# NSF Postdoctoral Research Fellowship

### Eamon Quinlan-Gallego, receives a Mathematical Sciences Postdoctoral Research Fellowship from the National Science Foundation.

The three-year fellowship is awarded to support future leaders in mathematics and statistics by helping them participate in postdoctoral research that will enhance their development. “Receiving this fellowship is an incredible honor, and it will allow me to dedicate myself to research full-time for four semesters and extend my stay in Utah for an extra year. It will also give me funds to travel to conferences and visit collaborators,” he said.

Quinlan-Gallego studies solutions to polynomial equations and their singularities. For example, in pre-calculus, the equation y = x^2 defines a parabola in the plane. This parabola is smooth—it doesn’t have any sharp corners; however, occasionally, polynomial equations can fail to be smooth. These non-smooth points, called singularities, are ubiquitous across mathematics, and their study is a fundamental problem.

## "Receiving this fellowship is an incredible honor, and it will allow me to dedicate myself to research full-time for four semesters and extend my stay in Utah for an extra year."

Typically, Quinlan-Gallego uses two different techniques to study these singularities. First, he can associate certain differential equations to them whose behavior allows them to be classified in different ways. Second, he can study singularities using “modulo-p.” He fixes a prime number (usually denoted by p, but in this case, for example, we could use p = 5). Working “modulo-5” means that when he looks at a polynomial equation, like y^3 = x^2, instead of thinking about it in the real-number system (as you would in pre-calculus), he thinks of it in clock arithmetic. This means that he does all of the algebra using a clock with 5 hours. For example, if our clocks had 5 hours, and it was 4 o’clock and 2 hours pass, it is 1 o’clock. In clock arithmetic, we would say that 4 + 2 = 1. Similarly, 4 x 2 is usually 8 but in our clock, we have 4 x 2 = 3. “By working in this clock arithmetic, we lose all of the “geometry,” but we gain a host of other tools we can use, and the hope is that as the prime p selected gets larger and larger, the behavior of the singularity modulo-p approaches the real behavior,” he said. He also likes to combine these two techniques and think about differential equations modulo-p.

He was good at math as a kid but until he was a senior in high school, he thought he would become a biologist. Then two things happened: he realized he only wanted to study biology because the idea of going to remote islands to look at creatures no one had seen before sounded cool, but learning about all the chemical reactions going on in the mitochondria wasn’t so exciting--and he read Stephen Hawking’s book ** A Brief History of Time** and became fascinated by how mathematics is used to learn about things that are far away in space and time. At that point, he switched from studying biology to physics. The jump from physics to mathematics was much more straightforward when he realized he was enjoying his math classes more than experimental physics.

Quinlan-Gallego was raised in Spain—his mother is Spanish and his father is American. After high school, he left Spain to study in Scotland at the **University of Glasgow**. “There was this great program for citizens of the European Union that allowed me to study in Scotland for free,” he said. “I had a wonderful time in Glasgow, and I was given so many amazing opportunities.” During his undergraduate years, he also participated in an exchange program at the **National University of Singapore** for a year.

Once he completed his bachelor’s degree, he knew he wanted to come to the U.S. for graduate school. He was accepted to the **University of Michigan** and began working under Professor **Karen Smith**, who serves as the William Fulton Distinguished University Professor of Mathematics. He also spent more than a year in Tokyo, again as an exchange graduate student, at the **University of Tokyo**.

He’s looking forward to continuing his work at the U. “The department has many experts in modulo-p methods and a host of other very interesting topics, so I’m looking forward to learning as much as possible from them and moving forward in my research.”

**by Michele Swaner, first published @ math.utah.edu**