Vaughan Jones, knotter and ultra-creative mathematician

The 1990 Fields Medal winner, the New Zealander passed away on September 6, 2020. Etienne Ghys pays tribute to him in his column in “Le Monde”.

Carte blanche. The mathematician Vaughan Jones died on September 6, 2020 in Tennessee, USA. He had received the Fields Medal in Kyoto in 1990. Sometimes a mathematician builds bridges between fields that were thought to be completely independent. These are moments of grace in the development of mathematics, reserved for the most creative, like Vaughan. However, it should not be thought that it is eureka! that suddenly appears. It almost always takes a long maturation, hardly compatible with the demand for immediacy of our current university system. The University of Geneva allowed Vaughan Jones to blossom and give the best of himself.
Vaughan arrived in Switzerland in 1974 from New Zealand to do a doctorate in physics. One day, with his thesis almost finished, he passes the door of the mathematics department and is fascinated by André Haefliger’s course: he abandons physics to do a thesis in mathematics (although, of course, his training as a physicist will remain fundamental). He works on “von Neumann algebras”, a field so abstract that the spaces studied have non-integer dimensions. Imagine for example a space whose dimension is 3.14 ! Haefliger – his thesis supervisor – is not a specialist in this subject, which is a sign of the great originality of the student and the open-mindedness of his master.
The Swiss Pierre de la Harpe, who knows the subject well, will become a friend and a “big mathematical brother” of Vaughan. At that time, the small department of Geneva was a breeding ground animated by a few exceptional senior mathematicians who fought against any form of exaggerated specialization. Algebra, geometry and analysis were spoken about a lot, very often in the small Italian bistro on the first floor. On the day of Vaughan’s defense in 1979, he was dressed in a tuxedo, which contrasted with the way the jury was dressed. In 1990, during the Fields Medal ceremony, in the presence of very formal Japanese authorities, he had insisted on wearing the All Blacks jersey, out of attachment to his New Zealand origins.

Sideration of the specialists

After his thesis, he settled in the United States but he often returned to Geneva. One day, after one of his lectures, someone pointed out to him, perhaps at the Italian bistro, an analogy between a relationship he wrote on the board and what is called the “group of braids”, which Vaughan did not know. That was all it took to glimpse a link between the subject of his thesis and a theme that was new to him: the theory of knots. All this led to a major discovery in 1984: the “Jones polynomial” associated with a knot. Knots, in mathematics, are those we imagine, like those of sailors. The mathematical theory of knots dates back to the 19th century and had a priori nothing to do with von Neumann’s algebras. Vaughan’s announcement of an important application of these algebras in the field of nodes will generate a kind of astonishment among topology specialists. He was awarded the Fields Medal but was also elected Life Vice-President of the International Knotmakers Guild, something he was very proud of.
The rest of his career was admirable. For about twenty years, the Ecole normale supérieure de Lyon has organized a mathematics weekend for about fifty students and an experienced mathematician. In 2012, Vaughan Jones literally charmed the young students. We have not only lost a brilliant mathematician, but also a model of generosity and openness for young people.

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