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Vaughan Jones, who made fundamental contributions in operator algebras and knot theory (in particular developing a surprising connection between the two), died this week, aged 67.

Vaughan and I grew up in extremely culturally similar countries, worked in adjacent areas of mathematics, shared (as of this week) a coauthor in Dima Shylakhtenko, started out our career with the same postdoc position (as UCLA Hedrick Assistant Professors, sixteen years apart) and even ended up in sister campuses of the University of California, but surprisingly we only interacted occasionally, via chance meetings at conferences or emails on some committee business. I found him extremely easy to get along with when we did meet, though, perhaps because of our similar cultural upbringing.

I have not had much occasion to directly use much of Vaughan’s mathematical contributions, but I did very much enjoy reading his influential 1999 preprint on planar algebras (which, for some odd reason has never been formally published). Traditional algebra notation is one-dimensional in nature, with algebraic expressions being described by strings of mathematical symbols; a linear operator $T$, for instance, might appear in the middle of such a string, taking in an input $x$ on the right and returning an output $Tx$ on its left that might then be fed into some other operation. There are a few mathematical notations which are two-dimensional, such as the commutative diagrams in homological algebra, the tree expansions of solutions to nonlinear PDE (particularly stochastic nonlinear PDE), or the Feynman diagrams and Penrose graphical notations from physics, but these are the exception rather than the rule, and the notation is often still concentrated on a one-dimensional complex of vertices and edges (or arrows) in the plane. Planar algebras, by contrast, fully exploit the topological nature of the plane; a planar “operator” (or “operad”) inhabits some punctured region of the plane, such as an annulus, with “inputs” entering from the inner boundaries of the region and “outputs” emerging from the outer boundary. These algebras arose for Vaughan in both operator theory and knot theory, and have since been used in some other areas of mathematics such as representation theory and homology. I myself have not found a direct use for this type of algebra in my own work, but nevertheless I found the mere possibility of higher dimensional notation being the natural choice for a given mathematical problem to be conceptually liberating.

On Thursday, UCLA hosted a “Fields Medalist Symposium“, in which four of the six University of California-affiliated Fields Medalists (Vaughan Jones (1990), Efim Zelmanov (1994), Richard Borcherds (1998), and myself (2006)) gave talks of varying levels of technical sophistication. (The other two are Michael Freedman (1986) and Steven Smale (1966), who could not attend.) The slides for my own talks are available here.

The talks were in order of the year in which the medal was awarded: we began with Vaughan, who spoke on “Flatland: a great place to do algebra”, then Efim, who spoke on “Pro-finite groups”, Richard, who spoke on “What is a quantum field theory?”, and myself, on “Nilsequences and the primes.” The audience was quite mixed, ranging from mathematics faculty to undergraduates to alumni to curiosity seekers, and I severely doubt that every audience member understood every talk, but there was something for everyone, and for me personally it was fantastic to see some perspectives from first-class mathematicians on some wonderful areas of mathematics outside of my own fields of expertise.

Disclaimer: the summaries below are reconstructed from my notes and from some hasty web research; I don’t vouch for 100% accuracy of the mathematical content, and would welcome corrections.