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,Â the discovery of the BirmanWenzl algebra and the discovery of relations between Vassiliev invariants and knot polynomials.Â

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 wellknown 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, 2manifolds generalize to the branched 2manifolds of dif ferentiable dynamical systems, where Birman and Williams showed that in Lorentz?s wellknown 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 3dimensional flow. As for one more application in mathematics, the work of Birman and Series had an application to the wellknown McShane identity in number theory.



