Professors disprove original hypothesis in math research

Allegheny College conducts research across every department at all times of the year. The mathematics department is no different as Harald Ellers, professor of mathematics, and Craig J. Dodge, assistant professor of mathematics, carried out research with students over the summer of 2015.

The field of mathematics that the research focuses on deals with abstract algebra, which solely centralizes around how algebra behaves and works.

“What we are interested in, the kind of research we do, the problems we investigate are motivated by trying to understand the principles that govern the behavior of algebraic structures,” Ellers said. “[The questions] are not motivated by [the immediate needs of] scientists or engineers…I am thinking about just the algebra and how does it behave and how does it work.”

Specific to this research, Ellers and Dodge are investigating the algebraic analysis of symmetry. Essentially, thinking of every possible way of picking up a equilateral triangle and putting it down on itself, according to Ellers.

“Imagine a structure that has some symmetry, like an equilateral triangle,” Ellers said. “If I had a model, I could pick it up and turn it 120 degrees [a third of the way around] and if you didn’t see me do that, you wouldn’t know that I’d done it because the triangle looks the same from several points of view.”

The research Ellers and Dodge conducted delved deeper into this analysis of symmetry. The research focused on multiplication rules, which give a precise view of the symmetry.

“If you perform a rotation of a triangle through 120 degrees clockwise, represented by A, and follow it by a reflection in the vertical line of symmetry, represented by B, the result is the same as a single reflection in a different line of symmetry, represented by C; this fact is expressed algebraically by the equation AB=C,” Ellers said.

Each of these different multiplications are called groups. Groups measure symmetry much in the same way that numbers measure the size of something. Ellers and Dodge turned this problem around in their research. They would start with an algebraic model divorced from any type of symmetry, just the rules of the algebra. The point is to try to construct all the things in the world that have that kind of symmetry. The whole research idea is called representation theory, according to Ellers.

Ellers and Dodge enlisted the help of Kelly Pohland, ’16, and Yuki Nakada, ’16, to further this research over the past summer.

Pohland and Nakada tested a construction, much like an algorithm, developed by Ellers and Dodge, which would break down an important class of representations into the smallest possible pieces. This is analogous to breaking a large molecule into atoms. The construction would break down the centralizer representations into simple representations. A simple representation is a mathematical structure that has nice, clean properties, much like a single element from the periodic table of the elements, according to Pohland.

Before working with Pohland and Nakada, Ellers and Dodge conjectured that, the construction was able to fully break down every representation. In their work, Pohland and Nakada uncovered that sometimes the construction does fail.

“We started off over the summer testing simple cases,” Pohland said. “In the first couple cases, it did work, but very quickly we found the counter example. We showed explicitly in general a whole class of cases didn’t work. Even though we disproved what they wanted, we still found patterns that could lead us into a different direction to find different results.”

Since conducting the research the four researchers submitted a journal article to the Involve Undergraduate Journal of Mathematics. They also plan to attend the Joint Mathematics Meetings in Seattle, Washington in January 2016 if approved.

“[The students] will present in the student poster session,” Dodge said. “We will be going to tons of presentations, I will be introducing them to a number of colleagues in different fields…There is a number of different people from various colleges that they will be able to meet and talk to them about their various schools and programs.”

Although the construction does not break down every representation they were investigating at this point in time, Ellers, Dodge and the students involved in the research plan to find out a way to split those representations into simple representations.