Joan S. Lyttle Birman

Barnard College, Columbia University, New York, New York ~Research Professor; Professor Emerita of Mathematics. Specializes in low-dimensional topology. Known for work on braids and their relationship with knots, surface mappings, and 3-manifolds. Showed how braids and mapping class groups are related, culminating in her book Braids, Links, and Mapping Class Groups (1975). Work on knotted periodic orbits led to new ideas about chaos and its visualization in dynamical systems. Work on algebras associated with the Kauffman polynomial has been influential in algebras and their representation theory. One of the braid group representations defined by this algebra has been shown to be faithful, thereby proving the linearity of the braid groups. Understanding of Vassiliev's theory of knots created the subject of finite type invariants. Her study of links via closed braids uses powerful topological constructions to provide detail about links and closed braids that was thought impossible before her work, including a braid-based algorithm for recognizing the unknot. Ideas were fundamental to the development of topological quantum field theory, beginning with the Jones polynomial.