Proceedings of the 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F.R. Jones’ 60th birthday

Preface

Vaughan Jones has made landmark contributions to mathematics: indeed, he is a founder of the field of quantum topology, which can conveniently be described as "the Jones polynomial, and everything that came after".

While the Jones polynomial may be his most widely known work, it grew out of his fundamental work on von Neumann algebras. After classifying outer actions of finite groups on the hyperfinite $II_1$ factor, Vaughan proved his famous index rigidity theorem: if $N\subset M$ is a $II_1$ subfactor, then the index lies in the range
$$
\left\{ 4\cos^2(\pi/n) \middle| n\geq 3\right\} \cup [4, \infty].
$$
This result followed from analyzing the Temperley-Lieb-Jones algebras, in which he found his famous representations of braid groups.

Throughout this extraordinary career, Vaughan has always generously shared his ideas, helping to build a vibrant research community.  His generations of graduate students have gone on to make substantial contributions, in quantum groups, subfactors, tensor categories, and conformal field theory, amongst others.

Vaughan's Friday afternoon seminar at Berkeley attracted a regular stream of interesting visitors, and always concluded with a popular and convivial beer and pizza. Since then he's moved on to Vanderbilt, where he has trained a new crop of successful students.

In 2014 and 2015 a pair of conferences celebrated his (approximate) 60th birthday. The first of these was in Maui, Hawaii (a traditional site for hunting exotic subfactors), and the second was in Qinhuangdao, China. This proceedings volume, with articles contributed by participants from both of these conferences, reflects the breadth of Vaughan's mathematical interests, from statistical mechanics to skein theory to free probability.

The volume is being published as part of the Proceedings of the Centre for Mathematics and its Applications, based at the Australian National University; two of his former students now work there, and Vaughan has visited many times over the years.

As the editors, and on behalf of all the contributors and participants, we'd like to thank Vaughan for creating such a great community to work in. We're all looking forward to the next big birthday!

Contents

Chapter Page number
1 - About 30 Years of Integrable Chiral Potts Model, Quantum Groups at Roots of Unity and Cyclic Hypergeometric Functions (pdf 684.98 KB)
Helen Au-Yang and Jacques H H Perk
4
2 - The Relation between Subfactors arising from Conformal Nets and the Realization of Quantum Doubles (pdf 189.07 KB)
Marcel Bischoff
15
3 - Subfactors, Planar Algebras, and Rotations (pdf 1.05 MB)
Michael Burns
25
4 - Free analysis and planar algebras (pdf 678.42 KB)
S. Curran, Y. Dabrowski, and D. Shlyakhtenko
115
5 - Existence of the $AH+2$ subfactor (pdf 690.2 KB)
Pinhas Grossman
143
6 - $N\subset P \subset M$ planar algebras and the Guionnet-Jones-Shlyakhtenko construction (pdf 534.85 KB)
Michael Hartglass
169
7 - Mixing and Spectral Gap Relative to Pinsker Factors for Sofic Groups (pdf 583.3 KB)
Ben Hayes
193
8 - A Cuntz algebra approach to the classification of near-group categories (pdf 1.04 MB)
Masaki Izumi
222
9 - The generator conjecture for $3^G$ subfactor planar algebras (pdf 623.66 KB)
Zhengwei Liu and David Penneys
344
10 - A Formula for the Jones-Wenzl Projections (pdf 493.76 KB)
Scott Morrison
367
11 - Algebra structures coming from tangles (pdf 81.33 KB)
Sohan Lal Saini
379
12 - The Hubble diagram in a Bianchi I universe (pdf 935.85 KB)
Thomas Schücker
384
13 - Mednykh's Formula via Lattice Topological Quantum Field Theories (pdf 465.22 KB)
Noah Snyder
389
14 - From Dominoes to Hexagons (pdf 459.92 KB)
Dylan P. Thurston
399
15 - A minus sign that used to annoy me but now I know why it is there (pdf 1.56 MB)
Peter Tingley
415