathematical quandary solved

It's a rare opportunity for undergraduate students to contribute to research. But for a group of College of Lake County (CLC) students taking a special mathematics topics course, they not only got to conduct their own research, but they are getting their results published.

Akash Kumar, Patrick Rewers, Paul Shin and Khue To worked together on the research with CLC math instructor Dr. Jeff Mudrock. The class focuses on giving students a chance to learn about the research side of mathematics.

“When most people take mathematics classes, the goal is to understand, appreciate and apply mathematics that has been developed by great minds over hundreds of years,” Mudrock said.

Mudrock and Dr. Hemanshu Kaul, Mudrock’s PhD supervisor and an associate professor at the Illinois Institute of Technology, collaborate on research together. Kaul also helps Mudrock with the course. The two work together to select research problems that students work on to get experience with what Kaul calls the creative side of math.

“The class is an opportunity for students who enjoy math to challenge themselves and go beyond what they normally learn,” Rewers said. “You go from someone learning to someone furthering the field of mathematics.”

The problems selected are ones that undergraduates have the potential to work on and make progress since they don’t have much, if any, experience doing research. Because of this, progress is the goal, not results.

“I enjoyed working on the problem because there were no expectations,” Kumar said. “Research is about trying different things to see what happens. We were all happy we came up with something cool.”

The problem the students were tasked with solving is about the study of graph theory, specifically graph coloring. The questions came from Danish mathematician Carsten Thomassen’s work in 2009 on list coloring, a type of graph coloring where each vertex can be restricted to a list of allowed colors.

Despite the importance of the question in enumerative combinatorics, the study of counting the number of possibilities that meet a certain criterion, the mathematical community could not determine the answer to Thomassen's question.

The class starts with a catching up and background to get students up to speed with the problem they will work on and the kinds of math needed. Since each student can be at a different place with this, some students begin the research phase sooner.

The process took the entire summer of 2022. The group worked together, meeting up outside of class once a week to do work and research. Upon getting their results, all the students felt excitement and fulfillment.

“In other classes you learn the concepts and do the work to get familiar with it,” To said. “In this class, you get to think about a problem with an answer that never existed before.”

Mudrock said, “Akash, Khue, Patrick and Paul not only successfully discovered a mathematical property, but they used it to answer an open question from 2009 that was relatively well-known. Such an accomplishment would be a great success for a veteran mathematician, but the fact that they did it as undergraduates makes their work all the more remarkable.”