The Engaging World

Stefan Patrikis

For Stefan Patrikis, assistant professor of mathematics, an interest in math developed over a long period. He says two experiences particularly helped make him a research mathematician. The summer after ninth grade he attended an intensive math camp—the Ross Mathematics Program—for eight weeks, and for the first time was able to see what math outside the classroom is about. He went on to major in math in college but wasn’t sure about continuing with graduate school until his undergraduate advisor suggested a research problem, and he was hooked! Eventually, he stoppedgoing to classes and only worked on the problem, which he found thrilling.

What is Number Theory?

Ever since attending the Ross Program, studying mathematics has for Patrikis been all about number theory. At first it was simply the subject he had first fallen in love with. He then set out to learn the other things he would need in order to understand more number theory. “Ironically, what might sound like a narrow focus was in practice immensely broadening,” he said. “Number theory draws in deep ways on the widest range of modern mathematics: algebraic geometry, representation theory, topology, analysis, etc. That aspect of the subject—I think it’s the best single witness to the ‘unity of mathematics’—is one of the things I most appreciate.”

Number theory has deep historical roots, and the mathematical objects studied today have grown out of more classical questions that have been studied for centuries. Galois representations are an example of this. Named after 19th-century French mathematician Évariste Galois, these are ways of packaging “symmetries” of polynomial equations. “This is probably a surprising phrase,” said Patrikis. “We’re used to thinking of geometric objects as having ‘symmetries,’ but a fundamental discovery of the 19th-century is that equations do as well!” Galois representations encode all of classical algebraic number theory—many questions about prime numbers and ordinary integers—and vastly extend its scope, providing connections with algebraic geometry, representation theory, and more. “Much of the appeal in studying the modern subject is these astonishing connections that bind together what seem to be very different subjects of study,” he said. “We understand relatively little about these conjectural connections, so there is a great deal of freedom of exploration. Ultimately, this is what I enjoy and find most precious about mathematical research.”

Funding from the U and the National Science Foundation

Patrikis is grateful for funding provided both by the U’s John E. and Marva M. Warnock Presidential Chair in Mathematics and by a National Science Foundation (NSF) CAREER grant. Funding has given him the means to travel to conferences and invite visitors to Utah. However, just as important, funding has made a big difference in the training of his Ph.D. students, giving them more time to focus on their research and allowing them to travel to conferences and see some of the wider world of number theory. “This is crucial to their education—otherwise they would only be getting my perspective on the subject,” Patrikis said. The NSF CAREER grant, along with support from the U’s Mathematics Department, helps fund the department’s Summer Mathematics Program for High School Students. The program tries to give a group of local high school students an experience comparable to what Patrikis enjoyed and benefited from in the Ross Program.

Applications for Number Theory

“Number theory has important applications to cryptography and computing,” said Patrikis. “But I should add that the relationship between today’s basic science and tomorrow’s practical application can be so unexpected. In 1940, G.H. Hardy wrote a now-famous essay called ‘A Mathematician’s Apology.’ Heartbroken to see the world again at war, and disgusted by science’s contributions to modern warfare, he wrote a defense of ‘pure’ and even ‘useless’ mathematics. Hardy expressed relief that neither number theory nor Einstein’s theory of relativity would find any practical, let alone war-like, applications for many years. He was completely wrong! Today, number theory plays a key part in securing our internet communications, and the accuracy of GPS depends on understanding relativity.”

Move to the University of Utah

Born and raised near New Haven, Conn., Patrikis obtained a Ph.D. in mathematics from Princeton University in 2012. He spent time as an NSF postdoctoral fellow at Harvard University and as a Moore Instructor at MIT.

“There’s so much mathematical energy in Princeton,” he said, “and so many people who are so devoted to mathematics. The resulting intellectual camaraderie makes it a very special place.” He has equally fond memories of strolling along the Charles River and walking through Mount Auburn Cemetery in Cambridge. “I was relatively unconstrained by formal responsibilities in those years, so I could use that time to broaden my mathematical reach and think about new kinds of problems,” he said. “My own research has gained a great deal from this, and it has also made me a much more versatile doctoral advisor. I owe this to having the time to work with few distractions.”

He came to the University of Utah in 2015 and was struck by how extraordinarily welcome he was made to feel by the Math Department. When he isn’t teaching, doing research, or mentoring students, he enjoys cooking, reading fiction and poetry, playing the piano, and playing soccer.

As he continues with his research, Patrikis believes there are still peaks to climb. “It’s clear that other peaks will have to be discovered even to find a route up to those that we can see.”