Amie Wilkinson is an explorer. Instead of seeking uncharted land, she’s after undiscovered mathematical worlds — complex systems of motion that unfold in unexpected ways. As a professor at the University of Chicago, she’s known for discovering unique types of these “dynamical systems” that had been only conjectured to exist.

Wilkinson’s career has proceeded in a dynamical way, too. She now approaches her roles as a researcher, teacher and mentor very differently than she did when she was starting out. During her undergraduate years at Harvard University in the 1980s, there were no female professors in the math department, causing her to doubt whether she would ever belong in the ranks of top-flight mathematicians. Then when she was a young researcher, “I could be kind of hard on my students,” she said. “That was an extension of my own insecurities about my own abilities.” But, she says, “I think that I am much more visionary now with my students because I have much more confidence.”

Quanta Magazine spoke with Wilkinson about the emotional dimensions of mathematical discovery, bizarre examples of dynamical systems that she’s found, and the idea of a “safe space” in mathematics. Multiple interviews have been condensed and edited for clarity.

What are dynamical systems?

Dynamics is the study of motion. In particular, the motion of points in a space dictated by a fixed set of rules. For example, the evolution of the solar system, in its idealized form, unfolds exactly according to the rules of gravity. And so where our solar system is a minute from now is completely determined by where it is now.

What attracted you to this area of mathematics?

You can imagine a photograph of the current solar system. That’s a static thing. But when you add dynamics, that static picture comes to life. There are certain complicated fractal objects — like a Julia set, or the Koch snowflake — that are produced by dynamical systems. The actual geometric features of the objects carry the marks of the dynamics that produced them. I find that really beautiful. It’s like things have a history and a future, they’re not just these objects I’m studying.

What kinds of questions do mathematicians try to answer about dynamical systems?

You might ask: For this particular trajectory we’re on now, is it unstable? Or for how long might it be stable? How much time will it take before Mercury is ejected, for example? As it turns out, in this whole universe of solar systems there are some really weird long-term evolutions that can happen.

Even within the motion of our own solar system?

No. It’s when you consider the motion of all possible solar systems that feature a sun and planets with the same masses as our planets. The future of our solar system is just the trajectory of a single point through this massive dynamical system.

How’d you choose this field?

It could have been a coincidence. But I do think that dynamics appealed to me because of its dynamic nature. The idea of studying the notion of action, it just completely brought life into the subject for me. It literally animated the subject.

What have been the biggest or most provocative questions for you since then?

For me there are some interesting big questions like: What are the general mechanisms we can find for certain types of long-term behaviors in a dynamical system? How can we understand the rough features of a dynamical system with partial information? Those are the kinds of big organizing questions around things that I do.