by Dietmar Bisch, David E. Evans, Robion Kirby and Sorin Popa

Sir Vaughan Frederick Randal Jones, who died at age 67 on September 6, 2020, was one of the most influential and inspirational mathematicians of the last four decades. His original and penetrating analysis of inclusions of von Neumann algebras led to the creation of new fields of research, while reinvigorating old ones, thereby setting off an extraordinary interplay between disparate areas of mathematics, from analysis of operator algebras, to low-dimensional topology, statistical mechanics, quantum computing, and quantum field theory. Vaughan’s work had a major impact with unexpected, stunning applications, even outside of mathematics, for example to the study of knotted DNA strands and protein folding in biology. A crucial idea leading to these striking connections was his groundbreaking discovery in the early 1980s that the symmetries of a factor (a von Neumann algebra with trivial center), as encoded by its subfactors, are quantized. They generate “quantized groups,” a completely new type of structure, endowed with a dimension function given by a trace and an index that can be nonintegral.

This article gives a panoramic view of the scientific impact and enduring legacy of Vaughan’s work, as well as his personality and style of working through the contributions of colleagues and friends across mathematics and physics. Over the years, Vaughan’s countless mathematical interactions forged numerous lifelong friendships, and he will be sorely missed by all.
Vaughan was born on December 31, 1952, in Gisborne on the North Island of New Zealand to parents Jim Jones and Joan Jones (née Collins) and grew up in Auckland. Between the ages of eight and twelve he was educated at the boarding school St Peter’s School in Cambridge in rural North Island. Vaughan attended Auckland Grammar School until the age of sixteen and then studied Mathematics at Auckland University from 1969 to 1973. He left New Zealand in 1974 for graduate study at the University of Geneva with the intention of writing a thesis in Physics, but gradually moved in 1974–76 to work under the supervision of André Haefliger in Mathematics. It was in Switzerland where Vaughan met Martha (Wendy), who held a scholarship to study at the University of Fribourg and subsequently worked at the United Nations in Geneva. They married in 1979 and raised three children together, Bethany, Ian, and Alice.

Vaughan had several appointments in the USA until his death. However, the friendships he made during these formative years in New Zealand remained with him throughout his life. His love and loyalty to New Zealand would bring him back later, at least annually from 1994 on, to invigorate mathematics in his native land, often with summer schools, using his network of colleagues and friends worldwide and his scientific standing to attract other world-renowned stars to New Zealand.
In the fall of 1975, when Vaughan was switching from physics to
mathematics, he met
Alain Connes
at a conference in Strasbourg and
was very impressed. Connes had just finished his seminal work on
Masamichi Takesaki was impressed by Vaughan’s thesis and brought him to UCLA on a Hedrick assistant professorship in 1980. But after one year at UCLA, Vaughan returned to the East Coast to join his wife Wendy who was studying at Princeton. UPenn seized the opportunity and made him an offer. So during 1981–1985, Vaughan was at UPenn, first as a junior faculty member then as an associate professor, with 1984–1985 actually spent at MSRI. In 1985, he was appointed full professor at UC, Berkeley, where he remained until he retired in 2013 with the title Professor Emeritus. From 2011 on he held the Stevenson Distinguished Chair at Vanderbilt University. Vaughan was also a Distinguished Alumni Professor at the University of Auckland and Founding Director of the New Zealand Mathematics Research Institute from 1994 on. He kept in contact with Europe including spending one-year sabbaticals at the IHES during 1986–1987 and 1989–1990 and at the University of Geneva in 1993–1994 and 1998–99.
In his thesis, Vaughan developed a novel algebraic approach to the
classification of actions of finite groups on
But by November 1981, Vaughan made the amazing discovery that the index of a subfactor can take exactly the
values
In the summer of 1982, Vaughan realized that, because
of the algebraic relations they satisfy, the projections in the
tower of factors provide an unexpected family of semisimple quotients
of the Hecke algebras of type
The invariant
In 1988, Witten gave a physical interpretation for
In a parallel development which started in 1983, a connection was
made with calculations by
Temperley
and
Lieb
in solvable statistical
mechanics. This triggered yet another series of interactions with
physics, via statistical mechanics and conformal quantum field theory.
In the latter, a similar dichotomy of discrete and continuous parts
occurs for the central charge in the representations of the Virasoro
algebra which describes certain projective representations of the
diffeomorphism group of the circle. Subfactors provide a natural
framework for studying two-dimensional conformal quantum field
theories. Indeed the discrete series of the central charge in the
representation theory of the Virasoro algebra can be understood via
conformal nets of factors, as cosets of
Perhaps the deepest and most enduring of Vaughan’s revolutionary work
is within the theory of
One can hardly overstate the importance and depth of
these discoveries. This led right away to a huge number of
beautiful and exciting problems, such as the classification
of subfactor inclusions
Many outstanding results by a large number of people
have followed. Vaughan was much involved in these
developments, notably finding the best way to characterize the
objects
Planar algebras, together with a quest to produce a conformal theory from subfactors, led Vaughan to a study of the Thompson groups as discrete approximations to the diffeomorphism group of the circle, and again to unexpected spin-offs for the theory of knots and links (2015–2020).
More details of all these mathematical developments will be found in a forthcoming issue of the Bulletin of the AMS which is dedicated to Vaughan.

Vaughan was awarded the Fields Medal in Kyoto in 1990, and was elected Fellow of the Royal Society in the same year, Honorary Fellow of the Royal Society of New Zealand Te Apārangi in 1991, member of the American Academy of Arts and Sciences in 1993 and of the US National Academy of Sciences in 1999, and foreign member of national learned academies in Australia, Denmark, Norway, and Wales. He received the Onsager Medal in 2000 from the Norwegian University of Science and Technology. In 2002, he was made a Distinguished Companion of the NZ Order of Merit DCNZM, later redesignated Knight Companion KNZM. The same year, he became an honorary member of the London Mathematical Society. The Jones Medal of the Royal Society of New Zealand Te Apārangi is named in his honor.
Vaughan had a strong commitment of service to the community. In 1994, he was the principal founder and Director of the New Zealand Mathematical Research Institute, leading summer schools and workshops in New Zealand each January. He was Vice President of the American Mathematical Society in 2004–2006, and Vice President of the International Mathematical Union in 2014–2018.
Vaughan had an unusual and very personal style of doing research. He would freely share ideas about a project and discuss initial speculations and possible applications and concrete steps for how one might obtain the final result. Vaughan was a warm and gregarious individual whose humor and humility led to the generosity and openness from which the mathematical community drew substantial benefit. Vaughan had over 30 graduate students and was a sought-after doctoral advisor. His presence at mathematical events was stimulating for all who came in contact with him. He will be dearly missed by his family and his many friends all over the world.
[Editor’s note: The text above is from the Introduction of “Memories of Vaughan Jones” published in the Notices in October 2021. For the full article, click on the PDF link at the upper right of this page.]