# Passion for pure math theory fuels Lumen Scholar's research

### Peter Jakes '17 is among the recipients of the Lumen Prize, which provides selected students with a $15,000 scholarship and celebrates their academic and creative accomplishments.

*By Sarah Mulnick '17*

For most students, pure math theory is a mystical concept reserved for famous mathematicians like Alan Turing or Isaac Newton. But for Peter Jakes ‘17, it’s the focus of a passion he’s held since his sophomore year of high school.

At Elon University, he’s been able to realize that passion by studying the field of mathematics called Galois theory to determine the solvability of an equation. For the last two years, Jakes has worked to design an algorithm that could be presented to the greater mathematics community to sort equations by whether their solutions can be exactly expressed or merely approximated.

“Essentially, when you have an equation, you’re solving for when 'x' equals zero,” he said. “But when we get up to 'x' to the fifth power, a formula doesn’t always exist.” Jakes said that solutions for those polynomials cannot always be expressed exactly and added that it’s called “solvability by radicals,” when there’s a formula in existence. Formulas for the first through fourth power of polynomials have been known since the 1500s.

Galois group is a mathematical term for the grouping together of the properties of functions, or equations. Another way to think of it is as the characteristics of those equations – similar to how the characteristics of a person can tell you information about that person, such as where they’re from or what they do. Jakes’ research uses Galois groups for equations that go beyond the fifth degree, because sixth- and seventh-degree equations haven’t been studied as extensively.

That’s where Jakes’ research starts to fill the hole. “It expands on the current research,” he said. “There’s a pretty big gap, because there’s a lot of reducible polynomials and we didn’t have a method to find the Galois groups for those polynomials. It’s important to fill that gap within the specific group.”

Essentially, Jakes added, he uses Galois groups to figure out which equations are solvable by radicals, and which ones aren’t. He created an algorithm that takes into account whether or not an equation is reducible––that is, if it can be broken down into smaller equations.

His work is part of the Lumen Prize, Elon’s preeminent research grant. The $15,000 scholarship supports scholars and their faculty mentors, who collaborate on a two-year intensive research project.

Efforts include coursework, study abroad, research both on campus and abroad as well as during the regular academic year and summers, internships locally and abroad, program development and creative productions and performances. The name for the Lumen Prize comes from Elon’s historic motto, “Numen Lumen,” the Latin words for “spiritual light” and “intellectual light.”

Jakes is the first Lumen Scholar in mathematics since the program’s inception in 2008, and the first to conduct his research within pure math theory, rather than applied math or statistics.

He began his work by focusing on whether there was a more efficient way to compute the Galois groups of a sixth-degree polynomial. His eventual goal was to create a computer program that could allow any user to input a polynomial up to the seventh degree. The algorithm would determine the information available about the formula’s symmetries and characteristics. The process makes it possible for users to learn whether the formula is solvable by a radical – that is, whether they can find an exact solution to the equation.

Jakes said that for sixth- and seventh-degree equations, there had been a method to find the Galois group using what are known as resolvent polynomials, but that he’s found a more efficient way to do it. It reduces the number of steps necessary, and makes the algorithm he created work faster.

Although this research primarily deals with math theory, it has the potential to play a critical role in different areas such as cryptography or data analysis. His next step is to create a program that will be accessible for other mathematicians online, so that when they begin working with their polynomials, they have access to Jakes’ more efficient method.

Jakes worked with Chad Awtrey, associate professor of mathematics, to complete his project. Awtrey said that one of Jakes’ core strength is his ability to take ownership of a project and exhibit leadership in his responsibilities. “He’s very bright,” Awtrey added. “The ability to acquire new knowledge and make sense of it, and then be able to apply and communicate that knowledge — that’s something that’s very strong in Peter.”

Jakes has presented his research at several different conferences, including MathFest 2016 in Columbus, Ohio, where he became the first Elon student to win an award for Best Presentation. He’s presented at the Spring Undergraduate Research Forum twice already, and will again this year, an plans to present at the national math conference the Joint Mathematics Meeting and National Council on Undergraduate Research.

Some of his work has been published in the Minnesota Journal of Undergraduate Research, and he is a contributing author to a recent paper published by Awtrey. In addition to his research, Jakes is an Honors Fellow, a brother in Beta Theta Pi fraternity and is a member of the marching band, club ultimate Frisbee and InterVarsity.

Following graduation in May, Jakes will begin work at Allstate as an actuary in Chicago.