## Studying Links via Braids

### San Antonio, Texas 1987

**JOAN S. LYTTLE BIRMAN** received her BA in mathematics in 1948 from Barnard College of Columbia University and her MA in physics two years later from Columbia. After several years of working as a systems analyst in the aircraft industry, she took a temporary break and devoted her energies to raising three children. In 1961, she began part-time work with Wilhelm Magnus towards a PhD, which she received in 1968 from the Courant Institute of Mathematical Sciences. She is currently professor of mathematics at Columbia University (Barnard College).

Birman's mathematical work has focused on low-dimensional topology: braids, knots, surface mappings, and three-manifolds. "I had an instant love affair with knot theory," she says. "The moment I heard about the braid group I knew that's what I wanted to study, and twenty-six years later, I am still fascinated by its intricacies and beauty." Her numerous publications--including the influential book, *Braids, Links, and Mapping Class Groups* (Annals of Mathematics Studies #82)--have led to speaking engagements in thirteen countries, including Japan, Korea, Israel, and Spain. In 1990, she presented a lecture at the International Congress of Mathematicians in Kyoto, Japan on the work of Fields Medallist Vaughan Jones. She presented an AMS-MAA Joint Invited Address at the Joint Mathematics Meetings in Baltimore in 1992. She has been a Visiting Member at the Institute for Advanced Study in Princeton.

Birman's Noether Lecture discussed the classification of knots and links in the three-sphere, which is a fundamental problem in topology. One approach to this problem is to study links via the nested sequence of braid groups. Since each link can be represented as a closed braid, and since braids form a group, this approach allows one to utilize familiar group invariants, such as group characters. In 1984, this approach led to the discovery by Vaughan Jones of vast new families of polynomial invariants of links. Some of these invariants can be understood by very simple combinatorial arguments, which were presented during the lecture. As a first step in understanding the underlying meaning of Jones' knot invariants, she was able to show (in joint work with X. S. Lin, in 1990) that whereas the classical invariants all described properties of a single knot or link, the new ones related to a space of all knots, and encoded data about the way knots "fit together."

Jones' knot invariants have had applications to the work of molecular biologists who have been studying the knotted shapes of DNA. "Biologists really had a problem knowing if the coiled strands of DNA they observed were knotted or not, or if two knotted pieces of DNA were the same," said Birman in an article in Columbia, the university's alumni magazine (Spring 1990). "So they came to mathematicians and we said, 'Oh, we've been working on that for years.'" "I'm pleased when my work is useful," Birman went on, "but I'm more pleased by the beauty of it. It's a bit like art, and art is not necessarily useful."

Birman points out that she has done much collaborative work, which she enjoys because it makes mathematics easier and more fun, and also because her collaborators have been such fine people. Among her nonmathematical interests, family is at the top of the list, followed by cooking (she finds that chopping vegetables is relaxing and helps mathematical creativity), walking, and reading.