Since Vaughan Jones introduced the theory of subfactors in 1983, it has been an open problem to determine the set of Jones indices of irreducible, hyperfinite subfactors. Not much is known about this set.

My student Julio Caceres and I could recently show that certain indices between 4 and 5 are realized by new hyperfinite subfactors with Temperley-Lieb-Jones standard invariant. This leads to a conjecture regarding Jones' problem. Our construction involves 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. If there is time, I will mention a few connections to quantum Fourier analysis and quantum information theory.