Nathan Pflueger, assistant professor of mathematics, confesses that until around the age of 13, he really didn’t like math.
“It felt pretty routine,” he said, “like there’s not much room for creativity. It felt like something a computer could do.”
But that view of mathematics changed dramatically for Pflueger when he started studying algebra in middle school.
“In algebra, I started to see that there was all this alchemy you could do. You have a problem, and the way you solve it is to push it, pull it, and move it around until it gets to something very simple. That felt miraculous to me. It looked like there was a lot of room for invention in this process because there’s more than one way to pull these levers.”
In pulling those levers, Pflueger discovered a passion for mathematics. He went on to graduate from Stanford with a bachelor’s degree in math. Then it was on to Harvard for a Ph.D., and now to a faculty position at Brown.
Pflueger’s study of algebra has come a long way since those formative middle school years. These days, he studies algebraic geometry, the multidimensional shapes and spaces formed by complex algebraic equations. Those shapes are called algebraic varieties, and Pflueger studies a specific type of variety called an algebraic curve.
“They’re in some sense the simplest algebraic varieties in that they’re the smallest,” Pflueger said. “But they still have quite a bit of mysterious structure and personality. What makes these guys interesting is that you can write down a whole bunch of curves and discover they’re actually the same curve. You can form one into any other through a series of steps.”
Those groups of converging curves are the main characters in Pflueger’s research. His approach to studying them is not unlike the study of ecology, he says. Ecology involves the study of not just a single species, but collections of species living and interacting together. At times, that ecosystem itself can appear to function like a living organism. The same goes for the curves Pflueger is trying to understand.
“Instead of studying individual curves, I study the ecosystem as a whole,” he said. “You start to grasp that the whole ecosystem is also a geometric object — an algebraic variety with shape of its own.”
And it’s the structure of those larger ecosystems that Pflueger is trying to understand.
Working with these shapes is dizzyingly complex. Pflueger’s approach to studying them involves tearing them apart and looking at them piece by piece. Through a process called degeneration, terms in the equations that form the curves are pushed to infinity, causing the curves shatter, or as Pflueger puts it, “decompose and leave fossils.” Then it’s a matter of doing a bit of paleontology with what’s left behind.
“The second part of this process is called regeneration,” Pflueger explains. “We assemble the fossils, and try to reverse engineer what this thing looked like under ordinary circumstances.”
Work in algebraic geometry has the potential to be used to develop new forms of cryptography. It might also have applications in physics, helping to make calculations in the multidimensional world of string theory. But for Pflueger, it’s the pure beauty and logic of the work that provides the allure.
“Like literature and music,” he said, “I think it enriches culture.”
Pflueger also sees another underappreciated benefit to the study of math. Fundamentally, he says, math is a form of communication.
“In math you really have to justify every single thing you believe, and you do that by a logical argument. Once you convince yourself, you have to be able to write the proof in a way that other people can understand even though they haven’t necessarily done all of the computations you have. The process of writing a math paper is taking all of your steps and figuring out how to distill them to something that’s minimal yet will communicate all these things. I think it’s a good way to train yourself to think very carefully and write precisely.”
Pflueger looks forward to engaging his students here at Brown in that process.
“Part of the reason I wanted to go into math was that I wanted to get to the point where I could teach it. What I liked about math from the beginning is that it’s not only a bunch of ideas and facts, but it’s also a way of communicating.”