Art by Karen Lin.
On a snowy February evening, thirty undergraduates gather inside a brightly lit classroom on the first floor of Leet Oliver Memorial Hall. At the front of the room is Hee Oh, the Abraham Robinson Professor of Mathematics at Yale, drawing crisp white circles on the board with a brand-new stick of Hagoromo chalk. By the time she turns around, she has traced out three perfect circles, each touching two others at a single point.
Oh studies Apollonian circle packings, which are created by filling the space between three mutually tangent circles with successively smaller tangent circles. As she draws the smaller circles on the board, she talks about her fascination with circle packings, connecting them to centuries-old theorems of Greek geometers and to recent developments in hyperbolic geometry. The students listen with undivided attention, captivated by her every word.
Twenty-eight years earlier, Oh had been sitting in the classroom next door, similarly entranced as a grey-haired professor lectured about connections between geometry, number theory, and dynamics. At the time, Oh was a first-year graduate student at Yale who had just arrived from Korea; the professor, Gregory Margulis, was a Fields medalist who had come to Yale the previous year. Oh had taken a number of math courses as an undergraduate at Seoul National University, but Margulis’s class was the first time she saw how the three topics could be brought together in such unexpected ways. When it came time to choose a thesis advisor, Oh decided that she wanted to work with Margulis.
The problem Margulis gave Oh was to show that a certain class of discrete subgroups were arithmetic, meaning that they could only be constructed using number-theoretic methods. Margulis suggested an approach that would prove the result for some specific cases, but he was surprised to find that, by the time she had finished graduate school, she had proven arithmeticity in nearly all cases using a result called Ratner’s theorem. According to Margulis, Oh worked exceptionally hard. “It’s difficult to predict what will happen with any student in ten or fifteen years,” Margulis said. “But even at that time, it was clear that she was capable of coming up with the kind of ideas that are necessary to become a leader in the field.”
Despite her mathematical acumen, Oh hadn’t always intended to be a professor. When she first began studying math in college, she thought that being a mathematician meant working alone on obscure problems, as if she could disappear from the world and no one else would notice. Halfway through her undergraduate studies, she felt that she wanted her life to have more purpose. “If I spent my time helping the weak and oppressed, I thought that would be a meaningful existence,” she said.
This conviction led Oh to pursue a calling as a social activist, organizing protests against the military dictatorship that had taken over the South Korean government in the mid-1980s. Oh recalls standing in the front line of student demonstrations, facing off against a wall of armed police. At one point, she was knocked unconscious on the street. But, in spite of her sacrifices, she wasn’t able to make significant progress on the social issues she had set out to solve, at least by her own standards. “There were no single bulletproof solutions like the ones I was used to in math,” she said. “I liked the clarity of mathematics, and I missed it.”
After a year of working as an activist, Oh decided to return to studying math. She credits the experience of standing firm during protests with giving her the persistence necessary to reach key points of her academic career, from finishing her PhD in 1997 to navigating her first positions at Oklahoma State, the Hebrew University, and Princeton. By the time she received a tenure offer from the California Institute of Technology in 2003, she had started a family with her husband with a two-year-old son, and a daughter later to come. Once she became a mother, her respect for other female mathematicians grew immensely. “I didn’t know what they were going through before,” she said.
Oh learned about circle packings in 2007 from Peter Sarnak, a number theorist at Princeton she met while working as an assistant professor. Sarnak had been trying to count circles in circle packings that were no smaller than a given radius, having realized that the task was equivalent to counting points of an orbit on an infinite-volume hyperbolic manifold. Because Oh was working on similar counting problems in finite-volume spaces, he thought she would be well-equipped to look into the problem. One of Sarnak’s former students, Alex Kontorovich, had been trying to understand the structure of this space using partial differential equations, but Oh saw that they could use an approach from representation theory instead. Oh and Kontorovich spent the next several months working out the details of their proof, encountering roadblocks and false leads along the way. Nevertheless, they pursued the problem with what Kontorovich calls a “keen tenacity,” and by the end of 2008, they had arrived at a solution they felt confident enough to share.
