Pattern Recognition

Bryna Kra understands that mathematics isn’t everyone’s first language. She’s witnessed eyes glaze over when she waxes poetic about “ergodic theory,” the study of long-term behavior of abstract systems, and “combinatorics,” a field of math that focuses on counting, arranging and combining elements.

“I’ve been a mathematician for a while,” she says. “I do know how things work.”

But saying Kra is a mathematician is like saying Yo-Yo Ma plays the cello. The Sarah Rebecca Roland Professor of Mathematics at Northwestern’s Weinberg College of Arts and Sciences, Kra is globally recognized for her scholarship on dynamical systems and ergodic theory. In layperson’s terms, Kra works in the abstract terrain of a field that is itself steeped in abstraction, finding patterns in systems that form the building blocks of mathematics. By studying those patterns, Kra says, she can make long-term predictions on how those systems will act.

Take climate change, for example. Researchers use dynamical models to calculate and predict changes in weather patterns. Kra works on an abstract version that informs such work.

“The abstract theory always goes before the concrete application,” she says. “It forms the backbone of the applications.”

Her passion for this research has led to major accomplishments. In 2010, she won the Levi L. Conant Award from the American Mathematical Society for her expository work on the Green-Tao theorem showing the existence of arbitrarily long arithmetic progressions in prime numbers. She was named to the society’s board of trustees in 2015. The following year, she became a fellow of the American Academy of Arts and Sciences. Earlier this year, Kra was elected to the National Academy of Sciences, an honor bestowed on the country’s best researchers.

“I love finding abstract structures that imply there’s a lot more going on,” she says. “They're hidden, you don't see them when you first look at a seemingly random system, but then the pattern appears and you discover that the absolutely random is not really something that happens. To me, that’s fascinating.”

A ‘Beautiful’ Connection Between Random and Structured

It wasn’t always fascinating. The daughter of Irwin Kra, an award-winning mathematician himself, Kra grew up in Stony Brook, New York, with “a lot of math around the house,” she says. But after struggling through an advanced class during her first year at Harvard University, Kra changed her major to philosophy. A year later, dissatisfied academically and unsure of her future, Kra returned to mathematics and embraced its intellectual challenges.

After graduating with a bachelor’s degree in math, Kra enrolled in Stanford University, where she worked with professor Yitzhak Katznelson who became a supportive presence as she explored various disciplines. He also encouraged her to enjoy life beyond the classroom. Kra soon discovered ergodic theory, the study of dynamical systems. She found beauty in using this type of math to answer combinatorial questions and find structural properties, she says.

“In ergodic theory, there are general principles that control all of these things that seemingly look very, very different,” she says. “You’re looking for general principles that tell me, ‘Okay, here you have completely random behavior, and over there, you have extremely structured behavior.’ Finding these new, deeper connections between fields is really appealing and beautiful.”

Kra earned both her masters and doctorate degrees at Stanford before spending the next six years as a postdoctoral scholar at the Hebrew University of Jerusalem, Ohio State University, the Institut des Hautes Études Scientifiques (IHES), and the University of Michigan. She counts her experiences in Israel and France as the two most influential for her career. She chose to learn both Hebrew and French to speak with her colleagues. Her work in Israel gave her the time to delve further into ergodic theory, but it was at IHES that she met mathematician Bernard Host in Paris.

The two would work on what became a seminal paper in her field. Their 2005 publication, “Nonconventional ergodic averages and nilmanifolds,” answered a question that essentially proved one could find the average within abstract systems by identifying structures controlling long-term behavior. Those structures — called nilsystems — could be applied to other fields of mathematics, including combinatorics and number theory.

In the process, she and Host solved a proof that had been open since the 1970s.

“Everyone likes answering an open question,” Kra says. “But it was more than that. You’re finding out what controls them in a way you can apply to other problems.”

Championing Minorities in Math

While working with Host, Kra joined the Northwestern faculty in 2004. Since then, she has continued to research abstract systems while also serving in various administrative roles, including chairing the Department of Mathematics from 2009 to 2012, becoming the first woman to do so.

Recently, her work has moved toward symbolic dynamics, a mathematical pursuit that involves finding patterns in infinite sequences that model other systems but are interesting in themselves. Kra began exploring this field after meeting Van Cyr, a postdoctoral scholar at Northwestern. Cyr is taking a year-long sabbatical as an associate professor from Bucknell University to work with Kra on Nivat’s Conjecture, a question about bi-infinite sequences, along with several other questions. Again, by studying that seemingly wild behavior portion of integers, the researchers hope to discern recurring patterns that can be found on both sides, which could have possible applications in symbolic as well as topological dynamics.

It’s something the two have worked on since 2011. While his focus was in thermodynamics — very different from Kra’s work — the two found it easy to engage with one another.

“She was happy to hear what I was working on and what kinds of questions I was interested in,” says Cyr. “I knew she was someone who was easy to talk to and could quickly understand the key features and the intricacies of them.”

Just as important to her as her research is Kra’s advocacy on behalf of women and people of color in mathematics. She spearheads multiple initiatives at Northwestern, including the department’s Women In Mathematics mentoring group, Graduate Research Opportunities for Women (GROW), a weekend-long conference for undergraduates interested in mathematics, and a one-year post-baccalaureate program for minorities that will begin in summer 2020.

These programs are essential to the future of math, Kra says. She’s proven that she can examine the numbers and see a pattern, but with respect to lack of inclusion in her field she wants to change the pattern. “Math has had low numbers of minorities and women working in research and they haven’t budged in a long time,” she says. “That’s not acceptable.”

By Glenn Jeffers