Proceedings of the National Academy of Sciences of the United States of America

About the PNAS Member Editor
Name Birman, Joan S.
Location Barnard College
Primary Field Mathematics
 Election Citation
Joan Birman is known for the Birman exact sequence, the study of Lorenz knots, the discovery of the Birman-Wenzl algebra and the discovery of relations between Vassiliev invariants and knot polynomials. 
 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.

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