In 2014, Maryam Mirzakhani became the first woman to win a Fields Medal, commonly described as mathematics’ equivalent of the Nobel Prize. Another mathematician to have won the medal is Vaughan Jones who received this honour in 1990. Mirzakhani was an Iranian geometer and Jones an algebraist from New Zealand, but one thing linked their work: A connection to Joan Birman, an American mathematician who turns 97 on 30 May.
Birman’s research primarily deals with topology, the mathematical study of objects that can be twisted, stretched, or deformed in some way. Her work has deeply impacted several areas; particularly knot and braid theory, mapping class groups of surfaces and three-dimensional topology. These, in turn, are fundamental in other areas such as singularity theory, complexity theory, chaos, statistical mechanics as well as quantum computing, which is all the rage today.
Among topologists, there is a famous account of how a young Jones walked into Birman’s office one fateful day 40 years ago. Over the next couple of weeks, there was a flurry of brainstorming between the two mathematicians that culminated in the discovery of a mathematical expression that came to be known as the ‘Jones Polynomial’. The polynomial was a breakthrough that, to put it very simply, could help mathematicians answer the question “Is this knot the same as that knot?”
Though the discovery of the Jones Polynomial excited her tremendously, Birman took the call to not join Jones as he proceeded in this line of work. She continued the research she had previously committed to doing with her British collaborator Carolyn Series. This was not an easy decision for Birman, as she candidly describes in a 1997 interview. “I missed out on a whole lot of mathematics where I really had an inside track. I don’t know if that was the right decision.”
Jones, on his part, was always disarmingly transparent about Birman’s role in this breakthrough. In a famous paper, he singled out Birman, calling her contributions to the topic “of inestimable importance”. The mathematics community, too, recognised the significance of their partnership. When Jones won the Fields Medal in 1990, it was Birman who introduced his work at the International Congress of Mathematicians.
As it turned out, Birman’s choice to continue her research with Series paid off. Decades later, the duo’s work on simple closed curves formed the basis for some of Maryam Mirzakhani’s most important mathematical achievements, winning her a Fields Medal in 2014.
Unbeaten path
If Birman had a LinkedIn profile, it would go against all our preconceived notions of how to achieve success in academia.
After a Bachelor’s in mathematics, she pursued a Master’s in physics and then worked for five years in the aircraft industry. Birman was 34 when she returned to mathematics by enrolling for graduate school at New York University. By this time, she was married to physicist Joseph Birman and a mother of three. It was only then that she got acquainted with topology, the field that she would quickly come to master. At the ages when Mirzakhani (37) and Jones (38) won their Fields Medals, Birman had not even completed her PhD. By the time she did, she was 41. Her PhD thesis titled ‘Mapping Class Groups and Their Relationship to Braid Groups’ officially kicked off her career as a research mathematician.
But a LinkedIn profile could never tell a complete and honest story of a mathematician like Birman: The electrifying enthusiasm with which she and other students at her all-girls high school would debate solutions to geometry problems on the telephone every single night. When they began to find their syllabus too tepid, they even campaigned with the teachers for more geometry.
Or the daunting feeling of being a young woman attending courses at Columbia College, which was, at the time, still refusing to formally admit women. In a 2007 interview, Birman mentioned that most women who ventured into this male bastion gave up. “Eventually, I was the only girl in my classes, and I caught the idea that maybe maths was not for girls,” she’d said.
No CV would adequately describe the growing hopelessness of her first academic job hunt. She had graduated at a time when the job market was poor, and it did not help that she was older than most other PhDs. “There was a lot of prejudice against that,” she said. When she finally got a job (“by accident,” according to her), she would be the only woman among the 160 faculty members at Stevens Institute of Technology in New Jersey. In Birman’s own words: “I led a very wandering and undirected life! It amazes me that I got a career out of it — and it has been a really good career!”
In his 2019 profile of Birman, mathematician Dan Margalit noted that one of the standout aspects of her mathematical life is her “knack for pursuing and embracing unlikely collaborations across mathematical disciplines, and for uncovering and revitalising hidden or forgotten fields”. “Because of this,” he continued in his article, “her work has often been ahead of its time, with important implications and applications found years or decades after the original discoveries.”
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For all women in STEM
In 1990, Birman donated funds to institute the biennial Ruth Lyttle Satter Prize in mathematics, in memory of her botanist sister. Since then, the Satter Prize has been quietly recognising outstanding contributions to mathematics research specifically by women. Mirzakhani was one of the awardees — this was before she won the Fields Medal. In 2019, the Satter Prize went to Maryna Viazovska. Three years later, Viazovska became the second woman to win the Fields Medal.
The low representation of women among Nobel Prize and Fields Medal laureates is often attributed to the historical absence of women in STEM. Does that mean the only thing to do is sit and wait for the numbers to improve? Maybe not. The fact that the only two women Field Medallists have previously won Satter Prizes indicates that smaller awards and fellowships for women can play a major role in narrowing the gender gap.
More recently, in 2017, the Birman couple established the Joan and Joseph Birman Fellowship for Women Scholars, which provides a $50,000 fellowship to support the research careers of mid-career mathematicians based in the United States. Special care is taken by the jury to be inclusive of candidates who have had to navigate varied personal circumstances. The idea, according to the American Mathematical Society, is “to ensure that the fellowship will make a real difference in recipients’ trajectories”.
At the brink of 97, Birman seems to be as active as ever. When we wrote to her to inform her about this article, we were pleasantly surprised to hear back within hours. Not only did she provide helpful information, she also mentioned that she was particularly interested in “the large number of Indian women currently working toward an American PhD”.
This May marks not just the 97th anniversary of Joan Birman’s birth, but also the 40th anniversary of the discovery of the Jones Polynomial. What better time to commemorate her brilliant mathematics, her exceptional journey, and the many profound ways in which she is changing the game for all women in mathematics.
Shantha Bhushan is a faculty of mathematics and Nandita Jayaraj is a science communication consultant at Azim Premji University. Views are personal.
(Edited by Humra Laeeq)