There’s a popular image of a mathematician as a genius at a chalkboard who can look at a complicated problem and somehow see the answer right away. But for Marissa Kawehi Loving, mathematics is less about knowing the answer and more about figuring out how to find it.
An assistant professor of mathematics at UW–Madison, Loving studies geometric topology and geometric group theory, exploring the relationships between shapes and spaces. Her work uses the behavior of two-dimensional surfaces to answer questions about three-dimensional spaces. But before she can tackle the big questions, she often starts small.
When approaching a problem, Loving says that the first question she asks herself is, “What is the simplest example of this that I can come up with?” She might begin with a single symmetry of a surface, draw it out, and see what happens when she applies it repeatedly. Then she’ll try a slightly more complicated example, looking for patterns and testing whether her initial ideas hold up. Sometimes they do. Other times, she discovers something she hadn’t anticipated and has to rethink what she thought she understood.
Although theoretical mathematics doesn’t involve experiments in a laboratory, Loving considers the process surprisingly experimental. Working through examples helps her develop an intuition about what might be true before figuring out how to prove it. When the work is published, much of that messy process disappears.
“A paper tells you a very small part of the story,” she says. “It’s the cleaned-up, nice version from the problem to the solution.”
Loving rarely works through those questions alone. Much of her research involves talking with collaborators, sharing ideas, and challenging each other’s assumptions. “When I do have breakthroughs, it’s not when I’m doing it by myself,” she says. “It’s when I’m actually doing it with others.”
This is a lesson Loving passes on to her students. When an undergraduate recently worried that Loving was disappointed because a proof was taking so long, she was surprised. The student had been returning to their weekly meetings with new ideas, only to discover each time that something was still missing.
“Am I disappointed when you come, and you’ve thought, and you’ve wrestled with it, and you’ve struggled with it? Not even a little bit,” Loving says. “You’re developing your ability to persevere on something where you’re stuck and where you don’t quite understand.”
For Loving, struggling is not a sign that someone isn’t cut out for mathematics. It’s part of learning how to solve problems. She wishes more people understood that, particularly those who have come to think of themselves as not being “math people.”
“Everybody is capable of doing mathematics,” she says. That ability isn’t reserved for people who fit the familiar image of a mathematical genius.
“A lot of math is struggling,” Loving says, “and struggling with math doesn’t mean that you shouldn’t be doing it.”
