A Montana State University mathematics professor who is nationally recognized for his work to solve a problem that may advance an area of quantum physics has just received a highly coveted National Science Foundation CAREER grant, the first such award received by a member of the MSU Department of Mathematical Sciences.

David Ayala works in a mathematical field called topology, specifically higher category theory that has important applications in quantum physics. The bulk of his $400,000 NSF CAREER award, given over five years, will fund two mathematical physics conferences around Bozeman, as well as support graduate students in the MSU Department of Mathematical Sciences in the College of Letters and Science, he said.

Ayala explains that higher category theory is a system of techniques for identifying patterns, or structure, held by certain arrays of information. Ayala's work applies these techniques to data that are the observables of certain quantum systems, with the goal of addressing the long-time, long-distance behavior of the system.

Ayala, 37, just began a semester-long appointment as one of three leaders of a program that hosts 200 mathematicians from around the world during a four-month program at the Mathematical Sciences Research Institute in Berkeley, California. The fall semester Ayala will travel to several other locations to work on his ongoing project before returning to MSU in January 2021.


When Ayala returns, the NSF grant will enable him to integrate research and education between students and researchers in Montana and surrounding states. He also will plan conferences that will bring renowned researchers in mathematical physics to the state.

While his field of mathematics may sound esoteric, his colleagues use more concrete words to describe his interactions.

“David is fiercely committed to people’s enjoyment of mathematics, whether it is himself, his undergraduate students or his graduate students. And he is fearless about trying new things,” said Elizabeth Burroughs, head of the MSU Department of Mathematical Sciences. “That’s what makes him a great teacher, but it also makes him a great mathematician, taking his research field in new directions.”

Although Ayala’s work is solitary and his field is complex, he often likes to work communally in local coffee shops, where he jots down complicated equations in large, unlined art notebooks. He also frequently spends several hours each day video chatting with his collaborators, who are located around the world.

A prominent researcher in his field, mathematician Haynes Miller at Massachusetts Institute of Technology, said that Ayala and his collaborator, John Francis, have led the exploration of a field of mathematics called factorization homology, which connects high-dimensional geometry with higher-category theory. Miller said the impact of Ayala and Francis’s work is only just beginning to be felt.
 
While Ayala’s work is garnering him recognition from colleagues in mathematics and quantum physics around the world, people in Bozeman are likely to recognize his name for his accomplishments as a trail runner. He has twice won the Bridger Ridge Run, including last August’s race. His arduous 93-mile run through Death Valley in 2013 was featured in Trail Runner magazine.

Ayala said that he runs not to think and solve equations. Rather, trail running allows himself to turn off his active brain.

“I run so I can’t think of anything else,” he said.
 
In fact, the landscape and community surrounding Bozeman was an important factor in his coming to MSU. The son of medical providers who worked in Zion National Park, Ayala grew up in Rockville, in southern Utah. He attended a two-room schoolhouse in which the first three grades were in one room and the other grades in another room. There were five students in his grade.

“It was precious,” said Ayala of his upbringing. He said he remembers being embraced by his entire community, since there weren’t many children. He was bussed to high school in Hurricane, Utah.
 
Ayala recalls spending a lot of time alone, running and hiking and skiing. He taught himself to juggle and then later started reading books on math and physics for fun. He attended the University of Utah. “I didn’t even think of going any place else.”

And then his education and passion for math jumped into high gear. He earned a bachelor’s degree in math in just two years. In the next two years, he earned another bachelor’s degree in physics as well as a master’s degree in math. His doctorate in mathematics is from Stanford. He was a post-doctoral fellow at the University of California, Berkeley’s Mathematical Sciences Research Institute, the University of Southern California, Harvard University and the University of Copenhagen.

“I really, really wanted to return to the Intermountain West after that,” said Ayala, who has been at MSU since 2014.

Ayala said that in addition to allowing him to pursue his research, his position at MSU provides him the satisfaction of working with intriguing undergraduate students, several graduate students and collaborators.
 
“I enjoy sharing an idea and seeing in a student’s eyes the moment they understand it,” he said.
 
Burroughs said the fact that Ayala already has worked at such a high level of scholarship for many years and produced a steady output of strong and innovative mathematics is just part of the Ayala equation.
 
“He finds joy in what he does. And he hopes other people find joy in it, too,” Burroughs said. “He has chosen to be here at MSU in the mountains doing the work he loves. And we at MSU are so fortunate.”