**Beyond** the Numbers

## With a passion for math theory, **Peter Jakes ’17** explores the solvability of complex equations.

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, he realized that passion by studying a field of mathematics called Galois theory to determine the solvability of an equation. A mathematics and statistics major and Honors Fellow, 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.

His two-year project involved designing 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. Formulas for the first through fourth power of polynomials have been known since the 1500s, but for polynomials from the fifth power onward, solutions cannot always be expressed precisely. When a formula for those polynomials does exist, it’s called “solvability by radicals.”

Galois group is a mathematical term for the grouping together of the properties of functions, or equations. Analyzing an equation’s properties reveals specific information about that equation, in the same way that a person’s characteristics convey certain details about their personality. 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. His goal was to create a computer program that allowed any user to input a polynomial up to the seventh degree. His algorithm then determines 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.

Essentially, when you have an equation, you’re solving for when ‘x’ equals zero. But when we get up to ‘x’ to the fifth power, a formula doesn’t always exist.

A method to find the Galois group for sixth- and seventh-degree equations existed previously, but Jakes’ method reduces the number of steps necessary and makes his algorithm work faster. Although this research primarily deals with math theory, it has the potential to play a critical role in areas such as cryptography or data analysis. His next step is to create a program that will be accessible to other mathematicians online, so when they begin working with their polynomials, they have access to Jakes’ more efficient method.

Jakes 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 also received the Walt and Susan Patterson Award for best presentation at the 2017 Southeastern Sectional Meeting of the Mathematical Association of America. 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 Associate Professor of Mathematics Chad Awtrey.

Following graduation, Jakes began work as an actuary at Allstate in Chicago.