Knots in mathematics and physics

Abstract

Mathematical knot theory began in the nineteenth century as a result of questions about fluid flow and electromagnetism. In the twentieth century, it became a part of topology, the study of smooth deformations.

In recent years, knot theory has gone back to its physical origins in an interplay of mathematical analysis, topology, statistical mechanics and quantum field theory. It is far from clear what will emerge from these developments.

Fields Medal 1990 for the discovery of a 1-parameter representation of the classical braid groups into a family of C*-algebras, which led to a polynomial invariant of links. His work has greatly stimulated the recent development of research in C*-algebras, statistical mechanics, representations of quantum groups, knot theory and invariants of 3-dimensional manifolds.

Transcribed from the videotape of the talk by Carme Safont; revised and supplemented by the author.