Ask. The Advice That Changed a Career in Mathematics
Former AMS President Bryna Kra on Research, Leadership, and the Future of the Field
Mathematics is often presented as the most certain of disciplines—a world of final answers and tidy conclusions. But when we spoke with Bryna Kra, a different portrait emerged. Mathematics, she suggested, is not powered by certainty alone. It is powered by motion: ideas that evolve, recur, and return—sometimes from unexpected places.
We invited Bryna to talk about her work in dynamical systems and ergodic theory, and about how tools built to describe change can illuminate something that seems unchanging: the integers. What unfolded was a reflection on how mathematics moves—through ideas, institutions, mentorship, and the small acts of asking that quietly reshape what is possible.
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At first glance, Bryna’s field sounds paradoxical. Dynamical systems study how things evolve over time—how patterns recur and what long-term behavior must appear. The standard examples are physical: planets orbiting, weather shifting, populations fluctuating.
But Bryna broadened the definition. A dynamical system, she said, is simply a collection of objects together with a rule for how they change.
And then she made the surprising claim that numbers themselves contain a hidden dynamical life.
The idea, developed by Hillel Furstenberg, is to take a subset of integers—say, the even numbers—and repeatedly apply a rule like “add one.” Evens become odds. Add one again, and you return to evens. What looked static becomes a system with recurrence and structure.
The example is simple, but the framework is powerful. More complicated subsets produce dynamics rich enough to tackle deep questions in number theory. Instead of asking only what a number is, you ask what survives under repeated transformation. That shift in perspective, Bryna told us, is part of what makes mathematics feel alive. “Math is all one interconnected beast.”
Her own path reflects that interconnectedness. She entered graduate school intending to study number theory, only to be captivated by lectures on dynamics that reframed arithmetic in motion. The maneuver—taking something static and turning it into change—opened an entirely new toolkit. Ideas that once described planetary stability now illuminate patterns in integers. Mathematics advances, she suggested, when someone dares to move a problem into a new frame.
We expected to focus mostly on research. Instead, Bryna returned again and again to communication.
Much of pure mathematics has no immediate application. But mathematicians, she argued, owe the public—and each other—a clearer explanation of how abstract work becomes future infrastructure. Mathematics survives not just through theorems, but through the ecosystem that supports discovery: teaching, mentoring, institutions, and shared resources.
That conviction shaped her presidency of the American Mathematical Society. Many mathematicians, Noah admitted, barely know what the AMS president does. Bryna’s answer was revealing: the president does “sort of everything”—and alone can do nothing.
Professional societies run on collective effort. Job boards, databases, grants, conferences, accessibility initiatives—much of the discipline’s scaffolding operates quietly in the background. During her term, she championed small grants for mathematicians at primarily undergraduate institutions, where heavy teaching loads often limit research funding. The grants were modest, but the process mattered: ideas become real only when a community chooses to build them.
Leadership, in her view, is not about directing mathematicians. It is about building pathways that make their work possible.
In her farewell presidential address at the Joint Mathematics Meetings, she expanded the lens further. Mathematics faces tightening budgets, shifting classrooms, and the rise of AI in both research and education. Departments cannot assume they will remain untouched.
Her message was direct: mathematicians need to “get out of the silence.” They must communicate more—with each other and with the broader world—not as branding, but as responsibility. If mathematics underlies modern science and technology, then mathematicians must remain engaged in shaping its future, including the trajectory of AI.
For centuries, mathematical truth and human understanding were tightly linked. AI may loosen that connection, producing correct answers without traditional explanation. Bryna did not sound alarmist. She sounded resolute. The field must evolve.
As the episode comes out during Women’s History Month, we asked about being a woman in mathematics. Bryna gently reframed the question. She sees herself first as a mathematician who happens to be a woman.
Still, she acknowledged that women remain underrepresented in many areas. Her response has been to build spaces and programs that widen participation—founding Women in Math groups, launching graduate research workshops, and creating pathways for those who might not otherwise see mathematics as a career.
Barriers, she noted, often begin early. The goal is not to force outcomes but to open doors. Everything requires resources; what matters is how we choose to use them.
By the end of the conversation, Bryna’s mathematics and her leadership felt like variations on the same theme. Dynamical systems ask what persists over time, what returns, what structures emerge from repeated action.
Her career has followed similar patterns: collaborations sparked by chance conversations, long-term partnerships born from simple requests, institutional policies created because someone asked for change.
That became her closing advice to young researchers:
Ask.
Ask for what you need. Ask for support. Ask for policies that do not yet exist. The worst answer is often just “no.” But sometimes asking is the first step in making something real.
“If one doesn’t ask,” she said, “it doesn’t happen.”
Mathematics is often described as a finished structure. Bryna Kra offered something more dynamic: a discipline in motion, shaped not only by proofs but by pathways.
And those pathways begin, more often than we admit, with a simple, brave sentence:
Could we try this?
Always,
Autumn
The "just ask" advice sounds almost too simple but it's surprisingly hard to internalize. In math especially there's this culture where you're supposed to figure things out alone, like asking for help is admitting you're not good enough. Kra's point about leadership being a learned skill — not something you're born with — applies way beyond academia.