A Formula for Expanding the Numbers
She likes being on the top of mountains. It’s the vista that inspires her, the feeling of being able to see the world laid out before her, stretching toward the horizon. The sense of clarity is refreshing, perhaps because for so many years Dusa McDuff had a hard time seeing the way forward. Now McDuff is one of the most renowned mathematicians in the field of symplectic geometry and topology.
As she describes it, symplectic geometry is, like other kinds of geometry, the study of space. Yet, unlike the Euclidian geometry taught in high school, symplectic geometry is very abstract. Rather than measuring a single quantity, such as the length of an object, symplectic geometry studies the interactions of pairs of quantities via the measurement of two-dimensional areas. McDuff, with mathematician Dietmar Salamon, is the co-author of two textbooks that are classic references in the field: Introduction to Symplectic Topology (1998, 2nd edition) and J-Holomorphic Curves and Symplectic Topology (2012, 2nd edition). McDuff has also received numerous awards and honors.
McDuff fell in love with numbers growing up in post-World War II Edinburgh. “I loved doing sums,” she says. Turning her interest into a satisfying career path wasn’t as easy. “I come from a sort of intellectual family and was always brought up to have a career. The only career I ever imagined doing was something academic.” Though she had an idea of where she wanted to go, getting there was hardly simple—in part because McDuff knew few other female mathematicians. “I knew there were some women mathematicians. And my father, a professor of genetics at the University of Edinburgh, had had female scientists in his lab, so I knew it was possible to be a scientist and a woman,” she recalls. “But I had no immediate role models, and no immediate friends.” She adds, “I had great ambitions, but I thought of myself as different. …It took a lot of effort to become a mathematician.”
Another challenge: She was named Dusa, after her maternal grandmother, who left big shoes to fill. Amber “Dusa” Reeves was a feminist writer who scandalized society by having an affair with H.G. Wells. “It was a daring thing to do, and it wasn’t what was expected of her,” McDuff says. “Following her, I felt I couldn’t just be ordinary.”
However, McDuff’s academic direction, and the ultimate accomplishments it led to, hardly turned out to be ordinary. She pursued her doctorate and post-doctorate at the University of Cambridge, where she solved a well-known problem, becoming the first person to prove the existence of infinitely many type II sub one factors, a problem related to the mathematical structures called von Neumann algebras. Her doctoral thesis on the subject appeared in the prestigious Annals of Mathematics. “That gave me the confidence that I could do mathematics on an international level,” McDuff says.
Yet McDuff felt adrift with her work. Two significant experiences abroad, one in Russia and one in the U.S., changed her course. In the fall of 1969, during the last year of her PhD work, she accompanied her husband to Russia, where she met the charismatic and influential mathematician Israel Gelfand. “He was the first person who captured my imagination as a mathematician,” she says. With his help, she immersed herself in the abstract, conceptual side of mathematics and realized that the discipline could be both creative and beautiful. “Gelfand used to read me poetry and call that teaching me mathematics,” she says. “He thought it was all part of the same experience.”
Returning to the U.K., McDuff completed her PhD in 1971 and began teaching at The University of York, all the while pursuing her new study of topology, a form of geometry concerned with the properties of space. “Most people do their thesis and build from there: The research grows in some sort of organic way,” she says. “But I switched directions completely when I went to Moscow.”
Though she was making a name for herself professionally, the personal road wasn’t easy. She found herself in her mid-20s, supporting her husband (a translator of poetry) and a young child, while working full time. In the late 1960s and early 1970s, there were very few provisions made for female students, and virtually no support for married women with young children. The best-paying fellowships were reserved for men.
Her new academic trajectory eventually led her to take a year off from her position at York and assume a visiting position at the Massachusetts Institute of Technology in 1974, a position specifically reserved for a woman.
Over time, McDuff’s horizons widened, and she realized it was crucial to be more proactive about promoting her career. She also moved more toward the study of symplectic geometry.
In recognition of her work, she received the first ever Ruth Lyttle Satter Prize (Ruth Lyttle ’44) from the American Mathematical Society in 1991. Three years later, she was named a Fellow of the Royal Society.
After teaching at The University of York, The University of Warwick, MIT, and Stony Brook University, McDuff joined the Barnard faculty in 2007 as Helen Lyttle Kimmel Chair (Helen Lyttle ’42) and professor of mathematics. Now, McDuff teaches classes ranging from basic calculus to advanced courses in topology and geometry. Though her own research tends toward the abstract, McDuff urges students to study both the theoretical and the concrete. “I had a student who nearly dropped the mathematics major before discovering that she loved statistics,” McDuff says. “Mathematics as a way of thought is very broad, and it can be useful in many different circumstances. There are a lot of people who have strong math talent who don’t like traditional, proof-based, abstract mathematics.”
Supporting their passion is a practical matter. McDuff is working to diversify Barnard’s mathematics offerings to encourage students to explore topics such as statistics and computer science. This semester, the department launched a new computer-science workshop, giving students a strong foundation for future programming classes. Equally important is inspiring students to interact with one another, she says. In October, McDuff and the math department organized an evening party, in the computer-science help room, a space where Barnard students interested in the field can meet each other, in addition to getting help with their questions. The event honored mathematician Ada Lovelace, daughter of Lord Byron and a countess, who was born in 1815. A pioneer in computer science, Lovelace is sometimes described as the first computer programmer.
While still few, the number of female mathematicians has grown significantly during McDuff’s career; she can imagine even more—given the right support and camaraderie. “If you’re a young woman who’s very talented in math, you end up being isolated,” McDuff says. “There still aren’t enough senior faculty members or graduate students to serve as mentors.” That’s why McDuff divides her time between working as a faculty advisor to math-minded students at Barnard and helping run an annual summer program for women in mathematics through the Institute for Advanced Study and Princeton University.
“In mathematics, you have to work incredibly hard, and most of the time you feel as though you don’t understand,” she says. “You just keep working, though, and gradually things become clearer. Most of the time, you’re in a fog. But sometimes, it’s instant, and you just see.”