Mathematics

# Math Students Play Crucial Role in Proving Theorem

Photo of Emma Morgan, Yifei Zhao, and Mike Cassel by Jeffrey Hakim.

During the summer of 2011, Mike Cassel, BS mathematics ’11 and current MA mathematics student, and Emma Morgan, BS mathematics ’12, and one student in Shanghai, Yifei Zhao, worked together to prove a new mathematical theorem. Communicating via e-mail, a wiki, and Skype, and working under the supervision of Professors Jeffrey Adler and Joshua Lansky, the students tackled a problem in the character theory of finite reductive groups.

Finite reductive groups are systems of symmetries that are relevant to number theory, combinatorics, and other areas. Characters are extremely complicated functions that play the same role as sine waves in sound processing. Just as it is natural to decompose a sound into a sum of pure tones, or sinusoids, in other contexts it can be natural to decompose functions into characters. The theorem doesn’t directly relate to the real world, but it is important for certain areas of mathematics. The students investigated a relationship, conjectured by Adler and Lansky as part of a research grant from the National Science Foundation, between the characters of different symmetry systems.

Though the students were chosen to work on the theorem because of their strong math skills, that didn’t mean they could jump right in. “At the beginning of the summer, we spent a month learning new concepts to understand what the problem was,” Morgan says.

Lansky acknowledges the amount of prep work needed to solve the theorem. “In the process of preparing to work on the project, they learned about a lot of deep and beautiful ideas in a number of different branches of mathematics,” he says. “This will be very useful in any future mathematical study.”

Initially, the students were only asked to prove a small case within the conjecture that Adler and Lansky were trying to prove. “But then two things happened,” Adler says. “They kept extending their methods to cover more and more general situations, eventually addressing the entire conjecture; and our conjecture, as we originally formulated it, turned out to be wrong. Fortunately, the students knew how to address the key question: if an expected statement is false, then what is true? They found out.”

Both Cassel and Morgan emphasize how collaborative and unusual an opportunity it was to work on the theorem as students. “It’s not like a usual math class where the professors have all the answers,” Cassel says. “We’re doing this right alongside them.”

Working on the theorem also introduced the students to parts of the math world that they had yet to experience. “In writing the theorem, we got to learn how to use LaTeX, which is like a type of programming code,” Morgan says. “I consider that a really useful skill, because any type of math paper that I write later is going to be using this. You don’t learn that in class.”

Because of the size of AU’s Department of Mathematics and Statistics, students are able to form close bonds with their professors, which can expose students to opportunities they wouldn’t have found on their own. “Dr. Adler knows exactly what I can do, and based on that kind of personal connection, I get invited to do things like this,” Cassel says.

“I would say that every math professor I’ve had knows who I am,” Morgan says. “You get opportunities like this, but you can also work in the Mathematics and Statistics Tutoring Lab, you can TA for someone, you can be a grader for math papers. There are so many opportunities and so much interaction that you can develop. I think it lets you get more engaged in the subject, because you can have conversations with your professors.”