Vaughan Jones

1952–2020, Gisborne, New Zealand

AnalysisTopologyFields Medal 1990

New Zealand mathematician who founded subfactor theory and discovered the Jones polynomial knot invariant; awarded the Fields Medal in 1990.

Vaughan Frederick Randal Jones was born in Gisborne, New Zealand, in 1952 and educated at the University of Auckland before taking his doctorate at the University of Geneva in 1979 under André Haefliger, on the index theory of von Neumann algebras. After positions in the United States he settled at the University of California, Berkeley, in 1985.

In 1983 Jones proved that the index of a subfactor of a $II_1$ factor is either at least 4 or lies in the discrete sequence $4\cos^2(\pi/n)$ — values that can actually all be realized. Analyzing the towers of algebras generated by iterating the basic construction, he arrived at a braid group representation that, combined with a Markov trace, produced in 1984 the Jones polynomial: a link invariant whose discovery electrified knot theory and, through the work of Witten, Reshetikhin–Turaev and Khovanov, opened the whole field of quantum topology.

Jones received the Fields Medal at the 1990 ICM in Kyoto, famously wearing an All Blacks rugby shirt at the ceremony. He later moved to Vanderbilt University, founded the New Zealand Mathematics Research Institute, and continued to work on subfactors, planar algebras and their links to conformal field theory until his death in 2020.

Workplaces: University of California, Berkeley, Vanderbilt University

New Zealand, United States

Award-winning work

Fields Medal 1990Jones discovered the index theorem for subfactors of von Neumann algebras: the index of a

II_1

subfactor is constrained to the discrete set

\{4\cos^2(\pi/n)\}

below 4, a completely unexpected rigidity. Tracing the associated tower of algebras led him to a representation of the braid group and, through a Markov-trace construction, to the Jones polynomial

V_L(t)

— a new knot and link invariant computed by a simple skein relation that distinguished knots no earlier invariant could separate. The invariant and its descendants (via Khovanov homology, TQFT and quantum groups) reshaped low-dimensional topology and created the field of quantum topology. The medal also honored his foundational theory of subfactors, later shown to encode topological quantum field theories and conformal field theories.

Contributions, linked to the library

Library topicKnot theoryThe Jones polynomial revolutionized knot theory and launched quantum topology.Library topicOperator theoryHis index theorem and subfactor theory are cornerstones of the theory of von Neumann algebras.Great problemUnknotting problem (complexity of recognising the unknot)Whether the Jones polynomial detects the unknot — i.e., whether a knot with

V_K(t) = 1

must be unknotted — is a famous open question descending directly from his invariant.Great problemVolume conjecture (knots)The conjecture relates the asymptotic growth of the colored Jones polynomial of a knot to the hyperbolic volume of its complement.