Vaughan Jones DCNZM DSc FRS FRSNZ

Vaughan Jones is a Professor of Mathematics at the University of California at Berkeley and Distinguished Alumni Professor of the University of Auckland, and a Co-Director (with Marston Conder) of the NZ Institute of Mathematics & its Applications (NZIMA).

Vaughan Jones was born and educated in New Zealand, at St Peter's School (Cambridge), Auckland Grammar School, and the University of Auckland, where he graduated MSc with First Class Honours in 1973.

He was awarded a Swiss Government Scholarship and an FWW Rhodes Memorial Scholarship, which enabled him to study for his doctorate at the University of Geneva. In 1979 he was awarded the degree of Docteures Sciences (Mathematique) and the following year the Vacheron Constantin Prize for his doctoral thesis.

He spent his first postdoctoral year as ER Hedrick Assistant Professor of Mathematics at the University of California at Los Angeles (UCLA), and from 1981 worked at the University of Pennsylvania as Assistant and then Associate Professor, before being appointed as Professor of Mathematics at the University of California at Berkeley in 1985.

During the 1980s his research focussed on von Neumann algebras, and in the course of this work he discovered a new polynomial invariant for knots which led to surprising connections between apparently quite different areas of mathematics.

Vaughan Jones was awarded a Fields Medal at the 1990 International Congress in Kyoto (Japan) for his remarkable and beautiful mathematical achievements. The Fields Medal is awarded every four years, and is regarded as the equivalent of a Nobel Prize. (Some say that Nobel's wife had an affair with a mathematician and that this is the reason why there is no Nobel Prize for Mathematics.) Vaughan Jones is the only New Zealander ever to have won this prestigious award, and until 2006 he was the only winner from the Southern Hemisphere.

Since 1990 he has gone on to receive numerous awards and honours, for example: awarded a Guggenheim Fellowship 1986; elected a Fellow of the Royal Society (of London) 1990; awarded the Rutherford Medal 1991; awarded an honorary DSc from the University of Auckland 1992; appointed as a Distinguished Alumni Professor at the University of Auckland 1992; awarded an honorary DSc from the University of Wales 1993; elected to the American Academy of Arts and Sciences 1993; elected to the US National Academy of Sciences 1999; awarded the Onsager medal of Trondheim University (Norway) 2000; elected as a foreign member to the Norwegian Royal Society of Letters and Sciences 2001; made a Distinguished Companion of the Order of New Zealand 2002; elected Honorary Member of the London Mathematical Society 2002; awarded a Doctor Honoris Causa, Universite du Littoral, Cote d'Opale 2002; awared the Prix Mondial Nessim Habif 2007.

He has published in research in leading international scientific journals, and has been invited to lecture at numerous international conferences, including the following: International Association of Mathematical Physicists (Swansea) 1988 ,International Congress of Mathematicians (Kyoto) 1990, International Association of Mathematical Physicists (Brisbane) 1997.

He has also served as editor or associate editor of several international journals, including the Transactions of the American Mathematical Society, the Pacific Journal of Mathematics, the New Zealand Journal of Mathematics, Reviews in Mathematical Physics, and the Journal of Mathematical Chemistry.

In addition, he has been a member of the Scientific Advisory Boards of several leading mathematical institutes around the world, including the Fields Institute for Mathematics (Canada), the Erwin Schrodinger Institute for Mathematical Physics (Vienna, Austria), the Mathematical Sciences Research Institute (Berkeley, USA), the Center for Communications Research (USA), the Institut Henri Poincare (Paris, France).

Although based at the University of California, since 1992 he has returned to New Zealand at least once each year (in his role as Distinguished Alumni Professor at the University of Auckland) and he continues to engage with and stimulate the mathematical sciences community in NZ.

He is principal founder and a director of the New Zealand Mathematics Research Institute, which is a virtual institute (without premises) established in the 1990s to promote and foster mathematical research of the highest quality in NZ. Since 1994 the NZMRI has run summer research workshops in various parts of NZ, on mathematical themes of current importance, at which students and senior researchers can interact and learn from each other and from overeseas experts who are invited to give courses of lectures.

