Vaughan F.R. Jones

Abstract:   The suitably normalised limit of the average trace of a word on n independent random matrices defines a trace (the Voiculescu trace) on the algebra of noncommutative polynomials on n self-adjoint variables. In joint work with Guionnet and Shlyakhtenko we showed that this trace may be generalised to the case of an arbitrary planar algebra and one obtains interesting von Neumann algebras on completion. Of perhaps even greater interest is when the random matrices are distributed according to the Gaussian plus a higher order perturbation. It is possible that a judicious choice of this perturbation may lead to a model that is solvable by generalisations of Conway's skein theory method from knot theory.