JONES, VAUGHAN FREDERICK RANDAL

born Dec. 31, 1952, Gisborne, N.Z. New Zealand mathematician who was awarded the Fields Medal in 1990 for his study of functional analysis and knot theory. Jones attended the School of Mathematics, University of Geneva, Switz. (Ph.D., 1979), and became a professor at the University of California, Berkeley, U.S., in 1985. Jones was awarded the Fields Medal at the International Congress of Mathematicians in Kyoto, Japan, in 1990. In his study of von Neumann algebras (algebras of bounded operators acting on a Hilbert space) Jones came across polynomials that were invariant for knots and links in three-dimensional space. Initially it was suspected that these were essentially Alexander polynomials, but this turned out not to be the case. In 1928 the American mathematician James W. Alexander had analyzed linkssimple closed curves in three-dimensional spaceand their associated unique polynomials. For any topological displacement (without cutting the loop) the associated Alexander polynomial is unchanged, or invariant. It turned out that both Alexander's polynomials and the new polynomials are specializations of the more general two-variable Jones polynomials. The Jones polynomials do have an advantage over the earlier Alexander polynomials in that they distinguish knots from their mirror images. Further, while these polynomials are useful in knot theory, they are also of interest in the study of statistical mechanics, Dynkin diagrams in the representation theory of simple Lie algebras, and quantum groups. Jones's publications include Actions of Finite Groups on the Hyperfinite Type II1 Factor (1980); with Frederick M. Goodman and Pierre de la Harpe, Coxeter Graphs and Towers of Algebras (1989); and Subfactors and Knots (1991).