Oh gave the first talk on their results in October of that year, and she later wrote a paper with Kontorovich, which would go on to be published in one of the leading journals of mathematics. Oh describes the process of finding the solution like climbing up a mountain—sometimes she wasn’t sure if she was looking at the right mountain, or if there was a path to the summit at all. “You often go halfway up, only to find that the path doesn’t take you where you thought it would,” she said. But once she and Kontorovich had completed the proof, the implications of what they had done became clear. They had made it to the top—and they couldn’t help but marvel at the view.
The techniques Oh used opened up new avenues of research, leading to numerous collaborations through which she sought to investigate related properties of circle packings. For her work on circle packings, Oh received several national honors and awards—the Satter Prize, a Guggenheim Fellowship, and the Ho-Am Prize, the last of which described her work as a “technical tour de force” that brought together an interplay of ideas from different areas of mathematics. Among the distinctions was an offer of a faculty position at Yale, which she accepted in 2013 to become the university’s first tenured female math professor.
In some ways, Oh’s career has come full circle since returning to Yale. She now runs the Group Actions and Dynamics seminar that Margulis started thirty years ago, and her office is located across the hall from the classroom where her interest in geometry and dynamics was first piqued. For the past several years, Oh has also mentored graduate students, who have become her frequent collaborators and coauthors. “Watching them grow mathematically has been very rewarding,” she said. “By proving some of the conjectures I’ve proposed, they’ve realized my own mathematical dreams.”
Wenyu Pan, who completed her thesis under Oh’s supervision in 2018, recalls how Oh helped her develop the same mathematical confidence that she says Oh exhibits in her own research. “Hee challenged me to solve some really difficult problems, like the ones she herself would solve,” Pan said. When Pan couldn’t figure out one of Oh’s problems the first time around, Oh would give her an easier problem to solve, or encourage her to discuss ideas with other mathematicians until she was ready to try again. “Even at my lowest points, she just kept encouraging me,” she said.
Perhaps one reason for Oh’s encouragement is that there are so few women studying math in the first place. About one-third of math graduate students in the United States are female, but at Yale, the gender disparity is even more stark—only one of the thirty-two PhD candidates in the math department last year was a woman. There is no clear-cut fix to the gender imbalance—the issue more closely resembles the social science problems that Oh dealt with as an activist than the math problems she thinks about now—but mentoring graduate students and conducting research are ways that Oh is trying to approximate a solution. “If you see female mathematicians doing great research around you, you begin to feel that you can be one of them too,” she said.
As a graduate student, Oh had been inspired by the work of a woman named Marina Ratner—the ergodic theorist who proved the central result she used in her thesis. Now, a younger generation of students is looking up to Oh. For Jingyi Cui, a former math and economics major who took Real Analysis with Oh last spring, Oh was the first female professor outside the humanities she had during her four years at Yale. “It was so refreshing to see someone who looked like me address a room full of students,” said Cui, who started her PhD in economics this fall. “It gives me motivation to imagine that I could be like that one day.”
Cui recalls seeing Oh in the dining hall, eating lunch with other math professors before class. What stood out mostly clearly to her, Cui said, was how self-assured Oh had been at the table full of men. She contributed frequently to the conversation, and when she spoke, they listened. In that moment, Cui was reminded of why she had chosen to pursue a career in academia. “She showed us that the ceiling could be broken,” Cui said.
About the Author
Mirilla Zhu is a rising junior in Saybrook College majoring in mathematics. Besides writing for YSM, she enjoys reading poetry and exploring New Haven in search of the city’s best bubble tea.
Acknowledgements
The author would like to thank Professor Oh for her time and generosity with the interviews, in addition to sharing her knowledge of measure theory, fractal dimensions, and drawing the perfect circle.
Further Reading
Kontorovich, A., & Oh, H. (2011). Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds. Journal of the American Mathematical Society, 24(1), 603-648.
Mackenzie, D. (2010). A Tisket, a Tasket, an Apollonian Gasket. American Scientist, 98(1), 10-14.