Vaughan Jones has been instrumental in attracting some of the world's best mathematicians to NZ, and the success of these workshops is largely attributable to his vision and energy in setting up the NZMRI. His own style of working is informal, encouraging the free and open interchange of ideas, and this has rubbed off on many others. His efforts have made it possible for graduate students to gain first-hand knowledge of developments at the leading edge of their discipline, here in NZ.

This activity has been expanded and enhanced through the newly formed New Zealand Institute of Mathematics and its Applications, one of the five Centres of Research Excellence selected in 2002 for special funding from the New Zealand Government. Vaughan Jones is co-Director of this Institute and head its International Scientific Advisory Board, and continues to make invaluable contributions to scientific research and postgraduate education in his home country.

AUCKLAND MATHEMATICS GRADUATE WEAVES KNOTTY TRAIL
FROM QUANTUM MECHANICS TO MOLECULAR BIOLOGY

Vaughan Jones, a graduate in Mathematics of the University of Auckland, discovered in the 1980s a new polynomial invariant of knots which represented the first advance of its type since 1928. This discovery enabled - for instance - molecular biologists to gain a new insight into how DNA can remove the tangles that result when replication and cell division firstly duplicates the DNA and subsequently has to pull the chromosomal mass into different cells. The result represents a landmark in modern mathematics whose ramifications still remain to be fully explored.

The discovery came indirectly by way of a branch of quantum mechanics called Von Neumann algebras. These were developed to handle quantum mechanical observables such as energy, position and momentum. The capacity of the operators representing such quantities to be added or multiplied results in them having the structure of an algebra. Von Neumann algebras can be built out of simpler structures called factors which have the intriguing property that they can have 'continuous dimensions' i.e. real numbers such as g or 1/27. Jones was studying subfactors when he discovered that, rather than having continuous dimensions the only dimensions less than 4 were 4cos2p/n.

While showing the proof to some friends at Geneva, it was remarked that sections resembled the group of a braid, which is like a knot except that it is a series of threads beginning at the top which are woven over and under before being realigned at the bottom. A braid can be converted into a knot by joining its ends together as shown below. But the reverse process is not so easy.

Vaughan Jones ended up having a meeting at Columbia University with knot theorist Joan Birman to see if his work might have some application in knot theory. When the two sat down together, the discovery was almost instantaneous. Jones proved that Von Neumann algebras are related to knot theory and provide a way to tell very complicated knots apart.

The original polynomial for knots derived by Alexander in 1928 fails to separate a left-handed trefoil from a right-handed one, showing that much remained undiscovered, since some of the simplest knots could not be distinguished.

It was quickly discovered that the Jones polynomial and the Alexander polynomial give complementary descriptions of knots which can be combined to make a two-variable polynomial giving a more complete representation than provided by either alone. In fact one of the most striking instances of multiple scientific discovery occurred in which, during a few days in October 1985 no less than four independent groups in Britain and the U.S. reported the same results. Below is shown the new polynomial for a selection of simple knots and links. It is shown in a homogeneous three variable form in which each term has zero net power.

yz^-1 + x^(-1)y^2z^(-1) - x^(-1)z x-^(1)y^(-1)z^2 - xy^(-1) - x^(-1)y - 1

x^(-2)z^2 - 2x^(-1)y - x^(-2)y^2 y^(-2)z^2 - 2xy^(-1) - x^2y^(-2)

These results were of considerable interest to molecular biologists studying the conformations of DNA, because the two polynomials correspond to two different means by which nucleic acids can be transformed. One is passage in which one segment of DNA is broken and another is passed through. This is the way supercoiling and links are made. In the other, two strands are broken and joined in the exchanged arrangement. "All other motions of DNA can be reduced to these ones" reported Nicholas Cozzarelli who succeeded in forming electron micrographs of knotted DNA by coating it with the protein RecA. "Before Jones, the math was incredibly arcane. The way the knots were classified had nothing to do with biology, but now you can calculate the things important to you".