Universal Lessons from the Life of Joan Birman

What do an Iranian geometer and an algebraist from New Zealand have in common apart from being Fields Medallists? Their research is linked to Joan Birman, an American mathematician who turned 97 last week.
By and | Published on Jun 8, 2024

In 2014, Maryam Mirzakhani became the first ever woman to win a Field’s Medal, commonly described as the equivalent of the Nobel Prize in mathematics. Another Field’s Medalist is Vaughan Jones, who had received this honour in 1990. Mirzakhani was an Iranian geometer and Jones was an algebraist from New Zealand, but there was one thing that linked their work: a connection to Joan Birman, an American mathematician who turned 97 on 30th May.

Joan Birman’s research primarily deals with topology: the mathematical study of objects or spaces 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

“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!” – Joan Birman

Among topologists, there is a famous account of how a young Vaughan Jones walked into Joan’s office one fateful day forty 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, 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, Joan took the call to not start a formal collaboration with Vaughan as he proceeded in this line of work. Joan carried on with the research she had previously committed to with her British collaborator Carolyn Series. This was not an easy decision for Joan, 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.” 

Vaughan Jones, on his part, was always disarmingly transparent about Joan’s role in this breakthrough. In his famous paper about it, he singled out Joan, calling her contributions to the topic “of inestimable importance”. The mathematics community too recognised the significance of their partnership. When Vaughan won the Field’s Medal in 1990, it was Joan who introduced his work at the International Congress of Mathematicians where the medal was presented.

As it turned out, Joan’s choice to continue her research with Carolyn Series after the discovery of the Jones Polynomial 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 that won her a Field’s Medal in 2014. 

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“I remember he [Louis Nirenberg] asked me, ‘Do you like inequalities?’ And I said, ‘No, I don’t like inequalities!’ He said, ‘Then you don’t want to study applied math.’ And he was right!” 

– Joan Birman, 2007 [Source]

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If Joan Birman had a LinkedIn profile, it would go against all our preconceived notions of how to achieve success in academia. Her first ‘break’, as the ruthless academic culture defines any deviation from the Bachelors-Masters-Doctorate rigmarole, came right after her undergraduate studies. After securing a Bachelor’s in mathematics from Barnard College in New York, she worked at an engineering firm for a little less than a year, before pursuing a Masters degree – not in mathematics, but in physics. By then, she was married. She then took up another job, working on early navigation computers for the aircraft industry. After about five years of doing this, the couple had their first child, then the second, and then the third. During this time, Joan continued to work when she could, but interruptions were understandably plenty for the mother of three.

“Eventually I was the only girl in my classes, and I caught the idea that maybe maths was not for girls” – Joan Birman

The LinkedIn timeline would easily reveal that Joan was 34 when she returned to mathematics by enrolling for graduate school at New York University. It was only then that she got properly acquainted with topology, the field that she would quickly come to master. She finished her PhD in Mapping Class Groups and Their Relationship to Braid Groups when she was 41, and since then, has not looked back. 

What the LinkedIn profile could never reveal is a complete and honest story of a mathematician like Joan Birman: for example, 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, at that time, was still refusing to formally admit women. In a 2007 interview, Joan 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 had remarked.  

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 people who had just completed their PhDs. “There was a lot of prejudice against that,” she had 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.

Throughout her life, Joan Birman had to make choices based not just on where her interests lay, but also what made most sense for her family and financial obligations at the time. Take the case of her PhD: she could only consider it thanks to the fellowship that would cover childcare. In Joan’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 Joan, mathematician Dan Margalit noted that one aspect that stands out in 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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“…that was the unique way that I could help other women — simply by taking an interest, working with them when it was appropriate, and being open to their conflicts and sensitive to their concerns.” – Joan Birman, 2007 [source] 

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Neither George nor Lillian Lyttle – Joan’s parents – had completed high school, but they raised their daughters Helen, Ruth, Joan and Ada to prioritise it. Moreover, Joan recalls that they were encouraged to do something bigger with the education they got – something beyond earning money. Perhaps it was this that drove Joan Birman to live as richly and humbly as she has. Perhaps this also explains why she didn’t stop with just doing brilliant mathematics, but went one step ahead to pave the way for younger women who may be facing similar challenges as she once did.

…in 2017, Joan and her husband 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 USA. Special care is taken by the jury to be inclusive of candidates who have had to navigate varied personal circumstances.

In 1990, Joan donated funds to institute the biennial Ruth Lyttle Satter Prize in Mathematics, in the memory of her botanist sister who had died of leukaemia the previous year. Since then, the Satter Prize has been quietly recognising outstanding contributions to mathematics research specifically by women. Mirzakhani was one of the winners of the prize – this was before she won the Fields Medal. In 2019, the Satter Prize went to Maryna Viazovska. Three years later, Viazovska became the second ever woman to win the Field’s Medal. 

The low representation of women among Nobel Prize and Field’s Medal laureates is often attributed to the historical absence of women in STEM. So 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 diversify the pool from which bigger prizes like Field’s Medal makes its picks. This, in turn, may play a major role in narrowing the gender gap in the more prestigious/high-profile prizes. 

More recently, in 2017, Joan and her husband 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 USA. 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, Joan 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”.

The past month marked 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.

Note: A shortened version of this story first appeared on ThePrint.in. The featured image was compiled  by Nandita Jayaraj using images sourced from the public domain or with permission. The photograph of Joan Birman is by C.J  Mozzochi; Maryam Mirzakhani’s was on Wikimedia Commons and Vaughan Jones’s image is by David Monniaux. The letter in the background is from Joan Birman’s archives available on Celebratio Mathematica.

About the author(s)
Shantha Bhushan

Shantha Bhushan is a faculty of mathematics at Azim Premji University, Bangalore.

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