Joan S. Birman
Barnard College
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Primary Section: 11, Mathematics Membership Type:
Member
(elected 2021)
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Biosketch
Joan S. Birman is a mathematician whose primary area is low-dimensional topology. She is known for her discoveries of unexpected connections between braid groups and other parts of mathematics, for example the dynamical systems that underlie chaos. She was born in New York City and has spent most of her life in New York City and its suburbs (Lawrence on Long Island and New Rochelle in Westchester). Birman received her BA from Barnard College in 1948. During the years 1948 to 1961 she worked for engineering firms in the New York area, both full time and part time, on aircraft navigation computers. During this period she married Dr. Joseph L. Birman and cared for their 3 young children. In 1961 she began graduate studies part-time at the Courant Institute of New York University, receiving her PhD from NYU in 1968. Her career in academia began in 1968, when she was appointed as an Assistant Professor at Stevens Institute of Technology. In 1974 she moved to a permanent position as Professor of Mathematics at Barnard College, Columbia University. Over the years she received Fellowships from the Sloan and Guggenheim Foundations, became a Fellow of the American Mathematical Society, and was elected to both the American Academy of Arts and Sciences and the National Science Foundation. She holds the degree of Doctor Honoris Causa from the Israel Institute of Technology in Haifa, Israel.
Research Interests
A central theme in Joan Birman’s research has been braid groups and areas of mathematics where braiding plays an important (and often unexpected) role. In the most well-known application, connecting the k ends of a braid to its k starting points leads to knots and links. This aspect of her work played a central role in the discovery of the Jones polynomial and closely related quantum invariants of links. Another fruitful way to look at the classical braid group is to think of it as a group of motions of k distinct points on the Euclidean plane, i.e. a surface mapping class group. Here applications abound, because the k points might be the coefficients or roots of a polynomial of degree k, or k obstacles on a factory floor, or k autonomous vehicles moving through the streets of a city. The ‘Birman exact sequence’ and the ‘point pushing maps’ of her PhD thesis have played a central role in geometric group the- ory, when one seeks to understand the structure of surface mapping class groups. Under the right conditions, 2-manifolds generalize to the branched 2-manifolds of dif- ferentiable dynamical systems, where Birman and Williams showed that in Lorentz’s well-known differential equations, which describe the flow associated to a leaky water wheel, braiding and knotting was a key to giving structure to what had seemed to be a basic example of chaos in a 3-dimensional flow. As for one more application in mathematics, the work of Birman and Series had an application to the well-known McShane identity in number theory.
