Abstract

Francis Murray and John von Neumann discovered in the 1940's an intriguing algebra of operators on Hilbert space, which they called the hyperfinite II_1 factor R. Vaughan Jones showed in 1983 that R has a very rich theory of subfactors by defining a notion of index for a subfactor. It led to fascinating mathematical structures, including Jones' knot invariant and many non-classical tensor categories realized as subfactor representations.

It is an open problem to determine the set of Jones indices of irreducible, hyperfinite subfactors. Not much is known about this set, and one important question is what invariants subfactors with certain indices must have. Caceres and I have recently shown that all indices of finite depth subfactors between 4 and 5 are realized by hyperfinite subfactors with Temperley-Lieb-Jones standard invariant.

Our work leads to a conjecture and some results regarding Jones' index problem. The construction involves families of commuting squares, a graph planar algebra embedding theorem, and a few tricks that allow us to avoid solving large systems of linear equations to compute invariants of our subfactors.

I will try to make this talk accessible to non-experts in operator algebras.