2009 AWM Essay Contest:

Grand Prize Winner

### by Wai-Ting Lam

The American mathematician Dr. Joan Birman was born in 1927. She is a leading expert in topology and among today's foremost experts in braid and knot theory. Her book “Braids, Links, and Mapping Class Groups” was for many years the standard monograph for beginning researchers on the subject. As a testament to her contributions to her field, she has received numerous awards and much recognition for her work. Dr. Birman is currently a Research Professor Emerita at Barnard College, Columbia University, where she has been a Professor since 1973. Dr. Birman is a member of the European Academy of Sciences. She was a co-founder of the non-profit publishing house Mathematical Sciences Publishing, which publishes a number of mathematical journals. She has been actively involved in human rights issues, and is a member of the New York Academy of Sciences Human Rights of Scientists Committee.

Dr. Birman’s father emigrated from Russia and her mother was a native New Yorker. Dr. Birman started liking mathematics as early as elementary school. Both of her parents encouraged their four daughters to acquire an education, and all four completed college. Joan and her sister Helen were math majors, Ruth was a physics major (who later switched her interests to plant physiology) and Ada became a kindergarten teacher.

After earning the BA degree from Barnard in 1948, Dr. Birman was hired by a firm that manufactured microwave frequency meters and needed help in solving a mathematics problem they had encountered. That was the start of a fifteen-year detour away from academia. During that time she became aware of the needs of engineers, and learned that the more mathematics you knew, the better equipped you were to tackle the varied mix of industry-related math problems that arose. However, as her home responsibilities increased (she had 3 children) even part time work became difficult. It seemed natural to use whatever time she could spare to learn more math. With the support and encouragement of her husband, Professor of Physics Joseph Birman, she started graduate studies, part-time, in mathematics. Her first course (Linear Algebra) began in January 1961, when her youngest child was 3 weeks old. She went to New York University, where her husband was a faculty member (so that her tuition was free). There were some, but not many, women students. The faculty and staff in the department were very helpful, giving her a fellowship and giving her limited TA responsibilities.

After passing the basic qualifying exams, Dr. Birman had to take a series of more specialized exams for admission to research. She passed those exams too and began to look around for an advisor. Professor Wilhelm Magnus knew her and was particularly encouraging. He had noticed that Dr. Birman loved topology, but he was an algebraist, so he met her halfway and gave her a paper to read, by Fadell and Neuwirth, about mapping class groups of surfaces, including braid groups. It was a terrific topic.

At that time, Magnus had worked on the mapping class group of a twice punctured torus, and he suggested that perhaps his work could be generalized, through the work of Fadell and Neuwirth. Her thesis was about the mapping class group of a surface of any genus with any number of punctures. She proved that there is a homomorphism (the ‘point-filling homomorphism’) from the mapping class group of a punctured surface to that of a closed surface, induced by filling in the punctures. She worked out the exact sequence that identified the kernel of the homomorphism, which was closely related to braids. What is now known as the “point-pushing homeomorphism” and the “Birman exact sequence” had their origins in that work. She didn’t know a presentation for the image of the point-filling homomorphism, the mapping class group of a closed surface, and understood that it was an important open problem.

After being granted her Ph.D., she got a job at Stevens Institute of Technology. Her first year there, she began joint work with Hugh M. Hilden. That work proved to be very rewarding to Dr. Birman. For the entire year, she and Hilden talked about the image of the point-filling homomorphsm, and together, they finally solved the problem for the special case of genus 2. At that point, she was thoroughly involved in mathematics. Two years later, she was invited to give a lecture at Princeton on the work that Hilden and she had done together. Her lectures became the content of her book on braids, links, and mapping class groups.

Around this time, there was a very different paper, by Garside, also about braids, that greatly interested her. She knew that there was a major open problem of classifying knots via braids. When she saw that Garside had solved the conjugacy problem in the braid group, she thought his work might classify knots. She couldn’t have been more mistaken, but the problem grabbed her interest. She began working on Garside’s algorithm too. It led her to questions in what is now known as complexity theory.

Dr. Birman has been influential in theoretical mathematics and has contributed to fundamental developments in topology. Her work has focused on low-dimensional topology: braids, knots, surface mappings, and 3-dimensional manifolds. Mathematical knots are defined as embeddings of a circle in 3-dimension Euclidean space. The concept of a knot has also been extended to higher dimensions. Knot invariants have had a lot of applications to the work of molecular biologists who have been studying the knotted shapes of DNA. One of the problems that she is working on now concerns “Lorenz knots”, which are based on the systems of non-linear differential equations. The equations can only be integrated numerically, so her research makes heavy use of computers.

"Mathematics is very beautiful! There is something very lasting about it" has been said by Dr. Birman. These words hold a truth that is difficult to ignore for those who do not shy away from the truth. Mathematics is a beautiful universal language that enables those who practice it and delve into its depths to experience both beauty and immutable truth. The love of mathematics is at best, difficult for a woman who wishes to “have it all”, and yet even with her familial obligations, Dr. Birman managed to do what she loved. It was really not easy for her to return to school and get back to her field while raising a family. However, she did so and became a pioneer for women and mathematicians as a whole. She showed her dedication to mathematics and she is absolutely a great role model for aspiring young women mathematicians!