Vaughan Jones Obituary Operators Are Transformations of a Space Jones Grew Upinauckland, Newzealand

Obituary Vaughan Jones (1952–2020) Mathematician whose invention connected knots to quantum physics. n 1994, mathematician Vaughan Jones knots from their mirror images. It was the kind walked on stage to address the Italian of tool that topologists had been seeking for national academy in Rome’s Palazzo decades. Soon, the polynomial and its refine- Corsini, lit a cigar and began to blow smoke ments helped to solve long-standing problems, rings. With a mischievous grin, he told some of which had been formulated in the late Ithe assembled academics, “When I did this in nineteenth century by Peter Guthrie Tait — the Berkeley, I almost lost my job.” smoke-ring pioneer. Researchers also found a Jones had little time for the stuffy halls of plethora of connections between knot theory academia. He cared about the study of knots, a and physics. Jones investigated connections field launched by nineteenth-century physicists with statistical mechanics, the study of aver- experimenting with smoke rings. In 1990, he aged properties of large numbers of particles. became the first New Zealander to be awarded The most surprising link came in 1989. Theo- the Fields Medal — for an invention that revo- retical physicist Edward Witten at the Institute lutionized the field of topology, the branch of for Advanced Study in Princeton, New Jersey, maths that studies knots and other shapes. He showed that the polynomial can be interpreted attended the solemn ceremony wearing the as a feature of the quantum physics of a uni- jersey of New Zealand’s national rugby team. verse with simplified laws of nature and just “He was larger than life in many ways, but at the two dimensions of space. Other theoreticians same time he was very humble,” says mathema- then found a possible application for Witten’s tician Dietmar Bisch at Vanderbilt University in 2D universes. When electrons are confined to Nashville, Tennessee, a long-time collaborator one layer in certain ultracold devices, they can and friend. Born in 1952, Jones died suddenly form collective quantum states called anyons. OBERWOLFACH FORSCHUNGSINSTITUT M. BERGMAN/MATHEMATISCHES GEORGE on 6 September, aged 67, from complications towards that goal. One result, now known as Anyons can ‘remember’ how they have moved of an ear infection. the Jones index theorem, showed that when around each other, as if they formed braids in Mathematicians treasure discoveries that these algebras are nested within one another, space-time. Anyons and similar ‘topological reveal links between distant branches of their relative sizes conform to precise but enig- phases’ are now seen as a possible platform their discipline. Jones’s speciality, operator matic numerical ratios. for building future quantum computers. They algebras, provided a powerful tool in the study In 1984, while working at the University could run quantum algorithms essentially by of knots. His invention, which became known of Pennsylvania in Philadelphia, Jones came calculating Jones polynomials. as the Jones polynomial, seeded some of the across some formulas that his nested algebras Jones’s main focus continued to be Von ideas that physicists later developed into the satisfied. These were reminiscent of formulas Neumann algebras and their connections emerging field of topological quantum com- that appear in the study of braids. Mathemati- to other fields. He introduced a new tech- puting. Jones’s informal doctoral adviser, Alain nique, replacing the expressions of ordinary Connes, a Fields Medal winner at the Collège “Jones was a laid-back algebra — symbols arranged linearly — with a de France in Paris, called the discovery “one of 2D graphical arrangement. This planar algebra the great jewels of the unity of mathematics”. personality even enabled Jones and others to discover other- Jones grew up in Auckland, New Zealand. by the standards of wise elusive mathematical relationships. After studying maths and physics at his local mathematicians.” Jones was a laid-back personality even by university, he was awarded a scholarship to the standards of mathematicians. “He was con- do his doctorate at the University of Geneva, fident in his own powers and had a Southern Switzerland. There he became interested in the cians study braids as geometric objects, con- Hemisphere scorn for social hierarchies,” says theory of operators, which led him to work with sisting of curved lines twisted around each his former doctoral student Emily Peters at Connes, the pre-eminent expert in the field. other. Intrigued, Jones talked to the authority Loyola University in Chicago, Illinois. He was Operators are transformations of a space on braids, topologist Joan Birman at Columbia an avid windsurfer and kite surfer, and fre- with any number of dimensions — an example University in New York City. She explained that quently organized small conferences in places being the simple rotations of 3D space. They a braid can be ‘closed up’ end to end, to form a where participants could also go skiing or hik- are central to many areas of maths and to quan- knot. Both researchers began to suspect that ing. In 2011, he left the University of California, tum physics, where the geometric properties Jones’s formulas could be used to generate an Berkeley, where he had taught since 1985, to of operators encode all the possible results expression — a polynomial — that encoded a join Bisch at Vanderbilt. He was keen to foster of experimental measurements, such as the knot’s properties. mathematics in his native New Zealand, and spectrum of an element’s fluorescent emis- A week later, Jones showed Birman that this he spent the first months of every year down sions. By the 1970s, Connes had made strides was indeed the case. Birman recalls how she south, running a summer school. towards a complete classification of particular realized with astonishment that the ‘Jones systems of operators known as Von Neumann poly nomial’ was a powerful way of telling knots Davide Castelvecchi is a senior physical- algebras. Jones contributed important work apart. In particular, it could distinguish most sciences reporter at Nature. Nature | Vol 586 | 8 October 2020 | 195 ©2020 Spri nger Nature Li mited. All rights reserved. .

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